# On the functoriality of universal abelian factorizations

Luca Terenzi  Luca Terenzi
UMPA, ENS de Lyon
46 Allée d’Italie, 69007 Lyon (France<EMAIL_ADDRESS>and  missing

###### Abstract.

In this note, we discuss several aspects of the functoriality of universal
abelian factorizations associated to representations of quivers into abelian
categories. After recalling the general construction of universal abelian
factorizations, we review the canonical lifting procedures for exact functors
and natural transformations thereof (already studied by F. Ivorra in a
slightly different axiomatic framework) and we describe how these interact
with general categorical constructions; for sake of simplicity, we mostly
focus on the case where the quivers and representations considered are defined
by actual (additive) categories and (additive) functors. We then extend these
known results in two directions which have not been explored explicitly in the
existing literature: on the one side, to the setting of multi-linear functors;
on the other side, to the setting of abelian fibered categories. Combining
these two extensions, we are able to discuss the case of monoidal structures
on abelian fibered categories; the latter case has been successfully applied
in the author’s construction of the tensor structure on perverse Nori motives.

###### Contents

  1. 1 Recollections on universal abelian factorizations
  2. 2 Lifting exact functors
  3. 3 Lifting natural transformations
  4. 4 Extension to the multi-linear setting
  5. 5 Lifting abelian fibered categories
  6. 6 Lifting (external) tensor structures

## Introduction

### Motivation and goal of the paper

Let $\beta:\mathscr{D}\rightarrow\mathscr{A}$ be an additive functor from an
additive category $\mathscr{D}$ to an abelian category $\mathscr{A}$. As shown
in [BVP18], there exists an abelian category $\mathbf{A}(\beta)$ providing a
universal factorization of $\beta$ of the form

$\beta:\mathscr{D}\xrightarrow{\pi}\mathbf{A}(\beta)\xrightarrow{\iota}\mathscr{A}$

where $\pi:\mathscr{D}\rightarrow\mathbf{A}(\beta)$ is an additive functor and
$\iota:\mathbf{A}(\beta)\rightarrow\mathscr{A}$ is a faithful exact functor.
The abelian category $\mathbf{A}(\beta)$ is known as the universal abelian
factorization associated to the additive functor $\beta$: one can view it as
the finest possible non-full abelian subcategory of $\mathscr{A}$ keeping
track of all the information coming from $\mathscr{D}$ via the functor
$\beta$.

This categorical construction is motivated by the theory of mixed motives and
crucial for the emerging theory of perverse Nori motives introduced by F.
Ivorra and S. Morel in [IM19]. The results of [IM19] and [Ter23N] completely
describe the functoriality of the categories of perverse Nori motives with
respect to Grothendieck’s six functor formalism. The basic building blocks for
the six functor formalism on perverse Nori motives all come from the abstract
lifting properties of universal abelian factorizations with respect to exact
functors and natural transformations thereof.

The main purpose of this note is to study these abstract aspects of universal
abelian factorizations systematically. Some of the main results collected here
have been already obtained before, based on a different axiomatic framework,
and can be found in [Ara13, Ara19], [HMS17], [BVHP20], [Ivo17], and [IM19].
Our motivation to write the present note is twofold: firstly, in order to give
a uniform and self-contained treatment of the previously known results;
secondly, to fill some gaps in the existing literature. Several categorical
constructions recurring in [IM19] and [Ter23N] fit perfectly into the general
framework of the present note; in particular, certain technical proofs of
[Ter23T] become both considerably shorter and conceptually cleaner in light of
the abstract results collected here.

In view of the generality and usefulness of the concept of universal abelian
factorizations, we believe that the results of the present note will find
other applications within the theory of Nori motives and beyond.

### History and previous work

In the late 1990’s, M. Nori found an elegant way to construct a candidate for
the conjectural abelian category of mixed motives over a field. Nori’s
construction is categorical-theoretic at heart, since it relies on a general
result concerning representations of quivers into the abelian category of
finitely generated modules over a Noetherian commutative ring $R$. He showed
that any such quiver representation

$\beta:\mathscr{D}\rightarrow\mathsf{mod}_{R}^{f.g.}$

admits a universal factorization of the form

$\beta:\mathscr{D}\xrightarrow{\pi}\mathbf{A}(\beta)\xrightarrow{\iota}\mathsf{mod}_{R}^{f.g.}$

where $\mathbf{A}(\beta)$ is an $R$-linear abelian category, $\pi$ is a quiver
representation, and
$\iota:\mathbf{A}(\beta)\rightarrow\mathsf{mod}_{R}^{f.g.}$ is a faithful
exact $R$-linear functor. The universal property amounts to the fact that, for
any other factorization

$\beta:\mathscr{D}\xrightarrow{\pi^{\prime}}\mathscr{A}\xrightarrow{\iota^{\prime}}\mathsf{mod}_{R}^{f.g.}$

of the same form, there exists a unique faithful exact $R$-linear functor
$\mathbf{A}(\beta)\rightarrow\mathscr{A}$ making the diagram

${\mathbf{A}(\beta)}$${\mathscr{D}}$${\mathsf{mod}_{R}^{f.g.}}$${\mathscr{A}}$$\scriptstyle{\iota}$$\scriptstyle{\exists!}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\iota^{\prime}}$

commute. The category $\mathbf{A}(\beta)$ is constructed explicitly in terms
of the representation $\beta$ as

$\mathbf{A}(\beta):=2-\varinjlim_{\mathscr{D}^{\prime}\in\mathsf{Fin}(\mathscr{D})}\textup{End}(\beta|_{\mathscr{D}^{\prime}})-\mathsf{mod}_{R}^{f.g.},$

where $\mathsf{Fin}(\mathscr{D})$ denotes the filtered poset of all finite
full subquivers of $\mathscr{D}$ and, for each inclusion
$\mathscr{D}^{\prime}\subset\mathscr{D}^{\prime\prime}$ in
$\mathsf{Fin}(\mathscr{D})$, the corresponding functor
$\textup{End}(\beta|_{\mathscr{D}^{\prime}})-\mathsf{mod}_{R}^{f.g.}\rightarrow\textup{End}(\beta|_{\mathscr{D}^{\prime}})-\mathsf{mod}_{R}^{f.g.}$
in the above $2$-colimit is just the restriction-of-scalars functor induced by
the obvious $R$-algebra homomorphism
$\textup{End}(\beta|_{\mathscr{D}^{\prime\prime}})\rightarrow\textup{End}(\beta|_{\mathscr{D}^{\prime}})$.

It is clear from this description that the association
$\beta\mapsto\mathbf{A}(\beta)$ is functorial in the following sense: given
two quiver representations

$\beta_{1}:\mathscr{D}_{1}\rightarrow\mathsf{mod}_{R}^{f.g.},\quad\beta_{2}:\mathscr{D}_{2}\rightarrow\mathsf{mod}_{R}^{f.g.}$

together with a map of quivers
$\phi:\mathscr{D}_{1}\rightarrow\mathscr{D}_{2}$ fitting into a commutative
diagram of the form

${\mathscr{D}_{1}}$${\mathscr{D}_{2}}$${\mathsf{mod}_{R}^{f.g.}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\phi}$$\scriptstyle{\beta_{2}}$

there exists a unique exact $R$-linear functor
$\bar{\phi}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$ making the
whole diagram

${\mathscr{D}_{1}}$${\mathscr{D}_{2}}$${\mathbf{A}(\beta_{1})}$${\mathbf{A}(\beta_{2})}$${\mathsf{mod}_{R}^{f.g.}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\iota_{2}}$

commute. It is also clear that the association $\phi\mapsto\bar{\phi}$ is
compatible with composition of maps of quivers in the obvious way. Motivated
by his applications to mixed motives, Nori also described a general recipe to
construct a tensor product on the category $\mathbf{A}(\beta)$: the idea is to
start from a pairing
$-\times-:\mathscr{D}\times\mathscr{D}\rightarrow\mathscr{D}$ making the
diagram

${\mathscr{D}\times\mathscr{D}}$${\mathscr{D}}$${\mathsf{mod}_{R}^{f.g.}\times\mathsf{mod}_{R}^{f.g.}}$${\mathsf{mod}_{R}^{f.g.}}$$\scriptstyle{-\times-}$$\scriptstyle{\beta\times\beta}$$\scriptstyle{\beta}$$\scriptstyle{-\otimes_{R}-}$

commute up to natural isomorphism, and extend such a pairing along each step
of the construction of $\mathbf{A}(\beta)$ in order to get a bi-functor
$-\otimes-:\mathbf{A}(\beta)\times\mathbf{A}(\beta)\rightarrow\mathbf{A}(\beta)$
making the whole diagram

${\mathscr{D}\times\mathscr{D}}$${\mathscr{D}}$${\mathbf{A}(\beta)\times\mathbf{A}(\beta)}$${\mathbf{A}(\beta)}$${\mathsf{mod}_{R}^{f.g.}\times\mathsf{mod}_{R}^{f.g.}}$${\mathsf{mod}_{R}^{f.g.}}$$\scriptstyle{-\times-}$$\scriptstyle{\pi\times\pi}$$\scriptstyle{\pi}$$\scriptstyle{-\otimes_{R}-}$$\scriptstyle{\iota\times\iota}$$\scriptstyle{\iota}$$\scriptstyle{-\otimes_{R}-}$

commute up to natural isomorphism. In order to make this idea work, one has to
make sure that the representation
$\beta:\mathscr{D}\rightarrow\mathsf{mod}_{R}^{f.g.}$ takes values into flat
$R$-modules and that, in addition, iterated kernels of such $R$-modules are
still flat. Thus, in concrete applications, one is basically forced to assume
that either $R$ is a field or $R$ is a Dedekind domain and the $R$-module
$\beta(D)$ is torsion-free for all $D\in\mathscr{D}$. See [HMS17, § II] for a
thorough exposition of Nori’s constructions and results.

Until the last decade, the range of applications of Nori’s categorical
construction remained limited to representations of quivers into
$\mathsf{mod}_{R}^{f.g.}$, possibly with further restrictions on the ring $R$.
While this setting was certainly sufficient for Nori’s original applications
to mixed motives, it appeared to be too rigid for possible generalization of
Nori’s theory of motives to a theory of motivic sheaves. The basic idea would
be to mimic Nori’s construction starting from a representation of some quiver
into the abelian categories of constructible or perverse sheaves over
algebraic varieties, which are not of the form $\mathsf{mod}_{R}^{f.g.}$
despite being Noetherian (and even Artinian, in the case of perverse sheaves).

In the setting of ordinary constructible sheaves, Arapura introduced in
[Ara13] a clever yet notationally intricate variant of Nori’s construction:
roughly, one filters the category of constructible sheaves over a variety $S$
by finer and finer stratifications of $S$, considers collections of stalk
functors associated to the different strata, and compares the functors
resulting from different choices of stalks.

Of course, Arapura’s approach cannot be applied to the setting of perverse
sheaves over general varieties. In this setting, Ivorra observed in [Ivo17]
that Nori’s construction can be extended to representation of quivers into
arbitrary finite and hom-finite abelian categories: this works perfectly for
representations into categories of perverse sheaves (whereas it would not work
for representations into ordinary constructible sheaves over general
varieties, since these do not form Artinian categories). Furthermore, it makes
it very easy to understand the compatibility of the construction with respect
to morphisms of quiver representations: given two quiver representations into
finite and hom-finite abelian categories

$\beta_{1}:\mathscr{D}_{1}\rightarrow\mathscr{A}_{1},\quad\beta_{2}:\mathscr{D}_{2}\rightarrow\mathscr{A}_{2}$

together with a map of quivers
$\phi:\mathscr{D}_{1}\rightarrow\mathscr{D}_{2}$ and an exact functor
$\Phi:\mathscr{A}_{1}\rightarrow\mathscr{A}_{2}$ making the diagram

${\mathscr{D}_{1}}$${\mathscr{D}_{2}}$${\mathscr{A}_{1}}$${\mathscr{A}_{2}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\phi}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\Phi}$

commute up to natural isomorphism, there exists an essentially unique exact
functor $\bar{\phi}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$
making the whole diagram

${\mathscr{D}_{1}}$${\mathscr{D}_{2}}$${\mathbf{A}(\beta_{1})}$${\mathbf{A}(\beta_{2})}$${\mathscr{A}_{1}}$${\mathscr{A}_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\Phi}$

commute up to natural isomorphism. It is worth stressing that Ivorra’s result
is more precise than what we have just asserted, since it includes the datum
of a natural isomorphism
$\kappa:\Phi\circ\beta_{1}\xrightarrow{\sim}\beta_{2}\circ\phi$ into the input
and provides an explicit factorization of such a natural isomorphism as part
of the output. A similar result holds on the level of natural transformations;
again, in order to make the construction work, one needs to describe both the
input and the output very carefully.

It is natural to wonder whether Nori’s construction of universal abelian
factorizations can be further generalized to representations of quivers into
arbitrary abelian categories. This question was given an affirmative answer
thanks to O. Caramello’s investigations around the categorical logic of Nori
motives, which culminated in the paper [BVCL18]; in the subsequent paper
[BVP18], the same result was translated into the familiar language of
categories. The construction of the universal abelian factorization of a
general quiver representation $\beta:\mathscr{D}\rightarrow\mathscr{A}$ can be
described in a uniform and streamlined way as follows:

  1. (Step 1)

Make the quiver $\mathscr{D}$ into a category $\mathscr{D}^{cat}$, so that the
representation $\beta$ becomes a functor

$\beta^{cat}:\mathscr{D}^{cat}\rightarrow\mathscr{A}.$

  2. (Step 2)

Make the category $\mathscr{D}^{cat}$ into an additive category
$\mathscr{D}^{\oplus}$, so that the functor $\beta^{cat}$ extends to an
additive functor

$\beta^{\oplus}:\mathscr{D}^{\oplus}\rightarrow\mathscr{A}.$

  3. (Step 3)

Form the abelian hull $\mathbf{A}(\mathscr{D}^{\oplus})$, so that the additive
functor $\beta^{\oplus}$ induces an exact functor

$\beta^{+}:\mathbf{A}(\mathscr{D}^{\oplus})\rightarrow\mathscr{A}.$

  4. (Step 4)

Define the Serre quotient

$\mathbf{A}(\beta):=\mathbf{A}(\mathscr{D}^{\oplus})/\ker(\beta^{+}).$

Here, the first non-trivial passage is that from the additive category
$\mathscr{D}^{+}$ to its abelian hull $\mathbf{A}(\mathscr{D}^{+})$: the
latter is an abelian category on which $\mathscr{D}^{+}$ embeds fully
faithfully, with the universal property that every additive functor from
$\mathscr{D}^{+}$ to some abelian category extends uniquely to an exact
functor from the entire $\mathbf{A}(\mathscr{D}^{+})$. The existence of the
category $\mathbf{A}(\mathscr{D}^{+})$ was proved by P. Freyd in the classical
paper [Fre65]. Defining $\mathbf{A}(\beta)$ as a Serre quotient as in the last
passage amounts to rendering the exact functor
$\beta^{+}:\mathbf{A}(\mathscr{D}^{+})\rightarrow\mathscr{A}$ a faithful exact
functor in a universal way; with this explicit description, checking that the
category $\mathbf{A}(\beta)$ has the same universal property as Nori’s abelian
category essentially reduces to writing down the various universal properties
used in the passages above in sequence. In the case of quiver representations
arising from homological functors, a more streamlined description of the same
construction was given in [IM19], based on the study of Freyd’s abelian hulls
of triangulated categories from [Nee01]. Around the same time, the case of
representations into abelian tensor categories was explored in detail in
[BVHP20]. However, the compatibility of the latter construction of
$\mathbf{A}(\beta)$ with respect to functors and natural transformations has
not been explored yet in full generality.

### Main results

In the present note, we rephrase and extend the functoriality results of
[Ivo17] and [BVHP20] using the most general construction of universal abelian
factorization given in [BVP18]. Some of the fundamental results collected here
are just obvious adaptations of those presented in [Ivo17], using slightly
more streamlined notation.

For sake of simplicity, we refrain from considering the general framework of
representation of arbitrary quivers into abelian categories: all quivers and
maps of quivers that we consider in this paper are defined by actual
(additive) categories and (additive) functors. This choice is motivated by our
application to perverse Nori motives and has two advantages: on the one hand,
it allows us to avoid phrasing the main notions in the language of quivers and
discussing their (often straightforward) extension to the additive level in
detail; on the other hand, it allows us to study categorical notions (such as
adjunctions, equivalences, and $2$-colimits) that are needed for our
applications but lack a good analogue in the world of quivers.

The crucial functoriality results concern lifting of exact functors
(Proposition 2.5) and natural transformations thereof (Proposition 3.4);
combining these two fundamental lifting principles, we can easily study the
compatibility of universal abelian factorizations with several common
categorical notions, including adjunctions (Corollary 3.8) and equivalences
(Corollary 3.9).

Our main original contribution concern the extension of these functoriality
results to the case of multi-linear functors (Proposition 4.3 and Proposition
4.6) and to the setting of abelian fibered categories (Proposition 5.8 and
Proposition 5.10); this leads to a similar extension to the setting of
monoidal structures on abelian fibered categories (Proposition 6.4).

Along the way, we also collect useful observations about the interaction
between universal abelian factorizations and several common categorical
constructions, such as direct products (Corollary 2.12) and filtered
$2$-colimits (5.12).

### Structure of the paper

In Section 1 we review the construction of universal abelian factorizations
described in [BVP18] in the case of additive quiver representations; this
gives us the opportunity to fix the general notation used throughout the
paper.

The main technical results are collected in Section 2 and Section 3, where we
explain how to lift exact functors (Proposition 2.5) and natural
transformations thereof (Proposition 3.4) to the level of universal abelian
factorizations, respectively. In order to state our results in a precise form,
it is convenient to introduce informally a suitable ’$2$-category of quiver
representations’ where objects are single quiver representations into abelian
categories while $1$-morphisms (resp. $2$-morphisms) consist of pairs of
functors (resp. natural transformations) on the quiver level and on the
abelian level satisfying a certain compatibility condition (Definition 2.1 and
Definition 3.1). As a particular case, we show that the lifting procedure
naturally respects adjunctions (Corollary 3.8) and equivalences (Corollary
3.9). Moreover, we are able to study the compatibility of universal abelian
factorizations with diagram categories to some extent: most notably, we show
that universal abelian factorizations are compatible with finite direct
products of quiver representations (see Corollary 2.12).

This opens the way to extending the main lifting results to the case of multi-
exact functors and natural transformations thereof in Section 4. Arguing by
induction on the number of factors involved, we show that it is possible to
lift both multi-exact functors (Proposition 4.3) and their natural
transformations (Proposition 4.6) in the expected way. This represents our
first main original contribution to the existing literature.

