# On Krull–Schmidt bicategories

Ivo Dell’Ambrogio  Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé,
F-59000 Lille, France<EMAIL_ADDRESS>http://math.univ-
lille1.fr/$\sim$dellambr

###### Abstract.

We study the existence and uniqueness of direct sum decompositions in additive
bicategories. We find a simple definition of Krull–Schmidt bicategories, for
which we prove the uniqueness of decompositions into indecomposable objects as
well as a characterization in terms of splitting of idempotents and properties
of 2-cell endomorphism rings. Examples of Krull–Schmidt bicategories abound,
with many arising from the various flavors of 2-dimensional linear
representation theory.

###### Key words and phrases:

Bicategory, Krull–Schmidt property, 2-representation theory.

###### 2020 Mathematics Subject Classification:

20J05, 18B40, 18N10, 18N25

Author partially supported by Project ANR ChroK (ANR-16-CE40-0003) and Labex
CEMPI (ANR-11-LABX-0007-01).

###### Contents

  1. 1 Introduction and results
  2. 2 Preliminaries on direct sums
  3. 3 Characterizations of direct summands
  4. 4 Krull–Schmidt bicategories
  5. 5 Notions of strong indecomposability
  6. 6 Direct comparison of direct summands
  7. 7 Examples

##  1\. Introduction and results

A Krull–Schmidt category is an additive category in which every object admits
a decomposition into a direct sum of finitely many objects which are strongly
indecomposable, i.e. whose endomorphism rings are local. In fact, in a
Krull–Schmidt category strongly indecomposable objects coincide with those
which are indecomposable for direct sums. Most importantly, every object is a
direct sum of indecomposable ones in a unique way, up to isomorphism and
permutation of the factors. There is a nice characterization of Krull–Schmidt
categories: They are precisely those additive categories in which all
idempotent morphisms split and all endomorphism rings are semiperfect in the
sense of Bass. See e.g. [Kra15].

The goal of this note is to prove versions of the above eminently useful facts
in the setting of additive _bi_ categories. Perhaps surprisingly, the new
bicategorical theory is arguably more elegant and easier to use than its
classical counterpart.

The usual 1-categorical Krull–Schmidt property holds, in particular, for
categories of finite dimensional representations of groups and algebras. It is
a cornerstone of representation theory which helps alleviate the pain of
leaving the semisimple case for the modular case, where it becomes imperative
to understand the global structure of categories of representations. The
Krull–Schmidt property is also surprisingly subtle (e.g. it holds for finite
length modules over any ring but fails for artinian modules already over
certain semilocal noetherian commutative rings [FHLV95] [Fac03]), and it has
benefited from some serious investigation. Similarly, as _2-representation
theory_ (in its various directions) is beginning to venture beyond the
semisimple case too, it seems wise to prepare a suitable Krull–Schmidt theory.
We do this in Sections 2-6.

The final Section 7 discusses examples of Krull–Schmidt bicategories. They
include various bimodule bicategories of rings (such as rings with noetherian
center), bicategories of Mackey 2-motives in the sense of Balmer–Dell’Ambrogio
(7.9 – our original motivating example), semisimple 2-categories such as that
of finite dimensional 2-vector spaces à la Kapranov–Voevodsky–Neuchl (7.10) or
2-Hilbert spaces à la Baez (7.11), finite module categories over finite tensor
categories à la Etingof–Ostrik (7.18), the 2-category of finite dimensional
2-representations of any 2-group over any field (7.21), and the 2-category of
2-representations of a finitary linear 2-category à la Mazorchuk–Miemietz
(7.24). Underlying the representation-theoretic examples is 7.14, which
categorifies the classical fact that finite dimensional representations of any
algebra form a Krull–Schmidt category. In view of all this, we expect
Krull–Schmidt bicategories to enter the basic toolkit of any future modular
(i.e. nonsemisimple) linear 2-representation theory, where global structural
questions should gain in importance relative to combinatorial ones.

$*\;*\;*$

We now present our theoretical results in details.

We say a bicategory is _additive_ if it is locally additive, i.e. its Hom
categories are additive and its composition functors are additive in both
variables, and if it admits all finite direct sums of its objects; see Section
2 for details. For instance additive categories, additive functors and natural
transformations form a very large additive bicategory. A nonzero object $X$ in
an additive bicategory is _indecomposable_ if a direct sum decomposition
$X\simeq X_{1}\oplus X_{2}$ necessarily implies that $X_{1}\simeq 0$ or
$X_{2}\simeq 0$ (where $\simeq$ denotes equivalence). We say $X$ is _strongly
indecomposable_ if its 2-cell endomorphism ring
$\operatorname{End}_{\mathcal{B}(X,X)}(\mathrm{Id}_{X})$ is a connected
commutative ring (it is always commutative by the Eckmann–Hilton argument).
Here is our main result:

######  Theorem (See 4.3).

Suppose that an object $X$ of an additive bicategory admits two direct sum
decompositions $Y_{1}\oplus\cdots\oplus Y_{n}$ and $Z_{1}\oplus\cdots\oplus
Z_{m}$ into strongly indecomposable objects. Then $n=m$ and there are
equivalences $Y_{k}\simeq Z_{\sigma(k)}$ for all $k$ after some permutation
$\sigma\in\Sigma_{n}$ of the factors.

This suggests the following definition: An additive bicategory is
_Krull–Schmidt_ if every object is equivalent to a direct sum of finitely many
strongly indecomposable objects, in the above sense. We easily deduce (4.6)
that indecomposable and strongly indecomposable objects coincide in a
Krull–Schmidt bicategory, hence that every object admits a unique
decomposition into indecomposable ones.

We also obtain a very satisfying characterization of Krull–Schmidt
bicategories. Let us call a commutative ring _semiconnected_ if it is a direct
product of finitely many connected rings, i.e. if its Zariski spectrum has
finitely many topological connected components. We also say an additive
bicategory is _weakly block complete_ 111Unfortunately, “idempotent complete”
and “block complete” already have different meanings in this context. if every
decomposition $1_{X}=e_{1}+e_{2}$ of the identity 2-cell
$1_{X}\in\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$ as a sum of two
orthogonal idempotent 2-cells is induced by a direct sum decomposition
$X\simeq X_{1}\oplus X_{2}$ at the level of objects (see details in 2.10).
Then:

######  Theorem (See 4.8).

An additive bicategory is Krull–Schmidt if and only if it is weakly block
complete and the 2-cell endomorphism ring
$\operatorname{End}(\mathrm{Id}_{X})$ of every object $X$ is semiconnected.

Some comments are warranted.

As far as we can see, our theory does not recover its classical counterpart
because we cannot usefully view an additive category as an additive
bicategory. Thus our definitions and results are just analogs rather than
generalizations.

On its face, our 2-categorical definition of a strongly indecomposable object
$X$ may appear to be unrelated to the original 1-categorical notion. A direct
2-categorical analog of “$\operatorname{End}(X)$ is local” would be to require
that whenever $F$ and $G$ are two 1-cells $X\to X$ which are not equivalences,
then $F\oplus G\colon X\to X$ is not an equivalence either. But it turns out
that the latter property is a consequence of our definition, and as soon as
the category $\operatorname{End}(X)$ is idempotent complete (a mild
hypothesis) the two become equivalent. In fact, all reasonable options we
could think of become equivalent under the same hypothesis (5.4).

Obviously, in an additive 1-category endomorphism rings are rarely commutative
and there are no 2-cell endomorphism rings. This might explain why passing to
a higher-categorical setting actually appears to make indecomposables and the
uniqueness of decompositions _easier_ to handle—insofar as semiconnected rings
are easier than semiperfect ones. It also makes examples easier to recognize,
e.g. we do not feel any pressing need to discuss higher versions of bi-chain
conditions, finite length objects, etc. In order to recognize a Krull–Schmidt
bicategory using our characterization, it suffices to ensure weak block
completeness (which can always be implemented by [BD20, Thm. A.7.23]) and to
have a reasonable grasp of the 2-cell Homs (e.g. it suffices that they are not
unreasonably large).

We thought it instructive to give two different proofs of the uniqueness
theorem. The first proof (in Section 4), which seems not to have any
1-categorical analog, easily follows from two facts. Firstly, that a direct
summand $Y$ of $X$ is determined up to equivalence by the idempotent element
of $\operatorname{End}(\mathrm{Id}_{X})$ corresponding to $1_{Y}$; note that,
by definition, the summand is strongly indecomposable precisely when the
idempotent is primitive. Secondly, that in a commutative ring there cannot be
two different decompositions of $1$ as a sum of orthogonal primitive
idempotents. The second proof (in Section 6) is a calque of the usual
1-categorical proof, and proceeds by recursively comparing summands of two
decompositions of the same object via the given equivalences. Both proofs are
easy in principle but require a close inspection of the bicategorical notion
of “direct summand”, which we carry out in Section 3. To wit, in weakly block
complete bicategories we characterize a direct summand $Y$ of $X$ in a few
equivalent ways: as a special kind of ambijunction $Y\leftrightarrows X$, or
(forgetting the object $Y$) as a special kind of Frobenius monad on $X$, or
quite magically (forgetting the 1-cell $X\to X$) as an idempotent in
$\operatorname{End}(\mathrm{Id}_{X})$.

######  Acknowledgements.

I am grateful to Alexis Virelizier for pointing me to [DR18] and to Paul
Balmer and an anonymous referee for useful comments.

##  2\. Preliminaries on direct sums

Most of these preliminaries are also covered in [BD20, §A.7], where the reader
may find more details and context.