Our second main contribution concerns the setting of abelian categories
fibered over a base, which is the subject of Section 5. In this case, an easy
combination of the results of Section 2 and Section 3 allows us to deduce
analogous lifting results for single abelian fibered categories (Proposition
5.8) as well as for exact morphisms of such (Proposition 5.10).

The final Section 6 is devoted to the study of monoidal structures on
universal abelian factorizations. Applying the general results of Section 4,
we show that it is possible to lift exact monoidal structures (Proposition
6.4) together with associativity and commutativity constraints (6.8 and 6.10).
In view of our applications, we have chosen to phrase everything in terms of
the external tensor product following [Ter23T]; however, similar results hold
with respect to the usual internal tensor product (see [Ter23T, Thm. 6.1, Thm.
8.1] for the dictionary relating the two formulations). Due of lack of
applications, we did not treat the case of unit constraints (see [Ter23T, §
5]), but this could be easily included.

### Acknowledgments

The present note originated from an appendix of my Ph.D. thesis, written at
the University of Freiburg under the supervision of Annette Huber-Klawitter.
It is a pleasure to thank her for introducing me to the idea of universal
abelian factorizations and for many useful discussions around the contents of
this note, as well as for her constant support and encouragement.

##  1\. Recollections on universal abelian factorizations

We start by reviewing the construction of universal abelian factorizations of
representations of additive categories into abelian categories as described in
[BVP18], as well as the basic properties characterizing this construction. All
the results obtained in the next sections will essentially be formal
consequence of these properties.

###  1.1. Review of Freyd’s abelian hull

In the first place, we review the main properties of Freyd’s construction of
abelian hulls from [Fre65]. Recall that, for any category $\mathscr{C}$, a
presheaf $F\in\mathbf{Fun}(\mathscr{C}^{op},\mathbf{Ab})$ is called finitely
presented if it can be written as a cokernel of representable presheaves.

######  Notation 1.1.

Let $\mathscr{D}$ be an additive category.

  * •

We define $\mathbf{R}(\mathscr{D})$ as the full additive subcategory of
$\mathbf{Fun}(\mathscr{D}^{op},\mathbf{Ab})$ consisting of all finitely
presented presheaves.

  * •

Dually, we define
$\mathbf{L}(\mathscr{D}):=\mathbf{R}(\mathscr{D}^{op})^{op}$.

Note that any additive category $\mathscr{D}$ admits two canonical fully
faithful additive embeddings

$\mathscr{D}\hookrightarrow\mathbf{R}(\mathscr{D}),\qquad\mathscr{D}\hookrightarrow\mathbf{L}(\mathscr{D}),$

the first one induced by the Yoneda embedding
$\mathscr{D}\hookrightarrow\mathbf{Fun}(\mathscr{D}^{op},\mathbf{Ab})$ and the
second one arising from the embedding
$\mathscr{D}^{op}\hookrightarrow\mathbf{R}(\mathscr{D}^{op})$. Combining the
two constructions, we also obtain canonical fully faithful additive embeddings

$\mathscr{D}\hookrightarrow\mathbf{L}(\mathbf{R}(\mathscr{D})),\qquad\mathscr{D}\hookrightarrow\mathbf{R}(\mathbf{L}(\mathscr{D})).$

With this notation, we can summarize Freyd’s results as follows:

######  Theorem 1.2 (P. Freyd).

Let $\mathscr{D}$ be an additive category. Then the following statements hold:

  1. (1)

Define the additive category

(1) $\mathbf{A}(\mathscr{D}):=\mathbf{L}(\mathbf{R}(\mathscr{D})).$

Then:

    1. (i)

The category $\mathbf{A}(\mathscr{D})$ is abelian.

    2. (ii)

There exists a canonical additive equivalence of categories

(2)
$\mathbf{A}(\mathscr{D}):=\mathbf{L}(\mathbf{R}(\mathscr{D}))=\mathbf{R}(\mathbf{L}(\mathscr{D}))$

making the diagram

(3)
${\mathscr{D}}$${\mathbf{L}(\mathbf{R}(\mathscr{D}))}$${\mathbf{R}(\mathbf{L}(\mathscr{D}))}$

commute.

    3. (iii)

Every object of $\mathscr{D}$ is projective in $\mathbf{A}(\mathscr{D})$ via
the canonical embedding $\mathscr{D}\hookrightarrow\mathbf{A}(\mathscr{D})$.

  2. (2)

For every abelian category $\mathscr{A}$, the following statements hold:

    1. (i)

Every additive functor $\beta:\mathscr{D}\rightarrow\mathscr{A}$ extends
uniquely to an exact functor

$\beta^{+}:\mathbf{A}(\mathscr{D})\rightarrow\mathscr{A}.$

    2. (ii)

Given two additive functors
$\beta_{1},\beta_{2}:\mathscr{D}\rightarrow\mathscr{A}$, every natural
transformation $\kappa:\beta_{1}\rightarrow\beta_{2}$ extends uniquely to a
natural transformation between functors
$\mathbf{A}(\mathscr{D})\rightarrow\mathscr{A}$

$\kappa^{+}:\beta_{1}^{+}\rightarrow\beta_{2}^{+}.$

###### Proof.

  1. (1)

Statement (i) is proved in [Fre65, Thm 4.1]. Once one knows that
$\mathbf{L}(\mathbf{R}(\mathscr{D}))$ satisfies the universal property in
(2)(i), one deduces by symmetry that $\mathbf{R}(\mathbf{L}(\mathscr{D}))$
satisfies the same universal property, and this gives the canonical
equivalence in (ii). Via this equivalence, the fully faithful embedding
$\mathscr{D}\hookrightarrow\mathbf{A}(\mathscr{D})$ factors through the usual
Yoneda embedding
$\mathbf{L}(\mathscr{D})\hookrightarrow\mathbf{A}(\mathscr{D})$. As a
consequence of the Yoneda Lemma, all objects in $\mathbf{L}(\mathscr{D})$ are
projective in $\mathbf{A}(\mathscr{D})$, and this implies (iii).

  2. (2)

Statement (i) is proved in [Fre65, Thm. 4.1]. In order to prove (ii), let
$\mathscr{I}$ denote the category with two objects $1$ and $2$ and exactly one
non-identity arrow $1\rightarrow 2$. Given an abelian category $\mathscr{A}$
as in the statement, the category
$\mathscr{A}^{\mathscr{I}}:=\mathbf{Fun}(\mathscr{I},\mathscr{A})$ is again
abelian. Moreover, for every (additive) category $\mathscr{C}$, giving one
(additive) functor $F:\mathscr{C}\rightarrow\mathscr{A}^{\mathscr{I}}$ is the
same as giving two (additive) functors $F_{1},F_{2}\rightarrow\mathscr{A}$ and
a natural transformation $F_{1}\rightarrow F_{2}$; if in addition
$\mathscr{C}$ is abelian then, under this correspondence, $F$ is exact if and
only if $F_{1}$ and $F_{2}$ are both exact. Using this observation, we see
that (ii) follows by applying (i) to the abelian category
$\mathscr{A}^{\mathscr{I}}$.

∎

######  Corollary 1.3.

For every additive category $\mathscr{D}$, there exists a canonical
equivalence

(4) $\mathbf{A}(\mathscr{D}^{op})=\mathbf{A}(\mathscr{D})^{op}$

making the diagram

(5)
${\mathscr{D}^{op}}$${\mathbf{A}(\mathscr{D}^{op})}$${\mathbf{A}(\mathscr{D})^{op}}$

commute.

###### Proof.

Using Theorem 1.2(1)(ii), one can construct the equivalence in the statement
directly as

$\mathbf{A}(\mathscr{D}^{op}):=\mathbf{L}(\mathbf{R}(\mathscr{D}^{op}))=\mathbf{L}(\mathbf{L}(\mathscr{D})^{op})=\mathbf{R}(\mathbf{L}(\mathscr{D}))^{op}=\mathbf{A}(\mathscr{D})^{op},$

where the last passage is induced by the canonical equivalence (2); the
commutativity of the diagram (5) then follows directly from that of (3).
Alternatively, one can observe that the abelian categories
$\mathbf{A}(\mathscr{D}^{op})$ and $\mathbf{A}(\mathscr{D})^{op}$ satisfy the
same universal property described in Theorem 1.2(2)(i); of course, the
resulting equivalence
$\mathbf{A}(\mathscr{D}^{op})=\mathbf{A}(\mathscr{D})^{op}$ coincides with the
previous one. ∎

######  Remark 1.4.

The abelian category $\mathbf{A}(\mathscr{D})$ defined as in (1) is usually
called the abelian hull of the additive category $\mathscr{D}$. Note that this
terminology is slightly misleading, in that the abelian hull
$\mathbf{A}(\mathscr{A})$ of an additive category $\mathscr{A}$ is usually
much bigger than $\mathscr{A}$ itself! Indeed, in view of Theorem 1.2(2)(i),
the inclusion $\mathscr{A}\hookrightarrow\mathbf{A}(\mathscr{A})$ is an
equivalence if and only if every additive functor from $\mathscr{A}$ to an
abelian category is automatically exact; by considering the functors
$\textup{Hom}_{\mathscr{A}}(A,-):\mathscr{A}\rightarrow\mathbf{Ab}$ for
$A\in\mathscr{A}$, we deduce that this is the case if and only if all objects
of $\mathscr{A}$ are projective.

In the next two sections we will explain how to construct functors and natural
transformations on the level of universal abelian factorizations. These
constructions are based on the following easy consequence of Theorem 1.2:

######  Corollary 1.5.

Let $\mathscr{D}_{1}$ and $\mathscr{D}_{2}$ be two additive categories. Then:

  1. (1)

Every additive functor $\phi:\mathscr{D}_{1}\rightarrow\mathscr{D}_{2}$
extends uniquely to an exact functor

$\phi^{+}:\mathbf{A}(\mathscr{D}_{1})\rightarrow\mathbf{A}(\mathscr{D}_{2}).$

  2. (2)

Given two additive functors
$\phi_{1},\phi_{2}:\mathscr{D}_{1}\rightarrow\mathscr{D}_{2}$, every natural
transformation $\kappa:\phi_{1}\rightarrow\phi_{2}$ extends uniquely to a
natural transformation of functors
$\mathbf{A}(\mathscr{D}_{1})\rightarrow\mathbf{A}(\mathscr{D}_{2})$

$\kappa^{+}:\phi_{1}^{+}\rightarrow\phi_{2}^{+}.$

###### Proof.

  1. (1)

This follows by applying Theorem 1.2(2)(i) to the composite functor
$\mathscr{D}_{1}\xrightarrow{\phi}\mathscr{D}_{2}\hookrightarrow\mathbf{A}(\mathscr{D}_{2})$.

  2. (2)

This follows by applying Theorem 1.2(2)(ii) to the natural transformation of
additive functors $\mathscr{D}_{1}\rightarrow\mathbf{A}(\mathscr{D}_{2})$
induced by $\kappa$.

∎

######  Remark 1.6.

For every ring $R$ there exists an $R$-linear variant of Freyd’s construction:
given an $R$-linear additive category $\mathscr{D}$, it provides an $R$-linear
abelian category $\mathbf{A}_{R}(\mathscr{D})$ such that, for every $R$-linear
abelian category $\mathscr{A}$, every $R$-linear additive functor
$\mathscr{D}\rightarrow\mathscr{A}$ extends uniquely to an $R$-linear exact
functor $\mathbf{A}_{R}(\mathscr{D})\rightarrow\mathscr{A}$; see [BVP18, §
1.4] for more details.

For our applications we are mainly interested in the case where
$R=\mathbb{Q}$, which is particularly easy to treat. Indeed, for an additive
category, being $\mathbb{Q}$-linear is a property and not an additional
structure; moreover, for every $\mathbb{Q}$-linear additive category
$\mathscr{C}$, the categories $\mathbf{R}(\mathscr{C})$ and
$\mathbf{L}(\mathscr{C})$ are easily seen to be $\mathbb{Q}$-linear as well.
As a consequence, for every $\mathbb{Q}$-linear additive category
$\mathscr{D}$, the abelian category $\mathbf{A}(\mathscr{D})$ is
$\mathbb{Q}$-linear; since it obviously satisfies the universal property of
$\mathbf{A}_{\mathbb{Q}}(\mathscr{D})$, it is canonically isomorphic to it.

###  1.2. Universal abelian factorizations

It will prove convenient to treat representations of additive quivers into
abelian categories as being the objects of a certain "$2$-category of
representations". For the purposes of the present section, it suffices to
consider objects of this $2$-category; the notions of $1$-morphism and
$2$-morphism are deferred until Section 2 and Section 3, respectively.

######  Definition 1.7.

  1. (1)

We call (additive) representation a triple $(\mathscr{C},\mathscr{E};R)$
consisting of two (additive) categories $\mathscr{C}$, $\mathscr{E}$ and one
(additive) functor $R:\mathscr{C}\rightarrow\mathscr{E}$.

  2. (2)

We say that an additive representation $(\mathscr{C},\mathscr{E};R)$ is
abelian if the category $\mathscr{E}$ is abelian.

We are ready to define universal abelian factorizations of abelian
representations following [BVP18, § 1.4]:

######  Definition 1.8.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian representation. We let
$\mathbf{A}(\beta)$ denote the abelian category obtained as the Serre quotient
of the abelian hull $\mathbf{A}(\mathscr{D})$ by the Serre subcategory
$\ker\left\\{\beta^{+}:\mathbf{A}(\mathscr{D})\rightarrow\mathscr{A}\right\\}$,
and we call it the universal abelian factorization of $\beta$.

######  Notation 1.9.

Given an abelian representation $(\mathscr{D},\mathscr{A};\beta)$, we will
often use the following notation:

  * •

We let $\pi$ denote the canonical quotient functor
$\mathbf{A}(\mathscr{D})\rightarrow\mathbf{A}(\beta)$, and we also write $\pi$
for the composite additive functor
$\mathscr{D}\rightarrow\mathbf{A}(\mathscr{D})\rightarrow\mathbf{A}(\beta)$.

  * •

We let $\iota$ denote the induced faithful exact functor
$\mathbf{A}(\beta)\hookrightarrow\mathscr{A}$.

Whenever several abelian representations are involved, we adapt the above
notation in an obvious manner.

The following basic consequence of Theorem 1.2(2)(i) explains why universal
abelian factorizations deserve such a name:

######  Proposition 1.10.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian representation, with
universal abelian factorization

$\mathscr{D}\xrightarrow{\pi}\mathbf{A}(\beta)\xrightarrow{\iota}\mathscr{A}.$

Moreover, let $\mathscr{A}^{\prime}$ be another abelian category endowed with
a faithful exact functor $\Phi:\mathscr{A}^{\prime}\rightarrow\mathscr{A}$,
and suppose that there exists a representation
$\phi:\mathscr{D}\rightarrow\mathscr{A}^{\prime}$ making the diagram

${\mathscr{D}}$${\mathscr{A}^{\prime}}$${\mathscr{A}}$$\scriptstyle{\phi}$$\scriptstyle{\beta}$$\scriptstyle{\Phi}$

commute. Then there exists a unique faithful exact functor

(6) $\mathbf{A}(\beta)\rightarrow\mathscr{A}^{\prime}$

making the whole diagram

(7)
${\mathscr{D}}$${\mathbf{A}(\beta)}$${\mathscr{A}^{\prime}}$${\mathscr{A}}$$\scriptstyle{\phi}$$\scriptstyle{\beta}$$\scriptstyle{\pi}$$\scriptstyle{\iota}$$\scriptstyle{\Phi}$

commute.

###### Proof.

By Theorem 1.2(2)(i), the abelian representation
$\phi:\mathscr{D}\rightarrow\mathscr{A}^{\prime}$ extends canonically to an
exact functor

$\phi^{+}:\mathbf{A}(\mathscr{D})\rightarrow\mathscr{A}^{\prime}.$

By the universal property of Serre quotients, the latter factors though
$\mathbf{A}(\beta)$ if and only if, for an object
$A\in\mathbf{A}(\mathscr{D})$, the implication

$\phi^{+}(A)=0\implies\beta^{+}(A)=0$

holds; if this is the case, the induced exact functor (6) is uniquely
determined by this property and makes the whole diagram (7) commute. But we
have the chain of implications

$\phi^{+}(A)=0\implies\Phi(\phi^{+}(A))=0\iff\beta^{+}(A)=0,$

where the second passage follows from the equality
$\Phi\circ\phi^{+}=\beta^{+}$ (the latter, in turn, follows from the equality
$\Phi\circ\phi=\beta$, which holds by hypothesis). This proves the claim. ∎

######  Corollary 1.11.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian representation, and let
$\beta^{op}:\mathscr{D}^{op}\rightarrow\mathscr{A}^{op}$ denote the functor
induced by $\beta$ on the opposite categories. Then there exists a canonical
additive equivalence

$\mathbf{A}(\beta^{op})=\mathbf{A}(\beta)^{op}$

making the diagram

(8)
${\mathscr{D}^{op}}$${\mathbf{A}(\mathscr{D}^{op})}$${\mathbf{A}(\mathscr{D})^{op}}$${\mathbf{A}(\beta^{op})}$${\mathbf{A}(\beta)^{op}}$${\mathscr{A}^{op}}$$\scriptstyle{\pi^{op}}$$\scriptstyle{\pi^{op}}$$\scriptstyle{\iota^{op}}$$\scriptstyle{\iota^{op}}$

commute.

###### Proof.

This is completely analogous to Corollary 2.12. Namely, one can construct the
equivalence in the statement as

$\mathbf{A}(\beta^{op}):=\mathbf{A}(\mathscr{D}^{op})/\ker(\beta^{op,+})=\mathbf{A}(\mathscr{D})^{op}/\ker(\beta^{+})^{op}=:\mathbf{A}(\beta)^{op}$

where the middle equivalence is induced by (4); the commutativity of the
diagram (8) than follows directly from that of (5). Alternatively, one can
observe that the abelian categories $\mathbf{A}(\beta^{op})$ and
$\mathbf{A}(\beta)^{op}$ satisfy the same universal property described in
Proposition 1.10; of course, the resulting equivalence
$\mathbf{A}(\beta^{op})=\mathbf{A}(\beta)^{op}$ coincides with the previous
one. ∎

######  Remark 1.12.

The result of Proposition 1.10 seems to suggest a possible approach to recover
$\mathbf{A}(\beta)$ directly as a subcategory of $\mathscr{A}$. Namely,
consider the collection $\mathfrak{A}$ of all those additive subcategories
$\mathscr{A}^{\prime}$ of $\mathscr{A}$ which satisfy the following
properties:

  1. (i)

$\mathscr{A}^{\prime}$ contains $\beta(\mathscr{D})$;

  2. (ii)

$\mathscr{A}^{\prime}$ is abelian;

  3. (iii)

the inclusion functor $\mathscr{A}^{\prime}\subset\mathscr{A}$ is exact.

The collection $\mathfrak{A}$ is non-empty, since it contains $\mathscr{A}$
itself. If one could order $\mathfrak{A}$ by inclusion, then Proposition 1.10
would roughly state that the subcategory $\iota(\mathbf{A}(\beta))$ of
$\mathscr{A}$ is the minimal object of $\mathfrak{A}$; one would thus be
tempted to define $\mathbf{A}(\beta)$ to be the intersection of all members of
$\mathfrak{A}$. One issue with this approach is that, since the subcategories
$\mathscr{A}^{\prime}$ of $\mathscr{A}$ above are not assumed to be full, it
becomes somewhat problematic to check the stability of properties (ii) and
(iii) above under intersection: for example, the kernel of the same morphism
could be represented by distinct objects of $\mathscr{A}$ in two different
members of $\mathfrak{A}$. This issue cannot be removed by imposing that each
member of $\mathfrak{A}$ be stable under isomorphisms of its objects in
$\mathscr{A}$, since this would force one to automatically include all
automorphisms of each object in the image of $\beta$.

Before moving further, as a quick sanity check, let us note that the
construction of universal abelian factorizations is idempotent in a precise
sense:

######  Lemma 1.13.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian representation, with
associated factorization

$\beta:\mathscr{D}\xrightarrow{\pi}\mathbf{A}(\beta)\xrightarrow{\iota}\mathscr{A}.$

Regard the triple $(\mathscr{D},\mathbf{A}(\beta);\pi)$ as another abelian
representation, and let

$\pi:\mathscr{D}\xrightarrow{\rho}\mathbf{A}(\pi)\xrightarrow{\nu}\mathbf{A}(\beta)$

denote the corresponding factorization. Then the canonical faithful exact
functor $\nu:\mathbf{A}(\pi)\rightarrow\mathbf{A}(\beta)$ is in fact an
equivalence.

###### Proof.

It suffices to show that the two quotient functors

$\pi:\mathbf{A}(\mathscr{D})\rightarrow\mathbf{A}(\beta),\qquad\rho:\mathbf{A}(\mathscr{D})\rightarrow\mathbf{A}(\pi)$

have the same kernel. But, for every object $A\in\mathbf{A}(\mathscr{D})$, we
have the implications

$\pi(A)=0\iff\nu(\rho(A))=0\iff\rho(A)=0,$

where the second passage follows from the conservativity of $\nu$. This proves
the claim. ∎

##  2\. Lifting exact functors

In this section we explain how to define exact functors between universal
abelian factorizations. This is essentially an adaptation of the results of
[Ivo17] to the setting of [BVP18]; similar results are discussed in [IM19, §
1]. We firstly prove the analogue of [Ivo17, Prop. 6.3] in a quite rigid form
and then explain how to make it more flexible; the latter form turns out to be
a useful recognition principle in concrete applications.

###  2.1. Main result

We start by proving the general lifting result for exact functors. In order to
formulate it in a compact form, it is convenient to introduce the following
notion of $1$-morphism of (additive) representations:

######  Definition 2.1.

For $i=1,2$, let $(\mathscr{C}_{i},\mathscr{E}_{i};R_{i})$ be a representation
in the sense of Definition 1.7.

  1. (1)

We call $1$-morphism of representations from
$(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})$ to
$(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$ a triple $\alpha=(F,G;\kappa)$
consisting of two (additive) functors

$F:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2},\qquad
G:\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$

and a natural isomorphism of functors
$\mathscr{C}_{1}\rightarrow\mathscr{E}_{2}$

$\kappa:G\circ R_{1}\xrightarrow{\sim}R_{2}\circ F.$

We write it as
$\alpha:(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$
and we often depict it as a diagram of the form

${\mathscr{C}_{1}}$${\mathscr{C}_{2}}$${\mathscr{E}_{1}}$${\mathscr{E}_{2}}$$\scriptstyle{F}$$\scriptstyle{R_{1}}$$\scriptstyle{R_{2}}$$\scriptstyle{G}$

where, for notational simplicity, we do not write the natural isomorphism
$\kappa$ explicitly.

  2. (2)

In case the representations $(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})$ and
$(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$ are abelian, we say that a
$1$-morphism
$\alpha=(F,G;\kappa):(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$
is exact if the additive functor $G:\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$
is exact.

######  Remark 2.2.

Let us stress that, in particular, given two (additive) categories
$\mathscr{C}$ and $\mathscr{E}$ and two (additive) functors
$R_{1},R_{2}:\mathscr{C}\rightarrow\mathscr{E}$, $1$-morphisms of
representations of the form
$(\textup{id}_{\mathscr{C}},\textup{id}_{\mathscr{E}};\kappa):(\mathscr{C},\mathscr{E};R_{1})\rightarrow(\mathscr{C},\mathscr{E};R_{2})$
correspond to natural isomorphisms $\kappa:R_{1}\xrightarrow{\sim}R_{2}$
rather than just to natural transformations.

######  Remark 2.3.

There exists a natural notion of composition for $1$-morphisms of
representations: given two $1$-morphisms of representations
$\alpha_{1}=(F_{1},G_{1};\kappa_{1}):(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$
and
$\alpha_{2}=(F_{2},G_{2};\kappa_{2}):(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})\rightarrow(\mathscr{C}_{3},\mathscr{E}_{3};R_{3})$,
we define the composite $1$-morphism of representations as the triple

$(F_{2}\circ F_{1},G_{2}\circ
G_{1};\kappa_{2}\star\kappa_{1}):(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{1};R_{2}),$

where $\kappa_{2}\star\kappa_{1}$ denotes the natural isomorphism of functors
$\mathscr{C}_{1}\rightarrow\mathscr{E}_{3}$ defined by the formula

$(\kappa_{2}\star\kappa_{1})(C_{1}):=\kappa_{2}(F_{1}(C_{1}))\circ
G_{2}(\kappa_{1}(C_{1})):G_{2}G_{1}R_{1}(C_{1})\xrightarrow{\sim}R_{3}F_{2}F_{1}(C_{1}).$

We write it as $\alpha_{2}\circ\alpha_{1}$, and we often depict it as a
diagram of the form

${\mathscr{C}_{1}}$${\mathscr{C}_{2}}$${\mathscr{C}_{3}}$${\mathscr{E}_{1}}$${\mathscr{E}_{2}}$${\mathscr{E}_{3}}$$\scriptstyle{F_{1}}$$\scriptstyle{R_{1}}$$\scriptstyle{F_{2}}$$\scriptstyle{R_{2}}$$\scriptstyle{R_{3}}$$\scriptstyle{G_{1}}$$\scriptstyle{G_{2}}$

where, for notational simplicity, we do not write the natural isomorphisms
$\kappa_{1}$ and $\kappa_{2}$ explicitly.

######  Remark 2.4.

Clearly, the composite of two exact $1$-morphisms of representations is exact
as well.

We can now state our first main lifting result (analogous to [Ivo17, Prop.
6.3]) as follows:

######  Proposition 2.5.

For $i=1,2$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an abelian
representation. Suppose that we are given an exact morphism of abelian
representations
$\alpha=(\phi,\Phi;\kappa):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})$.
Then:

  1. (1)

There exists a unique exact functor
$\overline{\phi}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$
satisfying the equality of additive functors
$\mathscr{D}_{1}\rightarrow\mathbf{A}(\beta_{2})$

(9) $\pi_{2}\circ\phi=\overline{\phi}\circ\pi_{1}.$

We let $\hat{\alpha}$ denote the exact morphism of abelian representations

$(\phi,\overline{\phi};\textup{id}):(\mathscr{D}_{1},\mathbf{A}(\beta_{1});\pi_{1})\rightarrow(\mathscr{D}_{2},\mathbf{A}(\beta_{2});\pi_{2}).$

  2. (2)

There exists a unique natural isomorphism of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}_{2}$

(10)
$\tilde{\kappa}:\Phi\circ\iota_{1}\xrightarrow{\sim}\iota_{2}\circ\overline{\phi}$

making the diagram of functors $\mathscr{D}_{1}\rightarrow\mathscr{A}_{2}$

(11)
${\Phi\circ\beta_{1}}$${\beta_{2}\circ\phi}$${\Phi\circ\iota_{1}\circ\pi_{1}}$${\iota_{2}\circ\overline{\phi}\circ\pi_{1}}$${\iota_{2}\circ\pi_{2}\circ\phi}$$\scriptstyle{\kappa}$$\scriptstyle{\tilde{\kappa}}$

commute. We let $\tilde{\alpha}$ denote the exact morphism of abelian
representations

$(\overline{\phi},\Phi;\tilde{\kappa}):(\mathbf{A}(\beta_{1}),\mathscr{A}_{1};\iota_{1})\rightarrow(\mathbf{A}(\beta_{2}),\mathscr{A}_{2};\iota_{2}).$

###### Proof.

Since the abelian category $\mathbf{A}(\beta_{1})$ is generated under
(co)kernels by the image of $\mathscr{D}_{1}$, an exact functor
$\overline{\phi}$ as in (1) is uniquely determined by the equality (9),
provided it exists. For the same reason, a natural isomorphism
$\tilde{\kappa}$ as in (2) is uniquely determined by the commutativity of the
diagram (11), provided it exists. In order to show the existence of
$\overline{\phi}$ and $\tilde{\kappa}$, we argue as follows:

  1. (1)

Applying Corollary 1.5, we obtain a canonical exact functor

$\phi^{+}:\mathbf{A}(\mathscr{D}_{1})\rightarrow\mathbf{A}(\mathscr{D}_{2}).$

and a canonical natural isomorphism of functors
$\mathbf{A}(\mathscr{D}_{1})\rightarrow\mathscr{A}_{2}$

$\kappa^{+}:\Phi\circ\beta_{1}\xrightarrow{\sim}\beta_{2}\circ\phi^{+}.$

By the universal property of Serre quotients, a functor $\overline{\phi}$ as
in the statement exists if and only if, for an object
$A\in\mathbf{A}(\mathscr{D}_{1})$, the implication

$\pi_{1}(A)=0\implies\pi_{2}(\phi^{+}(A))=0$

holds. But we have the chain of implications

$\displaystyle\pi_{1}(A)=0\implies\iota_{1}(\pi_{1}(A))=0\implies$
$\displaystyle\Phi(\iota_{1}(\pi_{1}(A)))=0$ $\displaystyle\iff$
$\displaystyle\iota_{2}(\pi_{2}(\phi^{+}(A)))=0\implies\pi_{2}(\phi^{+}(A))=0,$

where the third passage follows from the fact that the objects
$\Phi(\iota_{1}(\pi_{1}(A)))=\Phi(\beta_{1}^{+}(A))$ and
$\iota_{2}(\pi_{2}(\phi(A)))=\beta_{2}^{+}(\phi(A))$ are isomorphic via
$\kappa^{+}(A)$, while the last passage follows from the conservativity of
$\iota_{2}$.

  2. (2)

As a consequence of the uniqueness property in Theorem 1.2(2)(ii), we see that
a natural isomorphism as in (10) makes the diagram (11) commute if and only if
makes the induced diagram of exact functors
$\mathbf{A}(\mathscr{D}_{1})\rightarrow\mathscr{A}_{2}$

${\Phi\circ\beta_{1}^{+}}$${\beta_{2}^{+}\circ\phi^{+}}$${\Phi\circ\iota_{1}\circ\pi_{1}}$${\iota_{2}\circ\overline{\phi}\circ\pi_{1}}$${\iota_{2}\circ\pi_{2}\circ\phi^{+}}$$\scriptstyle{\kappa}$$\scriptstyle{\tilde{\kappa}}$

commute. Since the quotient functor
$\pi_{1}:\mathbf{A}(\mathscr{D}_{1})\rightarrow\mathbf{A}(\beta_{1})$ is a
bijection on objects, we deduce that the only possible definition of (10) is
precisely via the latter diagram.

∎

######  Remark 2.6.

Clearly, the result of Proposition 1.10 is just a particular case of
Proposition 2.5.

######  Remark 2.7.

  1. (1)

Note that, while the existence of $\overline{\phi}$ follows from the existence
of the exact functor $\Phi$ (and of the natural isomorphism $\kappa$), the
actual expression of $\overline{\phi}$ only depends on $\phi$.

  2. (2)

As a consequence of the uniqueness in the two statements of Proposition 2.5,
we see that the assignments $\alpha\mapsto\hat{\alpha}$ and
$\alpha\mapsto\tilde{\alpha}$ are both compatible with composition of
$1$-morphisms of representations (as defined in Remark 2.3) in the following
sense: Given two exact $1$-morphisms of abelian representations
$\alpha_{1}=(\phi_{1},\Phi_{1};\kappa_{1}):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\kappa_{2})$
and
$\alpha_{2}=(\phi_{2},\Phi_{2};\kappa_{2}):(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})\rightarrow(\mathscr{D}_{3},\mathscr{A}_{3};\kappa_{3})$,
we have the equalities

$\widehat{(\alpha_{2}\circ\alpha_{1})}=\hat{\alpha}_{2}\circ\hat{\alpha}_{1},\quad\qquad\widetilde{(\alpha_{2}\circ\alpha_{1})}=\tilde{\alpha}_{2}\circ\tilde{\alpha}_{1}.$

Explicitly, this means that we have the equality of exact functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{3})$

$\overline{\phi_{2}\circ\phi_{1}}=\overline{\phi_{2}}\circ\overline{\phi_{1}}$

and the equality of natural isomorphisms of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}_{3}$

$\widetilde{\kappa_{2}\star\kappa_{1}}=\tilde{\kappa}_{2}\star\tilde{\kappa}_{1}.$

Before moving further, let us make an easy observation about conservativity on
universal abelian factorizations:

######  Lemma 2.8.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian representation, and let
$I$ be a (possibly infinite) index set. Suppose that we are given, for every
$i\in I$, an abelian representation
$(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ and an exact $1$-morphism
$\alpha_{i}=(\phi_{i},\Phi_{i};\kappa_{i}):(\mathscr{D},\mathscr{A};\beta)\rightarrow(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$.
Then the family of exact functors

(12)
$\left\\{\overline{\phi}_{i}:\mathbf{A}(\beta)\rightarrow\mathbf{A}(\beta_{i})\right\\}_{i\in
I}$

obtained via Proposition 2.5(1) is conservative as soon as the family of exact
functors

(13) $\left\\{\Phi_{i}:\mathscr{A}\rightarrow\mathscr{A}_{i}\right\\}$

is conservative.

###### Proof.

Recall that an exact functor between abelian categories is conservative if and
only if it is faithful. In our setting, for each $i\in I$ the functor
$\iota_{i}:\mathbf{A}(\beta_{i})\hookrightarrow\mathscr{A}_{i}$ is faithful
and exact be construction, and thus conservative. Therefore the family (12) is
conservative if and only if the family

$\left\\{\iota_{i}\circ\overline{\phi}_{i}:\mathbf{A}(\beta)\rightarrow\mathscr{A}_{i}\right\\}_{i\in
I}$

is conservative. Since the functors $\iota_{i}\circ\overline{\phi}_{i}$ and
$\Phi_{i}\circ\iota$ are naturally isomorphic for each $i\in I$, this holds if
and only if the family

$\left\\{\Phi_{i}\circ\iota:\mathbf{A}(\beta)\rightarrow\mathscr{A}_{i}\right\\}_{i\in
I}$

is conservative. Since $\iota:\mathbf{A}(\beta)\hookrightarrow\mathscr{A}$ is
conservative, this is the case as soon as the family (13) is conservative. ∎

###  2.2. Recognition principle

It is convenient to introduce a more flexible variant of Proposition 2.5 in
which the strict equality (9) is allowed to be a more general natural
isomorphism. This result turns out to be extremely useful in concrete
applications, where it serves as a recognition principle for exact functors
between universal abelian factorizations.

######  Corollary 2.9.

For $i=1,2$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an abelian
representation. Suppose that we are given an exact $1$-morphism of abelian
representations

$\alpha:(\phi,\Phi;\kappa):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2}).$

Moreover, suppose that we are given

  * •

an exact functor $\psi:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$,

  * •

a natural isomorphism of functors
$\mathscr{D}_{1}\rightarrow\mathbf{A}(\beta_{2})$

$\chi:\psi\circ\pi_{1}\xrightarrow{\sim}\pi_{2}\circ\phi,$

  * •

a natural isomorphism of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}_{2}$

$\xi:\Phi\circ\iota_{1}\xrightarrow{\sim}\iota_{2}\circ\psi$

satisfying the equality of $1$-morphisms of representations
$(\mathscr{D}_{1},\mathscr{D}_{2};\phi)\rightarrow(\mathscr{A}_{1},\mathscr{A}_{2};\Phi)$

$(\iota_{1},\iota_{2};\xi)\circ(\pi_{1},\pi_{2};\chi)=(\beta_{1},\beta_{2};\kappa).$

Then there exists a unique natural isomorphism of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

$\tilde{\chi}:\psi\xrightarrow{\sim}\overline{\phi}$

making the diagram of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}_{2}$

(14)
${\iota_{2}\circ\psi}$${\iota_{2}\circ\overline{\phi}}$${\Phi\circ\iota_{1}}$$\scriptstyle{\tilde{\chi}}$$\scriptstyle{\xi}$$\scriptstyle{\tilde{\kappa}}$

commute.

###### Proof.

Applying Proposition 2.5(2) to the $1$-morphism of abelian representations

$(\phi,\psi;\chi):(\mathscr{D}_{1},\mathbf{A}(\beta_{1});\pi_{1})\rightarrow(\mathscr{D}_{2},\mathbf{A}(\beta_{2});\pi_{2}),$

we obtain a canonical natural isomorphism of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

$\tilde{\chi}:\psi\circ\nu_{1}\xrightarrow{\sim}\nu_{2}\circ\bar{\phi}$

making the diagram of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

${\Phi\circ\pi_{1}}$${\pi_{2}\circ\phi}$${\Phi\circ\nu_{1}\circ\rho_{1}}$${\nu_{2}\circ\overline{\phi}\circ\rho_{1}}$${\nu_{2}\circ\rho_{2}\circ\phi}$$\scriptstyle{\kappa}$$\scriptstyle{\tilde{\kappa}}$

commute. By 1.13, the two faithful exact functors
$\nu_{1}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\pi_{1})$ and
$\nu_{2}:\mathbf{A}(\beta_{2})\rightarrow\mathbf{A}(\pi_{2})$ are in fact
isomorphisms of categories; we can use these isomorphisms to identify
$\mathbf{A}(\pi_{1})$ and $\mathbf{A}(\pi_{2})$ with $\mathbf{A}(\beta_{1})$
and $\mathbf{A}(\beta_{2})$, respectively. Thus we can interpret
$\tilde{\chi}$ simply as a natural isomorphism of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

$\tilde{\chi}:\psi\xrightarrow{\sim}\overline{\phi}$

making the diagram of functors
$\mathscr{D}_{1}\rightarrow\mathbf{A}(\beta_{2})$

(15)
${\psi\circ\pi_{1}}$${\overline{\phi}\circ\pi_{1}}$${\pi_{2}\circ\phi}$$\scriptstyle{\tilde{\chi}}$$\scriptstyle{\chi}$

commute. Let us show that the diagram (14) is commutative. Since all the
functors appearing in the diagram are exact functors on
$\mathbf{A}(\beta_{1})$, and the latter is generated under (co)kernels by the
image of $\pi_{1}:\mathscr{D}_{1}\rightarrow\mathbf{A}(\beta_{1})$, it
suffices to show that the diagram of functors
$\mathscr{D}_{1}\rightarrow\mathscr{A}_{2}$ obtained by restricting (14) along
$\pi_{1}$

${\iota_{2}\circ\psi\circ\pi_{1}}$${\iota_{2}\circ\overline{\phi}\circ\pi_{1}}$${\Phi\circ\iota_{1}\circ\pi_{1}}$$\scriptstyle{\tilde{\chi}}$$\scriptstyle{\xi}$$\scriptstyle{\tilde{\kappa}}$

is commutative. But the latter coincides with the outer part of the diagram

${\iota_{2}\circ\psi\circ\pi_{1}}$${\iota_{2}\circ\overline{\phi}\circ\pi_{1}}$${\iota_{2}\circ\pi_{2}\circ\phi}$${\beta_{2}^{+}\circ\phi}$${\Phi\circ\beta_{1}^{+}}$${\Phi\circ\iota_{1}\circ\pi_{1}}$$\scriptstyle{\tilde{\chi}}$$\scriptstyle{\xi}$$\scriptstyle{\tilde{\kappa}}$$\scriptstyle{\chi}$$\scriptstyle{\kappa}$

where the upper piece is the commutative diagram (15) above, the left-most
piece is commutative by hypothesis, and the right-most piece is commutative by
Proposition 2.5(2). This concludes the proof. ∎

###  2.3. Compatibility with diagram categories

We conclude this section by deriving a few remarks about the behavior of
universal abelian factorizations under categorical exponentiation; as a
particular case, we deduce that universal abelian factorizations are
canonically compatible with finite direct products.

######  Notation 2.10.

For every category $\mathscr{C}$ and every small category $\mathscr{L}$, we
let $\mathscr{C}^{\mathscr{L}}$ denote the functor category
$\mathbf{Fun}(\mathscr{L},\mathscr{C})$.

The basic observation is that, given an additive category $\mathscr{D}$ and a
small category $\mathscr{L}$, the two abelian categories
$\mathbf{A}(\mathscr{D}^{\mathscr{L}})$ and
$\mathbf{A}(\mathscr{D})^{\mathscr{L}}$ are related by a canonical exact
functor

$\omega_{\mathscr{L}}:\mathbf{A}(\mathscr{D}^{\mathscr{L}})\rightarrow\mathbf{A}(\mathscr{D})^{\mathscr{L}}$

making the diagram

${\mathscr{D}^{\mathscr{L}}}$${\mathbf{A}(\mathscr{D}^{\mathscr{L}})}$${\mathbf{A}(\mathscr{D})^{\mathscr{L}}}$$\scriptstyle{\omega_{\mathscr{L}}}$

commutative: both the existence and the uniqueness of $\omega_{\mathscr{L}}$
are guaranteed by Theorem 1.2(2)(i). In the next result, we extend this
picture to the case of arbitrary abelian representations of $\mathscr{D}$.

######  Lemma 2.11.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian representation. Given a
small category $\mathscr{L}$, let

$\beta^{\mathscr{L}}:\mathscr{D}^{\mathscr{L}}\rightarrow\mathscr{A}^{\mathscr{L}}$

denote the additive functor induced by post-composition with $\beta$. Consider
the two exact functors

$\pi^{(\mathscr{L})}:\mathbf{A}(\mathscr{D}^{\mathscr{L}})\rightarrow\mathbf{A}(\beta^{\mathscr{L}}),\quad\iota^{\mathscr{L}}:\mathbf{A}(\beta^{\mathscr{L}})\hookrightarrow\mathscr{A}^{\mathscr{L}}$

associated to the universal abelian factorization of the representation
$(\mathscr{D}^{\mathscr{L}},\mathscr{A}^{\mathscr{L}};\beta^{\mathscr{L}})$.
Then the following statements hold:

  1. (1)

There exists a unique faithful exact functor

$\omega_{\mathscr{L}}^{\beta}:\mathbf{A}(\beta^{\mathscr{L}})\rightarrow\mathbf{A}(\beta)^{\mathscr{L}}$

making the diagram

(16)
${\mathbf{A}(\mathscr{D}^{\mathscr{L}})}$${\mathbf{A}(\mathscr{D})^{\mathscr{L}}}$${\mathbf{A}(\beta^{\mathscr{L}})}$${\mathbf{A}(\beta)^{\mathscr{L}}}$${\mathscr{A}^{\mathscr{L}}}$$\scriptstyle{\pi^{(\mathscr{L})}}$$\scriptstyle{\omega_{\mathscr{L}}}$$\scriptstyle{\pi^{\mathscr{L}}}$$\scriptstyle{\iota^{(\mathscr{L})}}$$\scriptstyle{\omega_{\mathscr{L}}^{\beta}}$$\scriptstyle{\iota^{\mathscr{L}}}$

commute.

  2. (2)

For every object $l\in\mathscr{L}$, consider the two functors

$l_{\mathscr{D}}^{*}:\mathscr{D}^{\mathscr{L}}\rightarrow\mathscr{D},\quad
l_{\mathscr{A}}^{*}:\mathscr{A}^{\mathscr{I}}\rightarrow\mathscr{A}$

induced by evaluation at $l$. Then:

    1. (i)

There exists a unique exact functor

$\overline{l_{\mathscr{D}}^{*}}:\mathbf{A}(\beta^{\mathscr{L}})\rightarrow\mathbf{A}(\beta)$

satisfying the equality of functors
$\mathscr{D}^{\mathscr{L}}\rightarrow\mathbf{A}(\beta)$

(17) $\pi\circ
l_{\mathscr{D}}^{*}=\overline{l_{\mathscr{D}}^{*}}\circ\pi^{(\mathscr{L})}$

as well as the equality of functors
$\mathbf{A}(\beta^{\mathscr{I}})\rightarrow\mathscr{A}$

$\iota\circ\overline{l_{\mathscr{D}}^{*}}=l_{\mathscr{A}}^{*}\circ\iota^{(\mathscr{L})}.$

    2. (ii)

We have the equality

$\overline{l_{\mathscr{D}}^{*}}=l_{\mathbf{A}(\beta)}^{*}\circ\omega_{\mathscr{L}}^{\beta}.$

where
$l_{\mathbf{A}(\beta)}^{*}:\mathbf{A}(\beta)^{\mathscr{L}}\rightarrow\mathbf{A}(\beta)$
denotes the analogous functor induced by evaluation at $l$.

###### Proof.

  1. (1)

This follows applying Proposition 1.10 to the obvious commutative triangle

${\mathscr{D}^{\mathscr{L}}}$${\mathbf{A}(\beta)^{\mathscr{L}}}$${\mathscr{A}^{\mathscr{L}}.}$

  2. (2)

Statement (i) follows applying Proposition 2.5 to the exact morphism of
abelian representations
$(l_{\mathscr{D}}^{*},l_{\mathscr{A}}^{*};\textup{id}):(\mathscr{D}^{\mathscr{L}},\mathscr{A}^{\mathscr{L}};\beta^{\mathscr{L}})\rightarrow(\mathscr{D},\mathscr{A};\beta)$,
while (ii) follows from the fact that the functor
$\overline{l_{\mathscr{D}}^{*}}$ is uniquely characterized by the equation
(17).

∎

######  Corollary 2.12.

For $i=1,\dots,n$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an
abelian representation, and consider the product representation

$\beta_{1}\times\dots\times\beta_{n}:\mathscr{D}_{1}\times\dots\times\mathscr{D}_{n}\rightarrow\mathscr{A}_{1}\times\dots\times\mathscr{A}_{n},\quad(D_{1},\dots,D_{n})\mapsto(\beta_{1}(D_{1}),\dots,\beta_{n}(D_{n})).$

Then we have a canonical equivalence of abelian categories

$\mathbf{A}(\beta_{1}\times\dots\times\beta_{n})=\mathbf{A}(\beta_{1})\times\dots\times\mathbf{A}(\beta_{n}).$

###### Proof.

It suffices to prove the result for $n=2$; the general case will then follow
by induction. In the case $n=2$, applying Proposition 2.5 to the obvious
morphisms of representations

${\mathscr{D}_{1}\times\mathscr{D}_{2}}$${\mathscr{D}_{1}}$${\mathscr{A}_{1}\times\mathscr{A}_{2}}$${\mathscr{A}_{1}}$$\scriptstyle{pr_{1}}$$\scriptstyle{\beta_{1}\times\beta_{2}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{pr_{1}}$
${\mathscr{D}_{1}\times\mathscr{D}_{2}}$${\mathscr{D}_{2}}$${\mathscr{A}_{1}\times\mathscr{A}_{2}}$${\mathscr{A}_{2}}$$\scriptstyle{pr_{2}}$$\scriptstyle{\beta_{1}\times\beta_{2}}$$\scriptstyle{\beta_{2}}$$\scriptstyle{pr_{2}}$

we get canonical exact functors
$\mathbf{A}(\beta_{1}\times\beta_{2})\rightarrow\mathbf{A}(\beta_{i})$,
$i=1,2$. Together, they yield an exact functor

(18)
$\mathbf{A}(\beta_{1}\times\beta_{2})\rightarrow\mathbf{A}(\beta_{1})\times\mathbf{A}(\beta_{2})$

fitting into a commutative diagram of the form

${\mathbf{A}(\beta_{1}\times\beta_{2})}$${\mathbf{A}(\beta_{1})\times\mathbf{A}(\beta_{2})}$${\mathscr{A}_{1}\times\mathscr{A}_{2}}$$\scriptstyle{\iota_{1,2}}$$\scriptstyle{\iota_{1}\times\iota_{2}}$

Conversely, applying Proposition 2.5 to the obvious morphisms of
representations

${\mathscr{D}_{1}}$${\mathscr{D}_{1}\times\mathscr{D}_{2}}$${\mathscr{A}_{1}}$${\mathscr{A}_{1}\times\mathscr{A}_{2}}$$\scriptstyle{(\textup{id},0)}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\beta_{1}\times\beta_{2}}$$\scriptstyle{(\textup{id},0)}$
${\mathscr{D}_{2}}$${\mathscr{D}_{1}\times\mathscr{D}_{2}}$${\mathscr{A}_{2}}$${\mathscr{A}_{1}\times\mathscr{A}_{2}}$$\scriptstyle{(0,\textup{id})}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\beta_{1}\times\beta_{2}}$$\scriptstyle{(0,\textup{id})}$

we get canonical exact functors
$\mathbf{A}(\beta_{i})\rightarrow\mathbf{A}(\beta_{1}\times\beta_{2})$,
$i=1,2$. Together, they yield an exact functor

(19)
$\mathbf{A}(\beta_{1})\times\mathbf{A}(\beta_{2})\rightarrow\mathbf{A}(\beta_{1}\times\beta_{2})$

fitting into a commutative diagram of the form

${\mathbf{A}(\beta_{1})\times\mathbf{A}(\beta_{2})}$${\mathbf{A}(\beta_{1}\times\beta_{2})}$${\mathscr{A}_{1}\times\mathscr{A}_{2}.}$$\scriptstyle{\iota_{1}\times\iota_{2}}$$\scriptstyle{\iota_{1,2}}$

Regarding both $\mathbf{A}(\beta_{1}\times\beta_{2})$ and
$\mathbf{A}(\beta_{1})\times\mathbf{A}(\beta_{2})$ as (not necessarily full)
abelian subcategories of $\mathscr{A}_{1}\times\mathscr{A}_{2}$, we deduce
that the two functors (18) and (19) are canonically mutually quasi-inverse
equivalences. ∎

##  3\. Lifting natural transformations

In this section we explain how to define natural transformations between exact
functors on universal abelian factorizations. Again, the main result below is
essentially a reformulation of [Ivo17, Prop. 6.4] to the setting of [BVP18]
and a similar result appears in [IM19, § 1]. After discussing the general
result, we specialize this discussion to the cases of adjunctions and
equivalences, which is inspired by [IM19, § 2.7].

###  3.1. Main result

Following the approach of the previous sections, we formalize our main
definitions using a natural notion of $2$-morphism of (additive)
representations:

######  Definition 3.1.

For $i=1,2$ let $(\mathscr{C}_{i},\mathscr{E}_{i};R_{i})$ be a representation;
moreover, let $\alpha=(F,G;\kappa)$ and
$\alpha^{\prime}=(F^{\prime},G^{\prime};\kappa^{\prime})$ be two morphisms of
representations
$(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$.
We call $2$-morphism from $\alpha$ to $\alpha^{\prime}$ a pair
$(\lambda,\Lambda)$ consisting of

  * •

a natural transformation of functors
$\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}$

$\lambda:F(C_{1})\rightarrow F^{\prime}(C_{1}),$

  * •

a natural transformation of functors
$\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$

$\Lambda:G(E_{1})\rightarrow G^{\prime}(E_{1})$

such that the diagram of functors $\mathscr{C}_{1}\rightarrow\mathscr{E}_{2}$

${GR_{1}(C_{1})}$${G^{\prime}R_{1}(C_{1})}$${R_{2}F(C_{1})}$${R_{2}F^{\prime}(C_{1})}$$\scriptstyle{\Lambda}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa^{\prime}}$$\scriptstyle{\lambda}$

commutes. We write it as $(\lambda,\Lambda):\alpha\rightarrow\alpha^{\prime}$,
and we often depict it as a diagram of the form

${\mathscr{C}_{1}}$${\mathscr{C}_{2}}$${\mathscr{E}_{1}}$${\mathscr{E}_{2}}$${\Downarrow}$${\mathscr{C}_{1}}$${\mathscr{C}_{2}}$${\mathscr{E}_{1}}$${\mathscr{E}_{2}}$$\scriptstyle{F}$$\scriptstyle{R_{1}}$$\scriptstyle{R_{2}}$$\scriptstyle{G}$$\scriptstyle{F^{\prime}}$$\scriptstyle{R_{1}}$$\scriptstyle{R_{2}}$$\scriptstyle{G^{\prime}}$

where, for notational simplicity, we do not write the natural transformations
$\lambda$ and $\Lambda$ explicitly.

######  Remark 3.2.

Note that giving two (additive) functors
$F,F^{\prime}:\mathscr{C}_{1}\rightarrow\mathscr{E}_{1}$ together with a
natural transformation $\lambda:F\rightarrow F^{\prime}$ is the same a giving
a single (additive) functor

$(F\xrightarrow{\lambda}F^{\prime}):\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}^{\mathscr{I}},\quad
C_{1}\rightsquigarrow(F(C_{1}),F^{\prime}(C_{1});\lambda(C_{1})),$

where $\mathscr{I}$ denotes the category with two objects and exactly one non-
identity arrow between them. Similarly, giving two (additive) functors
$G,G^{\prime}:\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$ together with a
natural transformation $\Lambda:G\rightarrow G^{\prime}$ is the same as giving
a single (additive) functor

$(G\xrightarrow{\Lambda}G^{\prime}):\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}^{\mathscr{I}},\quad
E_{1}\rightsquigarrow(G(E_{1}),G^{\prime}(E_{1});\Lambda(E_{1})).$

Therefore we obtain two additive representations

$(G\circ R_{1}\xrightarrow{\Lambda}G^{\prime}\circ
R_{1}):\mathscr{C}_{1}\xrightarrow{R_{1}}\mathscr{E}_{1}\xrightarrow{(G\xrightarrow{\Lambda}G^{\prime})}\mathscr{E}_{2}^{\mathscr{I}},\qquad(R_{2}\circ
F\xrightarrow{\lambda}R_{2}\circ
F^{\prime}):\mathscr{C}_{1}\xrightarrow{(F\xrightarrow{\lambda}F^{\prime})}\mathscr{C}_{2}^{\mathscr{I}}\xrightarrow{R_{2}}\mathscr{E}_{2}^{\mathscr{I}}.$

Moreover, if the categories $\mathscr{E}_{1}$ and $\mathscr{E}_{2}$ are
abelian, then the functor $(G\xrightarrow{\Lambda}G^{\prime})$ is exact if and
only if $G$ and $G^{\prime}$ are both exact. By construction, saying that the
pair $(\lambda,\Lambda)$ defines a $2$-morphism
$\alpha\rightarrow\alpha^{\prime}$ is equivalent to saying that the two
natural isomorphisms

$\kappa:GR_{1}(C_{1})\xrightarrow{\sim}R_{2}F(C_{1}),\qquad\kappa^{\prime}:G^{\prime}R_{1}(C_{1})\xrightarrow{\sim}R_{2}F^{\prime}(C_{1})$

define a natural isomorphism of functors
$\mathscr{C}_{1}\rightarrow\mathscr{E}_{2}^{\mathscr{I}}$

$(\kappa,\kappa^{\prime}):(G\circ R_{1}\xrightarrow{\Lambda}G^{\prime}\circ
R_{1})\rightarrow(R_{2}\circ F\xrightarrow{\lambda}R_{2}\circ F^{\prime}).$

In this way, we obtain a $1$-morphism of representation

$(\textup{id}_{\mathscr{C}_{1}},\textup{id}_{\mathscr{E}_{2}^{\mathscr{I}}};(\kappa,\kappa^{\prime})):(\mathscr{C}_{1},\mathscr{E}_{2}^{\mathscr{I}};(G\circ
R_{1}\xrightarrow{\Lambda}G^{\prime}\circ
R_{1}))\rightarrow(\mathscr{C}_{1},\mathscr{E}_{2}^{\mathscr{I}};(R_{2}\circ
F\xrightarrow{\lambda}R_{2}\circ F^{\prime})).$

######  Remark 3.3.

There exists a natural notion of composition for $2$-morphisms of
representations: given three $1$-morphisms
$\alpha_{1}=(F_{1},G_{1};\kappa_{1})$, $\alpha_{2}=(F_{2},G_{2};\kappa_{2})$
and $\alpha_{3}=(F_{3},G_{3};\kappa_{3})$ between representations
$(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$
as well as two $2$-morphisms of representations
$(\lambda_{1},\Lambda_{1}):\alpha_{1}\rightarrow\alpha_{2}$ and
$(\lambda_{2},\Lambda_{2}):\alpha_{2}\rightarrow\alpha_{3}$, we define the
composite $2$-morphism of representations as the triple

$(\lambda_{2}\circ\lambda_{1},\Lambda_{2}\circ\Lambda_{1}):\alpha_{1}\rightarrow\alpha_{3},$

where $\lambda_{2}\circ\lambda_{1}$ and $\Lambda_{2}\circ\Lambda_{1}$ denote
the usual composite natural transformations between functors
$\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}$ and
$\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$, respectively.

Here is the main result of this section:

######  Proposition 3.4.

For $i=1,2$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an abelian
representation; in addition, let $\alpha=(\phi,\Phi;\kappa)$ and
$\alpha^{\prime}=(\phi^{\prime},\Phi^{\prime};\kappa^{\prime})$ be two exact
$1$-morphisms of abelian representations
$(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})$.
Suppose that we are given a $2$-morphism
$(\lambda,\Lambda):\alpha\rightarrow\alpha^{\prime}$. Then there exists a
unique natural transformation of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

$\overline{\lambda}:\overline{\phi}(M)\rightarrow\overline{\phi^{\prime}}(M)$

such that the pair $(\lambda,\overline{\lambda})$ defines a $2$-morphism
$\hat{\alpha}\rightarrow\hat{\alpha^{\prime}}$ and the pair
$(\overline{\lambda},\Lambda)$ defines a $2$-morphism
$\tilde{\alpha}\rightarrow\tilde{\alpha^{\prime}}$.

###### Proof.

Since the abelian categories $\mathbf{A}(\beta_{1})$ is generated by the image
of $\mathscr{D}_{1}$ under (co)kernels and the functors
$\overline{\phi},\overline{\phi^{\prime}}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$
are exact, a natural transformation $\overline{\lambda}$ as in the statement
is uniquely determined by the property that $(\lambda,\overline{\lambda})$
defines a $2$-morphism, provided it exists.

In order to show the existence of $\overline{\lambda}$, we let $\mathscr{I}$
denote the category with two objects $1$ and $2$ and exactly one non-identity
arrow $1\rightarrow 2$. Following the notation of 2.11, consider the abelian
representation
$(\mathscr{D}_{2}^{\mathscr{I}},\mathscr{A}_{2}^{\mathscr{I}};\beta_{2}^{\mathscr{I}})$
and its universal abelian factorization $\mathbf{A}(\beta^{\mathscr{I}})$.
Applying Proposition 2.5 to the exact morphism of abelian representations

$((\phi\xrightarrow{\lambda}\phi^{\prime}),(\Phi\xrightarrow{\Lambda}\Phi^{\prime});(\kappa,\kappa^{\prime})):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2}^{\mathscr{I}},\mathscr{A}_{2}^{\mathscr{I}};\beta_{2}^{\mathscr{I}})$

obtained as in Remark 3.2, we get a canonical exact functor

$\overline{(\phi\xrightarrow{\lambda}\phi^{\prime})}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2}^{\mathscr{I}})$

satisfying the equality of diagram representations
$\mathscr{D}_{1}\rightarrow\mathbf{A}(\beta_{2}^{\mathscr{I}})$

$\pi_{2}^{(\mathscr{I})}\circ(\phi\xrightarrow{\lambda}\phi^{\prime})=\overline{(\phi\xrightarrow{\lambda}\phi^{\prime})}\circ\pi_{1},$

together with a canonical natural isomorphism of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}^{\mathscr{I}}$

$\widetilde{(\kappa,\kappa^{\prime})}:(\Phi\xrightarrow{\Lambda}\Phi^{\prime})\circ\iota_{1}\xrightarrow{\sim}\iota_{2}^{(\mathscr{I})}\circ\overline{(\phi\xrightarrow{\lambda}\phi^{\prime})}$

which is uniquely characterized by the commutativity of the diagram of
functors $\mathscr{D}_{1}\rightarrow\mathscr{A}_{2}^{\mathscr{I}}$

${(\Phi\xrightarrow{\Lambda}\Phi^{\prime})\circ\beta_{1}}$${\beta_{2}^{\mathscr{I}}\circ(\phi\xrightarrow{\lambda}\phi^{\prime})}$${(\Phi\xrightarrow{\Lambda}\Phi^{\prime})\circ\iota_{1}\circ\pi_{1}}$${\iota_{2}^{(\mathscr{I})}\circ\overline{(\phi\xrightarrow{\lambda}\phi^{\prime})}\circ\iota_{1}}$${\iota_{2}^{(\mathscr{I})}\circ\pi_{2}^{(\mathscr{I})}\circ(\phi\xrightarrow{\lambda}\phi^{\prime}).}$$\scriptstyle{(\kappa,\kappa^{\prime})}$$\scriptstyle{\widetilde{(\kappa,\kappa^{\prime})}}$

Note that, as a consequence of 2.11(2), the two composite exact functors

$\mathbf{A}(\beta_{1})\xrightarrow{\overline{(\phi\xrightarrow{\lambda}\phi^{\prime})}}\mathbf{A}(\beta_{2}^{\mathscr{I}})\xrightarrow{\overline{1^{*}_{\mathscr{D}_{2}}}}\mathbf{A}(\beta_{2}),\qquad\mathbf{A}(\beta_{1})\xrightarrow{\overline{(\phi\xrightarrow{\lambda}\phi^{\prime})}}\mathbf{A}(\beta_{2}^{\mathscr{I}})\xrightarrow{\overline{2^{*}_{\mathscr{D}_{2}}}}\mathbf{A}(\beta_{2})$

coincide with $\overline{\phi}$ and $\overline{\phi^{\prime}}$, respectively.
Hence we can define the sought-after natural transformation
$\overline{\lambda}$ be declaring that its value on a given object
$M_{1}\in\mathbf{A}(\beta_{1})$ is determined by the equation

$(\omega_{\mathscr{I}}^{\beta_{2}}\circ\overline{(\phi\xrightarrow{\lambda}\phi^{\prime})})(M_{1})=(\overline{\phi}(M_{1}),\overline{\phi^{\prime}}(M_{1});\overline{\lambda}(M_{1})).$

It is clear by construction that the diagram of functors
$\mathscr{D}_{1}\rightarrow\mathbf{A}(\beta_{2})$

${\overline{\phi}\circ\pi_{1}}$${\overline{\phi^{\prime}}\circ\pi_{1}}$${\pi_{2}\circ\phi}$${\pi_{2}\circ\phi}$$\scriptstyle{\overline{\lambda}}$$\scriptstyle{\lambda}$

is commutative. Hence the pair $(\lambda,\overline{\lambda})$ defines a
$2$-morphism of representations. Moreover, as a consequence of the uniqueness
property of the natural transformation $\widetilde{(\kappa,\kappa^{\prime})}$,
we must have the equality

$\widetilde{(\kappa,\kappa^{\prime})}=(\tilde{\kappa},\tilde{\kappa^{\prime}}).$

By Remark 3.2, saying that the pair $(\tilde{\kappa},\tilde{\kappa^{\prime}})$
defines a natural transformation of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}^{\mathscr{I}}$ is equivalent to
saying that the pair $(\overline{\lambda},\Lambda)$ defines a $2$-morphism of
representations. ∎

######  Remark 3.5.

  1. (1)

Note that, while the existence of the natural transformation
$\overline{\lambda}:\overline{\phi}\rightarrow\overline{\phi^{\prime}}$
follows from the existence of $\Lambda$ (and from its compatibility with
$\lambda$), the actual expression of $\overline{\lambda}$ only depends on
$\lambda$.

  2. (2)

As a consequence of the uniqueness in the statement of Proposition 3.4, we see
that the assignment $(\lambda,\Lambda)\mapsto\bar{\lambda}$ is compatible with
composition of $2$-morphisms of representations (see Remark 3.3) in the
following sense: Given, for $i=1,2,3$, a morphism of abelian representations
$\alpha_{i}=(\phi_{i},\Phi_{i};\theta_{i}):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})$
and given, for $i=1,2$, a $2$-morphism
$(\lambda_{i},\Lambda_{i}):\alpha_{i}\rightarrow\alpha_{i+1}$, we have the
equality of natural transformations of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}(\beta_{2})$

$\overline{\lambda_{2}\circ\lambda_{1}}=\overline{\lambda_{2}}\circ\overline{\lambda_{1}}.$

  3. (3)

As a consequence of the previous two points, we deduce that the natural
transformation
$\overline{\lambda}:\overline{\phi}\rightarrow\overline{\phi^{\prime}}$ is
invertible as soon as $\lambda:\phi\rightarrow\phi^{\prime}$ is invertible.

###  3.2. The cases of adjunctions and equivalences

The results collected so far can be applied to lifting adjunctions between
exact functors; this method has been successfully used in the proof of [IM19,
§ 2.7], and the proof of our general result below is not harder than the
latter.

In order to state our result correctly, we need to use the notion of
adjunction in its precise form. Thus we say that an adjunction

$F_{1}:\mathscr{C}_{1}\rightleftarrows\mathscr{C}_{2}:F_{2}$

consists of two functors $F_{1}:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}$ and
$F_{2}:\mathscr{C}_{2}\rightarrow\mathscr{C}_{1}$ together with two natural
transformations

$\eta:\textup{id}_{\mathscr{C}_{2}}\rightarrow F_{1}\circ
F_{2},\quad\epsilon:F_{2}\circ F_{1}\rightarrow\textup{id}_{\mathscr{C}_{1}}$

called the unit and the co-unit of the adjunction, respectively, such that the
triangular identities are satisfied.

######  Definition 3.6.

Let $R_{1}:\mathscr{C}_{1}\rightarrow\mathscr{E}_{1}$ and
$R_{2}:\mathscr{C}_{2}\rightarrow\mathscr{E}_{2}$ be two functors. Suppose
that we are given two adjunctions

$F_{1}:\mathscr{C}_{1}\rightleftarrows\mathscr{C}_{2}:F_{2},\quad
G_{1}:\mathscr{E}_{1}\rightleftarrows\mathscr{E}_{2}:G_{2}$

fitting into morphism of representations

$\alpha_{1}=(F_{1},G_{1};\kappa_{1}):(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2}),\quad\alpha_{2}=(F_{2},G_{2};\kappa_{2}^{-1}):(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})\rightarrow(\mathscr{C}_{1},\mathscr{E}_{1};R_{1}).$

We say that the pair $(\alpha_{1},\alpha_{2})$ is adjoint if the unit
transformations

$\eta:\textup{id}_{\mathscr{C}_{1}}\rightarrow F_{2}\circ
F_{1},\qquad\eta:\textup{id}_{\mathscr{E}_{1}}\rightarrow G_{2}\circ G_{1}$

define a $2$-morphism of representations
$(\eta,\eta):(\textup{id}_{\mathscr{C}_{1}},\textup{id}_{\mathscr{E}_{1}};\textup{id})\rightarrow(F_{2}\circ
F_{1},G_{2}\circ G_{1},\kappa_{2}^{-1}\star\kappa_{1})$ and the co-unit
transformations

$\epsilon:F_{1}\circ
F_{2}\rightarrow\textup{id}_{\mathscr{C}_{2}},\qquad\epsilon:G_{1}\circ
G_{2}\rightarrow\textup{id}_{\mathscr{E}_{2}}$

define a $2$-morphism of representations $(\epsilon,\epsilon):(F_{1}\circ
F_{2},G_{1}\circ
G_{2};\kappa_{1}\star\kappa_{2}^{-1})\rightarrow(\textup{id}_{\mathscr{C}_{2}},\textup{id}_{\mathscr{E}_{2}};\textup{id})$.

######  Remark 3.7.

More explicitly, two morphisms $\alpha_{1}$ and $\alpha_{2}$ as in Definition
3.6 are adjoint if and only if the diagram of functors
$\mathscr{C}_{1}\rightarrow\mathscr{E}_{1}$

${R_{1}}$${R_{1}\circ F_{2}\circ F_{1}}$${G_{2}\circ G_{1}\circ
R_{1}}$${G_{2}\circ R_{2}\circ
F_{1}}$$\scriptstyle{\eta}$$\scriptstyle{\eta}$$\scriptstyle{\kappa_{2}}$$\scriptstyle{\kappa_{1}}$

and the diagram of functors $\mathscr{C}_{2}\rightarrow\mathscr{E}_{2}$

${G_{1}\circ R_{1}\circ F_{2}}$${R_{2}\circ F_{1}\circ F_{2}}$${G_{1}\circ
G_{2}\circ
R_{2}}$${R_{2}}$$\scriptstyle{\kappa_{2}}$$\scriptstyle{\kappa_{1}}$$\scriptstyle{\epsilon}$$\scriptstyle{\epsilon}$

are commutative. One can check that requiring the latter condition is
equivalent to requiring that the natural isomorphism $\kappa_{2}$ equal the
composite

$R_{1}F_{2}\xrightarrow{\eta}G_{2}G_{1}\circ
R_{1}F_{2}\xrightarrow{\kappa_{1}}G_{2}R_{2}\circ
F_{1}F_{2}\xrightarrow{\epsilon}G_{2}R_{2}$

or, equivalently, that the natural isomorphism $\kappa_{1}$ equal the
composite

$G_{1}R_{1}\xrightarrow{\eta}G_{1}R_{1}\circ
F_{2}F_{1}\xrightarrow{\kappa_{2}}G_{1}G_{2}\circ
R_{2}F_{1}\xrightarrow{\epsilon}R_{2}F_{1}.$

The following result asserts that compatible adjunctions between abelian
representations canonically induce adjunctions between the corresponding
universal abelian factorizations:

######  Corollary 3.8.

For $i=1,2$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an abelian
representation. Suppose that we are given two exact morphisms of abelian
representations

$\alpha_{1}=(\phi_{1},\Phi_{1};\kappa_{1}):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2}),\quad\alpha_{2}=(\phi_{2},\Phi_{2};\kappa_{2}^{-1}):(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})\rightarrow(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})$

such that the pair $(\alpha_{1},\alpha_{2})$ is adjoint in the sense of
Definition 3.6. Then the exact functors $\bar{\phi}_{1}$ and $\bar{\phi}_{2}$
fit in a unique way into an adjunction

(20)
$\overline{\phi_{1}}:\mathbf{A}(\beta_{1})\rightleftarrows\mathbf{A}(\beta_{2}):\overline{\phi_{2}}$

such that the pairs $(\hat{\alpha}_{1},\hat{\alpha}_{2})$ and
$(\tilde{\alpha}_{1},\tilde{\alpha}_{2})$ are adjoint as well.

###### Proof.

Applying Proposition 3.4, and taking into account Remark 2.7(2), we obtain a
natural transformation of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{1})$

$\overline{\eta}:\textup{id}_{\mathbf{A}(\beta_{1})}\rightarrow\overline{\phi_{2}\circ\phi_{1}}=\overline{\phi_{2}}\circ\overline{\phi_{1}}$

defining $2$-morphisms of abelian representations

$(\eta,\overline{\eta}):(\textup{id}_{\mathscr{D}_{1}},\textup{id}_{\mathbf{A}(\mathscr{D}_{1})};\textup{id})\rightarrow(\phi_{2}\circ\phi_{1},\overline{\phi_{2}}\circ\overline{\phi_{1}};\textup{id}),\qquad(\overline{\eta},\eta):(\textup{id}_{\mathscr{A}(\mathscr{D}_{1})},\textup{id}_{\mathscr{A}_{1}};\tilde{\kappa}_{2}\star\tilde{\kappa}_{1}).$

In the same way, we obtain a natural transformation of functors
$\mathbf{A}(\beta_{2})\rightarrow\mathbf{A}(\beta_{2})$

$\overline{\epsilon}:\overline{\phi_{1}}\circ\overline{\phi_{2}}=\overline{\phi_{1}\circ\phi_{2}}\rightarrow\textup{id}_{\mathbf{A}(\beta_{2})}$

defining $2$-morphisms of abelian representations

$(\epsilon,\overline{\epsilon}):(\phi_{1}\circ\phi_{2},\overline{\phi_{1}}\circ\overline{\phi_{2}};\textup{id}),\qquad(\overline{\epsilon},\epsilon):(\overline{\phi_{1}}\circ\overline{\phi_{2}},\Phi_{1}\circ\Phi_{2};\tilde{\kappa}_{1}\star\tilde{\kappa}_{2}^{-1}).$

As a consequence of the uniqueness part of Proposition 3.4, and taking into
account Remark 3.5(2), we see that the composite natural transformation of
functors $\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

$\overline{\phi_{1}}\xrightarrow{\overline{\eta}}\overline{\phi_{1}}\circ\overline{\phi_{2}}\circ\overline{\phi_{1}}\xrightarrow{\overline{\epsilon}}\overline{\phi_{1}}$

as well as the composite natural transformation of functors
$\mathbf{A}(\beta_{2})\rightarrow\mathbf{A}(\beta_{1})$

$\overline{\phi_{2}}\xrightarrow{\overline{\eta}}\overline{\phi_{2}}\circ\overline{\phi_{1}}\circ\overline{\phi_{2}}\xrightarrow{\overline{\epsilon}}\overline{\phi_{2}}$

are the identity natural transformations. Thus $\overline{\eta}$ and
$\overline{\epsilon}$ satisfy the triangular identities, and therefore define
the adjunction (20). ∎

In the same spirit, the previous results can be exploited to deduce
equivalences between universal abelian factorizations. By definition, giving
an equivalence between two categories $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$
means giving two functors

$F_{1}:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2},\qquad
F_{2}:\mathscr{C}_{2}\rightarrow\mathscr{C}_{1}$

and two natural isomorphisms

$\eta:\textup{id}_{\mathscr{C}_{1}}\xrightarrow{\sim}F_{2}\circ
F_{1},\qquad\epsilon:F_{1}\circ
F_{2}\xrightarrow{\sim}\textup{id}_{\mathscr{C}_{2}}.$

Using the Yoneda Lemma, one can see that this is exactly the same as giving an
adjunction

$F_{1}:\mathscr{C}_{1}\leftrightarrows\mathscr{C}_{2}:F_{2}$

where both $F_{1}$ and $F_{2}$ are fully faithful: indeed, the fully
faithfulness of $F_{1}$ is equivalent to the invertibility of the unit, while
the fully faithfulness of $F_{2}$ is equivalent to the invertibility of the
co-unit. In the following result, it is convenient to view equivalences in
this way.

######  Corollary 3.9.

Keep the notation and assumptions of Corollary 3.8, and suppose in addition
that the adjunction

$\phi_{1}:\mathscr{D}_{1}\leftrightarrows\mathscr{D}_{2}:\phi_{2}$

is an equivalence. Then the adjunction

$\overline{\phi_{1}}:\mathbf{A}(\beta_{1})\leftrightarrows\mathbf{A}(\beta_{2}):\overline{\phi_{2}}$

is an equivalence as well.

###### Proof.

We have to show that the two natural transformations

$\overline{\eta}:\textup{id}_{\mathbf{A}(\beta_{1})}\rightarrow\overline{\phi_{2}\circ\phi_{1}}=\overline{\phi_{2}}\circ\overline{\phi_{1}},\qquad\overline{\epsilon}:\overline{\phi_{1}}\circ\overline{\phi_{2}}=\overline{\phi_{1}\circ\phi_{2}}\rightarrow\textup{id}_{\mathbf{A}(\beta_{2})}$

constructed in the proof of Corollary 3.8 are invertible. Taking into account
Remark 3.5(3), this follows from the hypothesis that the two natural
transformations

$\eta:\textup{id}_{\mathscr{D}_{1}}\rightarrow\phi_{2}\circ\phi_{1},\qquad\epsilon:\phi_{1}\circ\phi_{2}\rightarrow\textup{id}_{\mathscr{D}_{2}}$

are invertible. ∎

###  3.3. Compatibility with diagram categories

We conclude by discussing the compatibility of Proposition 3.4 with families
of abelian representations indexed by some small category $\mathscr{L}$; as a
particular application, we show that natural exact sequences lift to natural
exact sequences.

For notational convenience, we use the following definition:

######  Definition 3.10.

Let $\mathscr{L}$ be a (small) category.

  1. (1)

We call $\mathscr{L}$-family of $1$-morphisms of representations from
$(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})$ to
$(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$ the datum of

     * •

for every $l\in\mathscr{L}$, a morphism of representations
$\alpha_{l}=(F_{l},G_{l};\kappa_{l}):(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$

     * •

for every arrow $\gamma:l\rightarrow l^{\prime}$ in $\mathscr{L}$, a
$2$-morphism
$(\lambda_{\gamma},\Lambda_{\gamma}):\alpha_{l}\rightarrow\alpha_{l^{\prime}}$

such that the assignments $\gamma\mapsto\lambda_{\gamma}$ and
$\gamma\mapsto\Lambda_{\gamma}$ are compatible with identities and composition
in the obvious sense. We write it as
$\alpha:\left\\{\alpha_{l}:(F_{l},G_{l};\kappa_{l})\right\\}_{l\in\mathscr{L}}$,
leaving the natural transformations $\lambda_{\gamma}$ and $\Lambda_{\gamma}$
implicit.

  2. (2)

In case the representations $(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})$ and
$(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$ are abelian, we say that an
$\mathscr{L}$-family of morphisms of representations
$\alpha:(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$
is exact of all the individual
$\alpha_{l}:\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$ for $l\in\mathscr{L}$
are exact in the sense of Definition 2.1(1)(b).

The lifting result of Proposition 2.5 is compatible with families of
$1$-morphisms in the following sense:

######  Lemma 3.11.

Let $(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})$ and
$(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})$ be two abelian representations.
Let $\mathscr{L}$ be a (small) category, and suppose that we are given an
$\mathscr{L}$-family of exact morphisms of abelian representations

$\left\\{\alpha_{l}=(\phi_{l},\Phi_{l};\kappa_{l}):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})\right\\}_{l\in\mathscr{L}}.$

Then the two collections of exact morphisms of abelian representations

$\left\\{\hat{\alpha}_{l}=(\phi_{l},\overline{\phi_{l}};\textup{id}):(\mathscr{D}_{1},\mathbf{A}(\beta_{1});\pi_{1})\rightarrow(\mathscr{D}_{2},\mathbf{A}(\beta_{2});\pi_{2})\right\\}_{l\in\mathscr{L}}$

and

$\left\\{\tilde{\alpha}_{l}=(\overline{\phi_{l}},\Phi_{l};\tilde{\kappa_{l}}):(\mathbf{A}(\beta_{1}),\mathscr{A}_{1};\iota_{1})\rightarrow(\mathbf{A}(\beta_{2}),\mathscr{A}_{2};\iota_{2})\right\\}_{l\in\mathscr{L}}$

canonically assemble into $\mathscr{L}$-families of exact morphisms of abelian
representations.

###### Proof.

For every fixed object $C\in\mathscr{C}$, we get by restriction a morphism of
abelian representations
$\alpha_{C}:=(\phi_{C},\Phi_{C};\kappa_{C}):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})$.
Applying Proposition 2.5, we obtain an exact functor

$\overline{\phi_{C}}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2}),$

uniquely determined by the equality of additive functors
$\mathscr{D}_{1}\rightarrow\mathbf{A}(\beta_{2})$

$\pi_{2}\circ\phi_{C}=\overline{\phi_{C}}\circ\pi_{1},$

as well as a natural isomorphism of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}_{2}$

$\tilde{\kappa}_{C}:\Phi_{C}\circ\iota_{1}\xrightarrow{\sim}\iota_{2}\circ\overline{\phi}_{C},$

uniquely determined by the commutativity of the diagram of functors
$\mathscr{D}_{1}\rightarrow\mathscr{A}$

${\Phi_{C}\circ\beta_{1}}$${\beta_{2}\circ\phi_{C}}$${\Phi_{C}\circ\iota_{1}\circ\pi_{1}}$${\iota\circ\overline{\phi_{C}}\circ\pi_{1}}$${\iota_{2}\circ\pi_{2}\circ\phi_{C}.}$$\scriptstyle{\kappa_{C}}$$\scriptstyle{\tilde{\kappa}_{C}}$

Moreover, for every morphism $\gamma:C\rightarrow C^{\prime}$ in
$\mathscr{C}$, the natural transformation of additive functors
$\mathscr{D}_{1}\rightarrow\mathscr{D}_{2}$

$\lambda_{\gamma}:\phi_{C}\rightarrow\phi_{C^{\prime}}$

and the natural transformation of exact functors
$\mathscr{A}_{1}\rightarrow\mathscr{A}_{2}$

$\Lambda_{\gamma}:\Phi_{C}\rightarrow\Phi_{C^{\prime}}$

define a $2$-morphism
$(\lambda_{\gamma},\Lambda_{\gamma}):\alpha_{C}\rightarrow\alpha_{C^{\prime}}$.
Applying Proposition 3.4, we obtain a natural transformation of exact functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

$\overline{\lambda_{\gamma}}:\overline{\phi_{C}}\rightarrow\overline{\phi_{C^{\prime}}},$

uniquely determined by the property that the pair
$(\lambda_{\gamma},\overline{\lambda_{\gamma}})$ defines a $2$-morphism
$\hat{\alpha}_{C}\rightarrow\hat{\alpha}_{C^{\prime}}$ and the pair
$(\overline{\lambda_{\gamma}},\Lambda_{\gamma})$ defines a $2$-morphism
$\tilde{\alpha}_{C}\rightarrow\tilde{\alpha}_{C^{\prime}}$. ∎

As a particular case, one can consider $\mathbb{Z}$-indexed families of
$1$-morphisms, for instance complexes of $1$-morphisms. We check that exact
complexes of $1$-morphisms of representations give rise to exact complexes
between universal abelian factorizations:

######  Corollary 3.12.

Let $(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})$ and
$(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})$ be two abelian representations.
Suppose that we are given, for every $n\in\mathbb{Z}$, an exact morphism of
abelian representations

$\alpha_{n}=(\phi_{n},\Phi_{n};\kappa_{n}):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2}).$

Suppose in addition that we are given, for every $n\in\mathbb{Z}$, a
$2$-morphism $(\lambda_{n},\Lambda_{n}):\alpha_{n}\rightarrow\alpha_{n+1}$,
such that the sequence of functors $\mathscr{A}_{1}\rightarrow\mathscr{A}_{2}$

$\dots\xrightarrow{\Lambda_{n-2}}\Phi_{n-1}\xrightarrow{\Lambda_{n-1}}\Phi_{n}\xrightarrow{\Lambda_{n}}\Phi_{n+1}\xrightarrow{\Lambda_{n+1}}\dots$

is exact. Then the sequence of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$

$\dots\xrightarrow{\overline{\lambda_{n-2}}}\overline{\phi_{n-1}}\xrightarrow{\overline{\lambda_{n-1}}}\overline{\phi_{n}}\xrightarrow{\overline{\lambda_{n}}}\overline{\phi_{n+1}}\xrightarrow{\overline{\lambda_{n+1}}}\dots$

is exact as well.

###### Proof.

Since the functor
$\iota_{2}:\mathbf{A}(\beta_{2})\hookrightarrow\mathscr{A}_{2}$ is exact and
faithful, it suffices to show that the sequence of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}_{2}$

$\dots\xrightarrow{\overline{\lambda_{n-2}}}\iota_{2}\circ\overline{\phi_{n-1}}\xrightarrow{\overline{\lambda_{n-1}}}\iota_{2}\circ\overline{\phi_{n}}\xrightarrow{\overline{\lambda_{n}}}\iota_{2}\circ\overline{\phi_{n+1}}\xrightarrow{\overline{\lambda_{n+1}}}\dots$

is exact. But the latter is naturally isomorphic to the sequence of functors

$\dots\xrightarrow{\Lambda_{n-2}}\Phi_{n-1}\circ\iota_{1}\xrightarrow{\Lambda_{n-1}}\Phi_{n}\circ\iota_{1}\xrightarrow{\Lambda_{n}}\Phi_{n+1}\circ\iota_{1}\xrightarrow{\Lambda_{n+1}}\dots,$

which is exact as an immediate consequence of the hypothesis. ∎

##  4\. Extension to the multi-linear setting

In this section, we explain how the universal property of universal abelian
factorizations can be used to define multi-exact functors and natural
transformations of such. The results below generalize those of Section 2 and
Section 3. In order to prove them, we need to combine the main results of the
previous sections, mostly via 3.11.

The problem of lifting multi-exact functors to the level of universal abelian
representations in a systematic way was addressed in [BVHP20]. The starting
point is [BVHP20, Prop. 1.4], which gives a canonical method to extend multi-
linear functors from additive categories to their abelian hulls and shows that
the resulting extension is right-exact in each variable. As pointed out in
[BVHP20, Ex. 1.12], the extension is not left-exact in each variable in
general; however, as explained in [BVHP20, Prop. 1.13], one can still use this
method to construct multi-linear functors between universal abelian
factorizations associated to certain representations, at least under suitable
hypotheses.

Here we describe a slightly different construction that is tailored to our
applications. The main difference with [BVHP20] is that we bypass the level of
abelian hulls and get directly a multi-exact functor on universal abelian
factorizations. Note that our construction works only if the given multi-
linear functor on the underlying abelian categories is multi-exact, while the
result of [BVHP20, Prop. 1.13] works under weaker assumptions (see [BVHP20,
Defn. 1.9]) and can be applied, for instance, to Nori motives with integral
coefficients (see [BVHP20, Thm. 2.20, Cor. 4.4]). In all those cases where
both methods apply, the multi-exact functors obtained via the two distinct
approaches necessarily coincide, since both of them satisfy the same
uniqueness property.

###  4.1. Lifting multi-exact functors

Given three categories $\mathscr{C},\mathscr{C}_{2}$ and $\mathscr{C}$, a
functor $F:\mathscr{C}_{1}\times\mathscr{C}_{2}\rightarrow\mathscr{C}$ can be
regarded in two equivalent ways:

  * –

as a $\mathscr{C}_{1}$-family of functors
$\left\\{F(C_{1},-):\mathscr{C}_{2}\rightarrow\mathscr{C}\right\\}_{C_{1}\in\mathscr{C}_{1}}$;

  * –

as a $\mathscr{C}_{2}$-family of functors
$\left\\{F(-,C_{2}):\mathscr{C}_{1}\rightarrow\mathscr{C}\right\\}_{C_{2}\in\mathscr{C}_{2}}$.

This observation generalizes to functors from arbitrary finite products of
categories. In the course of this section we systematically adopt this point
of view, and we often switch the roles of the two variables.

######  Definition 4.1.

  1. (1)

Let $\mathscr{C}_{1},\dots,\mathscr{C}_{n}$ and $\mathscr{C}$ be additive
categories. A functor

$F:\mathscr{C}_{1}\times\dots\times\mathscr{C}_{n}\rightarrow\mathscr{C}$

is multi-additive if it is additive separately with respect to each variable.

  2. (2)

Let $\mathscr{A}_{1},\dots,\mathscr{A}_{n}$ and $\mathscr{A}$ be abelian
categories. A multi-additive functor

$F:\mathscr{A}_{1}\times\dots\times\mathscr{A}_{n}\rightarrow\mathscr{A}$

is multi-exact if it is exact separately with respect to each variable.

######  Definition 4.2.

For each $i=1,\dots,n$ let $(\mathscr{C}_{i},\mathscr{E}_{i};R_{i})$ be an
additive representation; let $(\mathscr{C},\mathscr{E};R)$ be a further
additive representation.

  1. (1)

We call multi-linear $1$-morphism of representations from
$(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\times\dots\times(\mathscr{C}_{n},\mathscr{E}_{n};R_{n})$
to $(\mathscr{C},\mathscr{E};R)$ a triple $\alpha=(F,G;\kappa)$ consisting of
two multi-additive functors

$F:\mathscr{C}_{1}\times\dots\times\mathscr{C}_{n}\rightarrow\mathscr{C},\qquad
G:\mathscr{E}_{1}\times\dots\times\mathscr{E}_{n}\rightarrow\mathscr{E}$

and a natural isomorphism of functors
$\mathscr{C}_{1}\times\dots\times\mathscr{C}_{n}\rightarrow\mathscr{E}$

$\kappa:G\circ(R_{1},\dots,R_{n})\xrightarrow{\sim}R\circ F.$

  2. (2)

In case the representations $(\mathscr{C}_{i},\mathscr{E}_{i};R_{i})$,
$i=1,\dots,n$ and $(\mathscr{C},\mathscr{E};R)$ are all abelian, we say that a
$1$-morphism

$\alpha=(F,G;\kappa):(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})\times\dots\times(\mathscr{C}_{n},\mathscr{E}_{n};R_{n})\rightarrow(\mathscr{C},\mathscr{E};R)$

is multi-exact if the multi-additive functor
$G:\mathscr{E}_{1}\times\dots\times\mathscr{E}_{n}\rightarrow\mathscr{E}$ is
multi-exact.

Here is the multi-linear extension of Proposition 2.5:

######  Proposition 4.3.

For each $i=1,\dots,n$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an
abelian representation, and let $(\mathscr{D},\mathscr{A};\beta)$ be a further
abelian representation. Suppose that we are given a multi-exact $1$-morphism
of abelian representations
$\alpha=(\phi,\Phi;\kappa):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\times\dots\times(\mathscr{D}_{n},\mathscr{A}_{n};\beta_{n})\rightarrow(\mathscr{D},\mathscr{A};\beta)$.
Then:

  1. (1)

There exists a unique multi-exact functor
$\overline{\phi}:\mathbf{A}(\beta_{1})\times\dots\times\mathbf{A}(\beta_{n})\rightarrow\mathbf{A}(\beta)$
satisfying the equality of multi-additive functors
$\mathscr{D}_{1}\times\dots\times\mathscr{D}_{n}\rightarrow\mathbf{A}(\beta)$

(21) $\pi\circ\phi=\overline{\phi}\circ(\pi_{1},\dots,\pi_{n}).$

  2. (2)

There exists a unique natural isomorphism of functors
$\mathbf{A}(\beta_{1})\times\dots\mathbf{A}(\beta_{n})\rightarrow\mathscr{A}$

$\tilde{\kappa}:\Phi\circ(\iota_{1},\dots,\iota_{n})\xrightarrow{\sim}\iota\circ\overline{\phi}$

making the diagram of functors
$\mathscr{D}_{1}\times\dots\times\mathscr{D}_{n}\rightarrow\mathscr{A}$

(22)
${\Phi\circ(\beta_{1},\dots,\beta_{n})}$${\beta\circ\phi}$${\Phi\circ(\iota_{1},\dots,\iota_{n})\circ(\pi_{1},\dots,\pi_{n})}$${\iota\circ\overline{\phi}\circ(\pi_{1},\dots,\pi_{n})}$${\iota\circ\pi\circ\phi}$$\scriptstyle{\kappa}$$\scriptstyle{\tilde{\kappa}}$

commute.

###### Proof.

Since, for each $i=1,\dots,n$, the abelian category $\mathbf{A}(\beta_{i})$ is
generated under (co)kernels by the image of $\mathscr{D}_{i}$, a multi-exact
functor $\bar{\phi}$ as in (1) is uniquely determined by the equality (21),
provided it exists. For the same reason, a natural isomorphism
$\tilde{\kappa}$ as in (2) is uniquely determined by the commutativity of the
diagram (22), provided it exists. For simplicity, we only show the existence
of $\overline{\phi}$ and $\tilde{\kappa}$ in the case $n=2$; the general case
is just a notationally more intricate variant.

In the case $n=2$, regard the bi-additive functor
$\phi:\mathscr{D}_{1}\times\mathscr{D}_{2}\rightarrow\mathscr{D}$ as a
$\mathscr{D}_{2}$-family of additive functors

$\left\\{\phi_{D_{2}}:\mathscr{D}_{1}\rightarrow\mathscr{D},\;D_{1}\rightsquigarrow\phi(D_{1},D_{2})\right\\}_{D_{2}\in\mathscr{D}_{2}}.$

Similarly, regard the bi-exact functor
$\Phi:\mathscr{A}_{1}\times\mathscr{A}_{2}\rightarrow\mathscr{A}$ as an
$\mathscr{A}_{2}$-family of exact functors

$\left\\{\Phi_{A_{2}}:\mathscr{A}_{1}\rightarrow\mathscr{A},\;A_{1}\rightsquigarrow\Phi(A_{1},A_{2})\right\\}_{A_{2}\in\mathscr{A}_{2}},$

and, by restriction, as a $\mathscr{D}_{2}$-family of exact functors

$\left\\{\Phi_{\beta_{2}(D_{2})}:\mathscr{A}_{1}\rightarrow\mathscr{A},\;A_{1}\rightsquigarrow\Phi(A_{1},\beta_{2}(D_{2}))\right\\}_{D_{2}\in\mathscr{D}_{2}}.$

Together, they give rise to a $\mathscr{D}_{2}$-family of exact $1$-morphisms
of abelian representations

$\left\\{\alpha_{D_{2}}=(\phi_{D_{2}},\Phi_{\beta_{2}(D_{2})};\kappa_{D_{2}}):(\mathscr{D}_{1},\mathscr{A}_{1}\beta_{1})\rightarrow(\mathscr{D},\mathscr{A};\beta)\right\\}_{D_{2}\in\mathscr{D}_{2}}.$

Applying 3.11, we obtain a $\mathscr{D}_{2}$-family of exact functors

(23)
$\left\\{\bar{\phi}_{D_{2}}:\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta)\right\\}_{D_{2}\in\mathscr{D}_{2}}$

giving rise to a $\mathscr{D}_{2}$-family of exact morphisms of abelian
representations

(24)
$\left\\{\tilde{\alpha}_{D_{2}}=(\bar{\phi}_{D_{2}},\Phi_{\beta_{2}(D_{2})},\tilde{\kappa}_{D_{2}}):(\mathbf{A}(\beta_{1}),\mathscr{A}_{1};\iota_{1})\rightarrow(\mathbf{A}(\beta),\mathscr{A};\iota)\right\\}_{D_{2}\in\mathscr{D}_{2}}.$

We now switch the roles of the two variables, regarding (23) as an
$\mathbf{A}(\beta_{1})$-family of additive functors

$\left\\{\bar{\phi}_{M_{1}}:\mathscr{D}_{2}\rightarrow\mathbf{A}(\beta),\;D_{2}\rightsquigarrow\bar{\phi}_{D_{2}}(M_{1})\right\\}_{M_{1}\in\mathbf{A}(\beta_{1})}$

and (24) as an $\mathbf{A}(\beta_{1})$-family of morphisms of abelian
representations

$\left\\{\tilde{\alpha}_{M_{1}}:(\bar{\phi}_{M_{1}},\Phi_{\iota_{1}(M_{1})};\tilde{\kappa}_{M_{1}}):(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2})\rightarrow(\mathbf{A}(\beta),\mathscr{A};\iota)\right\\}_{M_{1}\in\mathbf{A}(\beta_{1})}.$

Applying 3.11 again, we obtain an $\mathbf{A}(\beta_{1})$-family of exact
functors

(25)
$\left\\{\bar{\phi}_{M_{1}}:\mathbf{A}(\beta_{2})\rightarrow\mathbf{A}(\beta)\right\\}_{M_{1}\in\mathbf{A}(\beta_{1})}$

giving rise to an $\mathbf{A}(\beta_{1})$-family of exact morphisms of abelian
representations (written in the same way, for notational convenience)

(26)
$\left\\{\tilde{\alpha}_{M_{1}}:(\bar{\phi}_{M_{1}},\Phi_{\iota_{1}(M_{1})};\tilde{\kappa}_{M_{1}}):(\mathbf{A}(\beta_{2}),\mathscr{A}_{2};\iota_{2})\rightarrow(\mathbf{A}(\beta),\mathscr{A};\iota)\right\\}_{M_{1}\in\mathbf{A}(\beta_{1})}.$

We can finally regard (25) as a bi-additive functor

$\overline{\phi}:\mathbf{A}(\beta_{1})\times\mathbf{A}(\beta_{2})\rightarrow\mathbf{A}(\beta)$

which is exact with respect to the second variable and, by construction,
satisfies the equality (21). The natural isomorphisms $\tilde{\kappa}_{M_{1}}$
in (26) assemble into a natural isomorphism of functors
$\mathbf{A}(\beta_{1})\times\mathbf{A}(\beta_{2})\rightarrow\mathscr{A}$

$\tilde{\kappa}:\Phi\circ(\iota_{1},\iota_{2})\xrightarrow{\sim}\iota\circ\overline{\phi}$

which, by construction, makes the diagram (22) commutative (with respect to
the bi-additive functor $\overline{\phi}$ just obtained). Since the underlying
bi-additive functor
$\Phi:\mathscr{A}_{1}\times\mathscr{A}_{2}\rightarrow\mathscr{A}$ is bi-exact,
using the commutativity of the diagram (22) together with the fact that
$\iota:\mathbf{A}(\beta)\rightarrow\mathscr{A}$ is faithful and exact we
deduce that the functor $\overline{\phi}$ is bi-exact as well. This concludes
the proof. ∎

######  Remark 4.4.

The analogue of Remark 2.7 applies, namely:

  1. (1)

Note that, while the existence of $\overline{\phi}$ follows from the existence
of the exact functor $\Phi$ (and of the natural isomorphism $\kappa$), the
actual expression of $\overline{\phi}$ only depends on $\phi$.

  2. (2)

As a consequence of the uniqueness in the two statements of Proposition 2.5,
we see that the assignments $\alpha\mapsto\hat{\alpha}$ and
$\alpha\mapsto\tilde{\alpha}$ are both compatible with composition of multi-
exact $1$-morphisms of representations in the expected way.

It is convenient to extend the flexible variant of Proposition 2.5 given by
Corollary 2.9 to the case of multi-linear representations.

######  Corollary 4.5.

For each $i=1,\dots,n$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an
abelian representation; let $(\mathscr{D},\mathscr{A};\beta)$ be a further
abelian representation. Suppose that we are given a multi-exact $1$-morphism
of abelian representations

$\alpha:=(\phi,\Phi;\kappa):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\times\dots\times(\mathscr{D}_{n},\mathscr{A}_{n};\beta_{n})\rightarrow(\mathscr{D},\mathscr{A};\beta).$

In addition, suppose that we are given

  * •

a multi-exact functor
$\psi:\mathbf{A}(\beta_{1})\times\dots\times\mathbf{A}(\beta_{n})\rightarrow\mathbf{A}(\beta)$,

  * •

a natural isomorphism of functors
$\mathscr{D}_{1}\times\dots\times\mathscr{D}_{n}\rightarrow\mathbf{A}(\beta)$

$\chi:\psi\circ(\pi_{1},\dots,\pi_{n})\xrightarrow{\sim}\pi\circ\phi,$

  * •

a natural isomorphism of functors
$\mathbf{A}(\beta_{1})\times\dots\times\mathbf{A}(\beta_{n})\rightarrow\mathscr{A}$

$\xi:\Phi\circ(\iota_{1},\dots,\iota_{n})\xrightarrow{\sim}\iota\circ\psi$

satisfying the equality of $1$-morphisms between multi-linear representations

$(\iota_{1}\times\dots\times\iota_{n},\iota;\xi)\circ(\pi_{1}\times\dots\times\pi_{n},\pi;\xi)=(\beta_{1}\times\dots\times\beta_{n},\beta;\kappa).$

Then there exists a unique natural isomorphism of functors
$\mathbf{A}(\beta_{1})\times\dots\mathbf{A}(\beta_{n})\rightarrow\mathbf{A}(\beta)$

$\tilde{\chi}:\overline{\phi}\xrightarrow{\sim}\psi$

making the diagram of multi-linear representations
$\mathbf{A}(\beta_{1})\times\dots\times\mathbf{A}(\beta_{n})\rightarrow\mathscr{A}$

${\iota\circ\psi}$${\iota\circ\overline{\phi}}$${\Phi\circ(\iota_{1}\times\dots\times\iota_{n})}$$\scriptstyle{\xi}$$\scriptstyle{\tilde{\chi}}$$\scriptstyle{\tilde{\kappa}}$

commute.

###### Proof.

This follows from Proposition 4.3 in the same way as Corollary 2.9 follows
from Proposition 2.5. We leave the details to the interested reader. ∎

###  4.2. Lifting multi-linear natural transformations

Here is the multi-linear extension of Proposition 3.4:

######  Proposition 4.6.

For each $i=1,\dots,n$ let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an
abelian representation, and let $(\mathscr{D},\mathscr{A};\beta)$ be a further
abelian representation; in addition, let $\alpha=(\phi,\Phi;\kappa)$ and
$\alpha^{\prime}=(\phi^{\prime},\Phi^{\prime};\kappa^{\prime})$ be two multi-
exact $1$-morphisms of abelian representations
$(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\times\dots\times(\mathscr{D}_{n},\mathscr{A}_{n};\beta_{n})\rightarrow(\mathscr{D},\mathscr{A};\beta)$.
Suppose that we are given a $2$-morphism
$(\lambda,\Lambda):\alpha\rightarrow\alpha^{\prime}$. Then there exists a
unique natural transformation of functors
$\mathbf{A}(\beta_{1})\times\dots\times\mathbf{A}(\beta_{n})\rightarrow\mathbf{A}(\beta)$

$\overline{\lambda}:\overline{\phi}(M_{1},\dots,M_{n})\rightarrow\overline{\phi^{\prime}}(M_{1},\dots,M_{n})$

such that the pair $(\lambda,\overline{\lambda})$ defines a $2$-morphism
$\hat{\alpha}\rightarrow\hat{\alpha^{\prime}}$ and the pair
$(\overline{\lambda},\Lambda)$ defines a $2$-morphism
$\tilde{\alpha}\rightarrow\tilde{\alpha^{\prime}}$.

###### Proof.

Since, for each $i=1,\dots,n$, the abelian category $\mathbf{A}(\beta_{i})$ is
generated under (co)kernels by the image of $\mathscr{D}_{i}$, and since the
functors $\overline{\phi}$ and $\overline{\phi^{\prime}}$ are multi-exact by
construction, a natural transformation $\overline{\lambda}$ as in the
statement is uniquely determined by the property that the pair
$(\lambda,\overline{\lambda})$ defines a $2$-morphism, provided it exists. In
order to show the existence of $\overline{\lambda}$, it suffices to apply
Proposition 4.3 to the induced multi-exact morphism of abelian representations

$((\phi\xrightarrow{\lambda}\phi^{\prime}),(\Phi\xrightarrow{\Lambda}\Phi^{\prime});(\kappa,\kappa^{\prime})):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\times\dots\times(\mathscr{D}_{n},\mathscr{A}_{n};\beta_{n})\rightarrow(\mathscr{D}^{\mathscr{I}},\mathscr{A}^{\mathscr{I}};\beta^{\mathscr{I}})$

and repeat the argument in the proof of Proposition 3.4. We leave the details
to the interested reader. ∎

######  Remark 4.7.

The analogue of Remark 3.5 applies, namely:

  1. (1)

While the existence of the natural transformation
$\overline{\lambda}:\overline{\phi}\rightarrow\overline{\phi^{\prime}}$
follows from the existence of $\Lambda$ (and from its compatibility with
$\lambda$), the actual expression of $\overline{\lambda}$ only depends on
$\lambda$.

  2. (2)

As a consequence of the uniqueness in the statement of Proposition 4.6, we see
that the assignment $(\lambda,\Lambda)\mapsto\bar{\lambda}$ is compatible with
composition of $2$-morphisms in the following sense: Given, for $i=1,2,3$, a
multi-exact morphism of abelian representations

$\alpha_{i}=(\phi_{i},\Phi_{i};\theta_{i}):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\times\dots\times(\mathscr{D}_{n},\mathscr{A}_{n};\beta_{n})\rightarrow(\mathscr{D},\mathscr{A};\beta)$

and given, for $i=1,2$, a $2$-morphism
$(\lambda_{i},\Lambda_{i}):\alpha_{i}\rightarrow\alpha_{i+1}$, we have the
equality of natural transformations of functors
$\mathbf{A}(\beta_{1})\rightarrow\mathscr{A}(\beta_{2})$

$\overline{\lambda_{2}\circ\lambda_{1}}=\overline{\lambda_{2}}\circ\overline{\lambda_{1}}.$

  3. (3)

As a consequence of the previous two points, we deduce that the natural
transformation
$\overline{\lambda}:\overline{\phi}\rightarrow\overline{\phi^{\prime}}$ is
invertible as soon as $\lambda:\phi\rightarrow\phi^{\prime}$ is invertible.

##  5\. Lifting abelian fibered categories

In this section, we apply the previous results to the setting of abelian
fibered categories. In the first place, we show how the results of Section 2
and Section 3 allow one to lift fibered categories to the level of universal
abelian factorizations in a natural way. We then derive some useful
applications to cofinality properties of universal abelian factorizations
indexed by a cofiltered category.

###  5.1. Main result

Throughout this section, we let $\mathscr{S}$ be a fixed small category. The
notion of fibered category over $\mathscr{S}$ that it is natural to consider
here corresponds to the notion of "catégorie clivée normalisée" in the
original reference [SGA1, Exp. VI, Defn. 7.1]. In detail:

######  Definition 5.1.

  1. (1)

An (additive) $\mathscr{S}$-fibered category $\mathscr{A}$ is the datum of

     * •

for every $S\in\mathscr{S}$, an additive category $\mathscr{A}(S)$,

     * •

for every morphism $f:T\rightarrow S$ in $\mathscr{S}$, an additive functor

$f^{*}:\mathscr{A}(S)\rightarrow\mathscr{A}(T),$

called the inverse image functor along $f$,

     * •

for every pair of composable morphisms $f:T\rightarrow S$ and $g:S\rightarrow
V$ in $\mathscr{S}$, a natural isomorphism of functors
$\mathscr{A}(V)\rightarrow\mathscr{A}(T)$

$\mathsf{conn}=\mathsf{conn}_{f,g}:(gf)^{*}A\xrightarrow{\sim}f^{*}g^{*}A$

called the connection isomorphism at $(f,g)$

such that the following conditions are satisfied:

    1. ($\mathscr{S}$-fib-0)

For every $S\in\mathscr{S}$, we have
$\textup{id}_{S}^{*}=\textup{id}_{\mathbb{H}(S)}$.

    2. ($\mathscr{S}$-fib-1)

For every triple of composable morphisms $f:T\rightarrow S$, $g:S\rightarrow
V$ and $h:V\rightarrow W$ in $\mathscr{S}$, the diagram of functors
$\mathscr{A}(W)\rightarrow\mathscr{A}(T)$

${(hgf)^{*}A}$${f^{*}(hg)^{*}A}$${(gf)^{*}h^{*}A}$${f^{*}g^{*}h^{*}A}$$\scriptstyle{\mathsf{conn}_{f,hg}}$$\scriptstyle{\mathsf{conn}_{gf,h}}$$\scriptstyle{\mathsf{conn}_{g,h}}$$\scriptstyle{\mathsf{conn}_{f,g}}$

is commutative.

  2. (2)

An additive $\mathscr{S}$-fibered category $\mathscr{A}$ as above is called
abelian if all additive categories $\mathscr{A}(S)$ are abelian and all
inverse image functors $f^{*}$ are exact.

######  Remark 5.2.

One could avoid to impose axiom ($\mathscr{S}$-fib-0) in such a strict form.
However, it is well-known that strict unitality of fibered categories can
always be enforced; see for example [Del01, Lemma 2.5].

In order to explain how to lift abelian $\mathscr{S}$-fibered categories, we
need to recall the notion of morphism between abelian $\mathscr{S}$-fibered
categories in the following form:

######  Definition 5.3.

Let $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ be two (additive)
$\mathscr{S}$-fibered categories.

  1. (1)

A $1$-morphism of (additive) $\mathscr{S}$-fibered categories
$R:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}$ is the datum of

     * •

for every $S\in\mathscr{S}$, an additive functor
$R_{S}:\mathscr{C}_{1}(S)\rightarrow\mathscr{C}_{2}(S)$,

     * •

for every morphism $f:T\rightarrow S$ in $\mathscr{S}$, a natural isomorphism
of functors $\mathscr{C}_{1}(S)\rightarrow\mathscr{C}_{2}(T)$

$\theta=\theta_{f}:f^{*}R_{S}(A)\xrightarrow{\sim}R_{T}(f^{*}A),$

called the $R$-transition isomorphism along $f$

such that the following condition is satisfied:

    1. (mor-$\mathscr{S}$-fib)

For every pair of composable morphisms $f:T\rightarrow S$ and $g:S\rightarrow
V$ in $\mathscr{S}$, the diagram of functors
$\mathscr{C}_{1}(V)\rightarrow\mathscr{C}_{2}(T)$

${(gf)^{*}R_{V}(A)}$${R_{T}((gf)^{*}A)}$${f^{*}g^{*}R_{V}(A)}$${f^{*}R_{S}(g^{*}A)}$${R_{T}(f^{*}g^{*}A)}$$\scriptstyle{\theta_{gf}}$$\scriptstyle{\mathsf{conn}^{(2)}_{f,g}}$$\scriptstyle{\mathsf{conn}^{(1)}_{f,g}}$$\scriptstyle{\theta_{g}}$$\scriptstyle{\theta_{f}}$

is commutative.

In case the $\mathscr{S}$-fibered categories $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$ are abelian, we say that a morphism of (additive)
$\mathscr{S}$-fibered categories $R:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}$
is exact if all functors $R_{S}$ are exact.

  2. (2)

Let $R_{1},R_{2}:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}$ be two morphisms
of (additive) $\mathscr{S}$-fibered categories. A $2$-morphism of (additive)
$\mathscr{S}$-fibered categories $\gamma:R_{1}\rightarrow R_{2}$ is the datum
of

     * •

for every $S\in\mathscr{S}$, a natural transformation of functors
$\mathscr{C}_{1}(S)\rightarrow\mathscr{C}_{2}(S)$

$\gamma_{S}:R_{1}(C)\rightarrow R_{2}(C)$

satisfying the following condition:

    1. ($2$-mor-$\mathscr{S}$-fib)

For every arrow $f:T\rightarrow S$ in $\mathscr{S}$, the diagram of functors
$\mathscr{C}_{1}(S)\rightarrow\mathscr{C}_{2}(T)$

${f^{*}R_{1,S}(C)}$${f^{*}R_{2,S}(C)}$${R_{1,T}(f^{*}C)}$${R_{2,T}(f^{*}C)}$$\scriptstyle{\gamma_{S}}$$\scriptstyle{\theta_{f}^{(1)}}$$\scriptstyle{\theta_{f}^{(2)}}$$\scriptstyle{\gamma_{T}}$

is commutative.

A $2$-morphism of $\mathscr{S}$-fibered categories $\gamma$ is called a
$2$-isomorphism if all natural transformations $\gamma_{S}$ are invertible.

######  Remark 5.4.

There exists an obvious notion of composition between morphisms of
$\mathscr{S}$-fibered categories. In detail, consider three (additive)
$\mathscr{S}$-fibered categories $\mathscr{C}_{1}$, $\mathscr{C}_{2}$ and
$\mathscr{C}_{3}$ and two morphisms
$R_{1}:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2}$ and
$R_{2}:\mathscr{C}_{2}\rightarrow\mathscr{C}_{3}$; for every morphism
$f:T\rightarrow S$ in $\mathscr{S}$, let $\theta_{f}^{(1)}$ and
$\theta_{f}^{(2)}$ denote the corresponding transition isomorphisms along $f$.
We define the composite morphism $R_{2}\circ
R_{1}:\mathscr{C}_{1}\rightarrow\mathscr{C}_{3}$ by assigning

  * •

to every $S\in\mathscr{S}$ the composite functor

$(R_{2}\circ
R_{1})_{S}:\mathscr{C}_{1}(S)\xrightarrow{R_{1,S}}\mathscr{C}_{2}(S)\xrightarrow{R_{2,S}}\mathscr{C}_{3}(S),$

  * •

to every arrow $f:T\rightarrow S$ in $\mathscr{S}$, the natural isomorphism of
functors $\mathscr{C}_{1}(S)\rightarrow\mathscr{C}_{3}(T)$

$\theta_{f}^{(3)}:f^{*}(R_{2}\circ
R_{1})_{S}(C):=f^{*}R_{2,S}R_{1,S}(C)\xrightarrow{\theta_{f}^{(2)}}R_{2,T}(f^{*}R_{1,S}(C))\xrightarrow{\theta_{f}^{(1)}}R_{2,T}R_{1,T}(f^{*}C)=:(R_{2}\circ
R_{1})_{T}(f^{*}C).$

It is easy to check that these data satisfy condition (mor-$\mathscr{S}$-fib)
and so define indeed a morphism of (additive) $\mathscr{S}$-fibered
categories. We leave the details to the interested reader.

With this notions at our disposal, we can define $\mathscr{S}$-fibered
representations as follows:

######  Definition 5.5.

  1. (1)

We call (additive) $\mathscr{S}$-fibered representation a triple
$(\mathscr{C},\mathscr{E};R)$ consisting of two (additive)
$\mathscr{S}$-fibered categories $\mathscr{C},\mathscr{E}$ and a morphism of
(additive) $\mathscr{S}$-fibered categories
$R:\mathscr{C}\rightarrow\mathscr{E}$.

  2. (2)

We say that the $\mathscr{S}$-fibered representation
$(\mathscr{C},\mathscr{E};R)$ is abelian if the $\mathscr{S}$-fibered category
$\mathscr{E}$ is abelian in the sense of Definition 5.1.

######  Definition 5.6.

For $i=1,2$, let $(\mathscr{C}_{i},\mathscr{E}_{i};R_{i})$ be an
$\mathscr{S}$-fibered representation in the sense of Definition 5.5.

  1. (1)

We call morphism of $\mathscr{S}$-fibered representations from
$(\mathscr{C}_{1},\mathscr{E}_{1};R_{1})$ to
$(\mathscr{C}_{2},\mathscr{E}_{2};R_{2})$ a triple $\alpha=(F,G;\kappa)$
consisting of two additive morphisms of $\mathscr{S}$-fibered categories

$F:\mathscr{C}_{1}\rightarrow\mathscr{C}_{2},\qquad
G:\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$

and a $2$-isomorphism between morphisms of $\mathscr{S}$-fibered categories
$\mathscr{C}_{1}\rightarrow\mathscr{E}_{2}$

$\kappa:G\circ R_{1}\xrightarrow{\sim}R_{2}\circ F.$

  2. (2)

In case the additive $\mathscr{S}$-fibered representations
$(\mathscr{C}_{1},\mathscr{E}_{1};\kappa_{1})$ and
$(\mathscr{C}_{2},\mathscr{E}_{2};\kappa_{2})$ are abelian, we say that a
$1$-morphism
$\alpha:(\mathscr{C}_{1},\mathscr{E}_{1};\kappa_{1})\rightarrow(\mathscr{C}_{2},\mathscr{E}_{2};\kappa_{2})$
is exact if the additive morphism of $\mathscr{S}$-fibered categories
$G:\mathscr{E}_{1}\rightarrow\mathscr{E}_{2}$ is exact.

######  Remark 5.7.

Requiring the commutativity condition of axiom (mor-$\mathscr{S}$-fib) in
Definition 5.3 is the same as asking that, for every pair of composable
morphisms $f:T\rightarrow S$ and $g:S\rightarrow V$ in $\mathscr{S}$, the
connection isomorphisms of $\mathscr{C}$ and $\mathscr{E}$ at $(f,g)$ define a
$2$-morphism between $1$-morphisms of representations
$(\mathscr{C}(V),\mathscr{E}(V);R_{V})\rightarrow(\mathscr{C}(T),\mathscr{E}(T);R_{T})$

$(\mathsf{conn}_{f,g},\mathsf{conn}_{f,g}):((gf)^{*},(gf)^{*};\theta_{gf})\rightarrow(f^{*},f^{*};\theta_{f})\circ(g^{*},g^{*};\theta_{g}).$

In the first place, we show that the universal abelian factorizations arising
from an abelian $\mathscr{S}$-fibered representation canonically assemble into
a similar abelian $\mathscr{S}$-fibered representation:

######  Proposition 5.8.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian $\mathscr{S}$-fibered
representation. Then:

  1. (1)

Associating

     * •

to every $S\in\mathscr{S}$, the abelian category $\mathbf{A}(\beta_{S})$,

     * •

to every morphism $f:T\rightarrow S$ in $\mathscr{S}$, the exact functor

$\overline{f^{*}}:\mathbf{A}(\beta_{S})\rightarrow\mathbf{A}(\beta_{T})$

obtained via Proposition 2.5(1),

     * •

to every pair of composable morphisms $f:T\rightarrow S$ and $g:S\rightarrow
V$ in $\mathscr{S}$, the natural isomorphism of functors
$\mathbf{A}(\beta_{V})\rightarrow\mathbf{A}(\beta_{T})$

(27)
$\overline{\mathsf{conn}}=\overline{\mathsf{conn}}_{f,g}:\overline{(gf)^{*}}\xrightarrow{\sim}\overline{f^{*}}\overline{g^{*}}$

obtained via Proposition 3.4

defines an abelian $\mathscr{S}$-fibered category $\mathbf{A}(\beta)$.

  2. (2)

Associating

     * •

to every $S\in\mathscr{S}$, the faithful exact functor
$\iota_{S}:\mathbf{A}(\beta_{S})\hookrightarrow\mathscr{A}(S)$,

     * •

to every morphism $f:T\rightarrow S$ in $\mathscr{S}$, the natural isomorphism
of functors $\mathbf{A}(\beta_{S})\rightarrow\mathscr{A}(T)$

(28)
$\tilde{\theta}=\tilde{\theta}_{f}:\overline{f^{*}}\iota_{S}(M)\xrightarrow{\sim}\iota_{T}(\overline{f^{*}}M)$

obtained via Proposition 2.5(2)

defines an exact morphism of abelian $\mathscr{S}$-fibered categories
$\iota:\mathbf{A}(\beta)\hookrightarrow\mathscr{A}$.

###### Proof.

Recall that the constructions of Proposition 2.5 and Proposition 3.4 are
compatible with composition by Remark 2.7(2) and Remark 3.5(2), respectively.
As a consequence, the isomorphisms (27) necessarily satisfy conditions
($\mathscr{S}$-fib-0) and ($\mathscr{S}$-fib-1) of Definition 5.1, which
implies (1). Similarly, the natural isomorphisms (28) satisfy condition
($m$-ETS) of Definition 6.1, which implies (2). ∎

######  Remark 5.9.

In fact, as $S$ varies in $\mathscr{S}$, the abelian representations
$(\mathscr{D}(S),\mathbf{A}(\beta_{S});\pi_{S})$ and
$(\mathbf{A}(\beta_{S}),\mathscr{A}(S);\iota_{S})$ assemble into abelian
$\mathscr{S}$-fibered representations.

Secondly, for sake of completeness, we also prove an enhanced version of
Proposition 2.5 in the $\mathscr{S}$-fibered setting:

######  Proposition 5.10.

For $i=1,2$, let $(\mathscr{D}_{i},\mathscr{A}_{i};\beta_{i})$ be an abelian
$\mathscr{S}$-fibered representation. Suppose that we are given an exact
morphism of $\mathscr{S}$-fibered representations

$(\phi,\Phi;\kappa):(\mathscr{D}_{1},\mathscr{A}_{1};\beta_{1})\rightarrow(\mathscr{D}_{2},\mathscr{A}_{2};\beta_{2}).$

Then, associating

  * •

to every $S\in\mathscr{S}$, the exact functor

$\overline{\phi_{S}}:\mathbf{A}(\beta_{1,S})\rightarrow\mathbf{A}(\beta_{2,S})$

obtained via Proposition 2.5,

  * •

to every morphism $f:T\rightarrow S$ in $\mathscr{S}$, the natural isomorphism
of functors $\mathbf{A}(\beta_{1,S})\rightarrow\mathbf{A}(\beta_{2,T})$

(29)
$\tilde{\kappa}=\tilde{\kappa}_{f}:\overline{f^{*}}\overline{\phi_{1,S}}(M)\xrightarrow{\sim}\overline{\phi_{2,T}}(\overline{f^{*}}M)$

obtained via Proposition 3.4

defines an exact $1$-morphism of $\mathscr{S}$-fibered categories
$\mathbf{A}(\beta_{1})\rightarrow\mathbf{A}(\beta_{2})$.

###### Proof.

Arguing as in the proof of the previous result, we see that the isomorphisms
(29) necessarily satisfy condition (mor-$\mathscr{S}$-fib), which implies the
thesis. ∎

######  Remark 5.11.

In fact, as $S$ varies in $\mathscr{S}$, the $1$-morphisms of abelian
representations $(\phi_{S},\overline{\phi_{S}};\textup{id})$ and
$(\overline{\phi_{S}},\Phi_{S};\tilde{\kappa}_{S})$ assemble into
$1$-morphisms of abelian $\mathscr{S}$-fibered representations.

###  5.2. Cofinality properties

We now specialize the discussion of the previous subsection to the case where
the base category is co-filtered; for sake of clarity, throughout this
subsection we let $\mathscr{U}$ denote a co-filtered small category. Our goal
is to study filtered $2$-colimits of universal abelian factorizations arising
from some abelian $\mathscr{U}$-fibered category.

Recall that, for every (additive) $\mathscr{U}$-fibered category
$\mathscr{C}$, it makes sense to define the filtered $2$-colimit

(30) $2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{C}(U),$

which is a well-behaved (additive) category whose objects and morphisms admit
the usual explicit description in terms of equivalence classes of
representatives. If $\mathscr{C}$ is an abelian $\mathscr{U}$-fibered category
in the sense of Definition 5.1, the $2$-colimit (30) is an abelian category.

######  Lemma 5.12.

Let $\mathscr{D}$ be an additive $\mathscr{U}$-fibered category. Then the
following statements hold:

  1. (1)

There exists a canonical equivalence

(31)
$2-\varinjlim_{U\in\mathscr{U}^{op}}\mathbf{A}(\mathscr{D}(U))\xrightarrow{\sim}\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U))$

making the diagram

${2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U)}$${2-\varinjlim_{U\in\mathscr{U}^{op}}\mathbf{A}(\mathscr{D}(U))}$${\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U))}$$\scriptstyle{\sim}$

commute.

  2. (2)

More generally, for every abelian $\mathscr{U}$-fibered representation
$(\mathscr{D},\mathscr{A};\beta)$, there exists a canonical equivalence

(32)
$2-\varinjlim_{U\in\mathscr{U}^{op}}\mathbf{A}(\beta_{U})\xrightarrow{\sim}\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\beta_{U})$

making the whole diagram

${2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U)}$${2-\varinjlim_{U\in\mathscr{U}^{op}}\mathbf{A}(\beta_{U})}$${\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\beta_{U})}$${2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{A}(U)}$$\scriptstyle{\sim}$

commute.

###### Proof.

  1. (1)

The functor (31) is induced by the $\mathscr{U}^{op}$-indexed system of exact
functors
$\mathbf{A}(\mathscr{D}(U))\rightarrow\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U))$
obtained by applying Theorem 1.2(2)(i) to the obvious $1$-morphisms of abelian
representations

$(\mathscr{D}(U),\mathscr{A}(U);\beta_{U})\rightarrow(2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U),2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{A}(U);2-\varinjlim_{U\in\mathscr{U}^{op}}\beta_{U}),\qquad
U\in\mathscr{U}.$

A canonical quasi-inverse to it is determined by the universal property of
$\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U))$ via Theorem
1.2(2)(i).

  2. (2)

As for the previous point, the functor (32) is induced by applying Proposition
2.5(1) for each $U\in\mathscr{U}$ and passing to the $2$-colimit, while a
canonical quasi-inverse is determined by the universal property of
$\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\beta_{U})$ via Proposition
2.5(1).

∎

We now derive an consequence of the previous result. In order to formulate it,
we need a suitable notion of denseness for (additive) $\mathscr{U}$-fibered
categories:

######  Definition 5.13.

Let $\mathscr{C}$ be an (additive) $\mathscr{U}$-fibered category, and let
$\mathscr{C}^{0}$ be an (additive) $\mathscr{U}$-fibered subcategory of
$\mathscr{C}$. We say that $\mathscr{C}^{0}$ is weakly dense in $\mathscr{C}$
if the canonical functor

$2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{C}^{0}(U)\rightarrow
2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{C}(U)$

is an equivalence.

######  Remark 5.14.

Concretely, saying that $\mathscr{C}^{0}$ is weakly dense in $\mathscr{C}$
amounts to requiring the following two conditions:

  1. (1)

For every $U\in\mathscr{U}$ and every object $C\in\mathscr{C}(U)$, there
exists an arrow $v:V\rightarrow U$ in $\mathscr{U}$ (depending on $C$) such
that the object $v^{*}C\in\mathscr{C}(V)$ lies in the essential image of
$\mathscr{C}^{0}(V)$.

  2. (2)

For every $V\in\mathscr{U}$, every pair of objects
$C,C^{\prime}\in\mathscr{C}(U)$, and every morphism $c:C\rightarrow
C^{\prime}$ in $\mathscr{C}(V)$, there exists an arrow $w:W\rightarrow V$ in
$\mathscr{U}$ (depending on $c$) such that the morphism
$w^{*}c:w^{*}C\rightarrow w^{*}C^{\prime}$ lies in $\mathscr{C}^{0}(W)$.

Note that, given an abelian representation $(\mathscr{D},\mathscr{A};\beta)$
and a (possibly non-full) additive subcategory $\mathscr{D}^{0}$ of
$\mathscr{D}$, the inclusion functor
$\mathscr{D}^{0}\hookrightarrow\mathscr{D}$ defines a canonical exact
$1$-morphism of abelian representations

$(\mathscr{D}^{0},\mathscr{A};\beta^{0})\rightarrow(\mathscr{D},\mathscr{A};\beta),$

where we have set $\beta^{0}:=\beta|_{\mathscr{D}^{0}}$. Applying Proposition
2.5, this yields a canonical exact functor

$\mathbf{A}(\beta^{0})\rightarrow\mathbf{A}(\beta)$

which is faithful by construction (but a priori not full nor essentially
surjective, even if one assumes that the inclusion
$\mathscr{D}^{0}\hookrightarrow\mathscr{D}$ is just full or just essentially
surjective, respectively). Thus we may naturally regard
$\mathbf{A}(\beta^{0})$ as an abelian subcategory of $\mathbf{A}(\beta)$.

As a consequence of Proposition 5.10, the analogous result holds for
$\mathscr{U}$-fibered representations: in this case, one starts from an
abelian $\mathscr{U}$-fibered representation $(\mathscr{D},\mathscr{A};\beta)$
together with a (possibly non-full) $\mathscr{U}$-fibered subcategory
$\mathscr{D}^{0}$ of $\mathscr{D}$, and one gets a canonical faithful exact
morphism of $\mathscr{U}$-fibered categories

$\mathbf{A}(\beta^{0})\rightarrow\mathbf{A}(\beta)$

expressing $\mathbf{A}(\beta^{0})$ naturally as a $\mathscr{U}$-fibered
subcategory of $\mathbf{A}(\beta)$.

As a consequence of the previous result, we get the following:

######  Corollary 5.15.

Let $(\mathscr{D},\mathscr{A};\beta)$ be an abelian $\mathscr{U}$-fibered
representation; in addition, let $\mathscr{D}^{0}$ be a (possibly non-full)
additive $\mathscr{U}$-fibered subcategory of $\mathscr{D}$, and let
$(\mathscr{D}^{0},\mathscr{A};\beta^{0})$ denote the restricted abelian
$\mathscr{U}$-fibered representation. Suppose that $\mathscr{D}^{0}$ is weakly
cofinal in $\mathscr{D}$ in the sense of Definition 5.13. Then
$\mathbf{A}(\beta^{0})$ is weakly dense in $\mathbf{A}(\beta)$ as well.

###### Proof.

Consider the canonical commutative diagram of abelian categories and exact
functors

${2-\varinjlim_{U\in\mathscr{U}^{op}}\mathbf{A}(\beta^{0}_{U})}$${2-\varinjlim_{U\in\mathscr{U}^{op}}\mathbf{A}(\beta_{U})}$${\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\beta^{0}_{U})}$${\mathbf{A}(2-\varinjlim_{U\in\mathscr{U}^{op}}\beta_{U})}$

$\scriptstyle\sim$

$\scriptstyle\sim$

in which the two vertical arrows are the equivalences of 5.12. Proving the
thesis amounts to showing that the upper vertical arrow is an equivalence, and
to this end it suffices to show that the lower horizontal arrow is an
equivalence. But, since $\mathscr{D}^{0}$ is weakly cofinal in $\mathscr{D}$
by hypothesis, the obvious $1$-morphism of abelian representations

$(2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}^{0}(U),2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{A}(U);2-\varinjlim_{U\in\mathscr{U}^{op}}\beta^{0}_{U})\rightarrow(2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{D}(U),2-\varinjlim_{U\in\mathscr{U}^{op}}\mathscr{A}(U);2-\varinjlim_{U\in\mathscr{U}^{op}}\beta_{U})$

is an equivalence (in the sense of Definition 3.6). Therefore, by Corollary
3.9, the lower horizontal arrow in the above diagram is an equivalence as
well. ∎

##  6\. Lifting (external) tensor structures

In this final section, we explain how to lift the structure of monoidal
fibered categories to the level of universal abelian factorizations; we
achieve this by combining the results of Section 4 on multi-exact functors
with those of Section 5 on abelian fibered categories.

Recall that in [Ter23T] we have shown how the usual structure and properties
of monoidal fibered categories, which are usually expressed in terms of the
internal tensor product, can be canonically rephrased in terms of the external
tensor product - provided the base category admits binary products, so that
the external tensor product can be actually defined.

In view of our applications to perverse Nori motives, where working with the
external tensor product is the most natural choice, we chose to formulate the
results of the present section in terms the external tensor product. The
interested reader can easily translate our statements and proofs to the usual
setting via the dictionary of [Ter23T].

###  6.1. Main result

As in the previous section, we work over a fixed small category $\mathscr{S}$,
but we now assume that $\mathscr{S}$ admits binary products.

For convenience, we start by recalling the notion of external tensor structure
on $\mathscr{S}$-fibered categories from [Ter23T, §§ 2, 8], adapted to the
additive and abelian settings:

######  Definition 6.1.

Let $\mathscr{C}$ be an additive $\mathscr{S}$-fibered category.

  1. (1)

An external tensor structure $(\boxtimes,m)$ on $\mathscr{C}$ is the datum of

     * •