######  2.1 Convention.

We will use the standard terminology for bicategories, as recalled e.g. in
[BD20, App. A]. However, whenever convenient and without further mention we
will pretend our bicategories are strict 2-categories in order to simplify
calculations and to more easily talk about (internal) monads; this is
justified by the coherence theorem for bicategories ([JY21, Ch. 8]). If
$\mathcal{B}$ is a bicategory, we write $\mathcal{B}(X,Y)$ for its Hom
categories and $\operatorname{End}_{\mathcal{B}(X,Y)}(F)$ or just
$\operatorname{End}_{\mathcal{B}}(F)$ or $\operatorname{End}(F)$ for the
2-cell endomorphism set of a 1-cell $F\colon X\to Y$. Internally to a
bicategory, $\cong$ and $\simeq$ stand for isomorphism and equivalence,
respectively; $\mathrm{Id}_{X}$ and $\mathrm{id}_{F}$ for identity 1-cells and
identity 2-cells, respectively; and also $1_{X}$ or just $1$ is short for
$\mathrm{id}_{\mathrm{Id}_{X}}$. Products, coproducts, initial and final
objects in a bicategory are understood in the sense of _pseudo_(co)limits
(a.k.a. bilimits [JY21, §5]), rather than strict ones; in particular, they are
only unique up to equivalence rather than isomorphism. We mostly ignore set-
theoretical size issues as irrelevant to this topic.

######  2.2 Definition.

A bicategory $\mathcal{B}$ is _locally additive_ if each Hom category
$\mathcal{B}(X,Y)$ is additive and each composition functor
$\mathcal{B}(Y,Z)\times\mathcal{B}(X,Y)\to\mathcal{B}(X,Z)$ is additive in
both variables.

######  2.3 Definition.

Let $\mathcal{B}$ be a locally additive bicategory. A _direct sum_ of two
objects $X_{1}$ and $X_{2}$ is a diagram of 1-cells of $\mathcal{B}$ of the
form

(2.4)
$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{1}}$$\scriptstyle{P_{2}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I_{2}}$

such that there exist isomorphisms $P_{i}I_{i}\cong\mathrm{Id}_{X_{i}}$ and
$P_{i}I_{j}\cong 0$ (for $i\neq j$) and $I_{1}P_{1}\oplus
I_{2}P_{2}\cong\mathrm{Id}_{X}$, the latter two being a zero object and a
direct sum in the additive categories
$\operatorname{Hom}_{\mathcal{B}}(X_{j},X_{i})$ and
$\operatorname{End}_{\mathcal{B}}(X)$, respectively. As usual, we will write
$X_{1}\oplus X_{2}$ to indicate the object $X$ or even the whole diagram;
direct sums of $n\geq 2$ objects are defined similarly; and the direct sum of
no objects is by definition a zero objet of $\mathcal{B}$, i.e. one which is
both initial and final.

######  2.5 Definition.

We say that a bicategory is _additive_ if it is locally additive and if it
admits arbitrary finite direct sums of its objects.

######  2.6 Example.

The (very large) 2-category $\mathsf{ADD}$ of (non-necessarily small) additive
categories, additive functors and natural transformations is an additive
bicategory. Direct sums are provided by the (strict) finite Cartesian products
of categories, with their projection functors
$P_{i}\colon\mathcal{A}_{1}\times\mathcal{A}_{2}\to\mathcal{A}_{i}$ and with
inclusion functors defined by $I_{1}\colon A_{1}\mapsto(A_{1},0)$ and
$I_{2}\colon A_{2}\mapsto(0,A_{2})$.

######  2.7 Definition.

An object $X$ of an additive bicategory $\mathcal{B}$ is _indecomposable_ if
whenever there is an equivalence $X\simeq X_{1}\oplus X_{2}$ then $X_{1}\simeq
0$ or $X_{2}\simeq 0$.

######  2.8 Remark.

One easily checks that any direct sum as in 2.3 is also a product
$(X,P_{1},P_{2})$ and a coproduct $(X,I_{1},I_{2})$ of $X_{1}$ and $X_{2}$
(and similarly with $n\geq 2$ factors). It follows that direct sums are unique
up to equivalence. Also, every direct sum decomposition $X\simeq X_{1}\oplus
X_{2}$ induces equivalences of additive categories

(2.9)
$\mathcal{B}(Y,X)\simeq\mathcal{B}(Y,X_{1})\times\mathcal{B}(Y,X_{2})\quad\textrm{
and
}\quad\mathcal{B}(X,Y)\simeq\mathcal{B}(X_{1},Y)\times\mathcal{B}(X_{2},Y)$

for all objects $Y$. It is therefore possible to use matrix notation in the
usual way for 1-cells as well as 2-cells between two directs sums, just as one
does inside each additive Hom category. In particular (cf. [BD20, Rem.
A.7.15]), we may decompose the category $\operatorname{End}_{\mathcal{B}}(X)$
into four factors. Then $\mathrm{Id}_{X}$ corresponds to the diagonal matrix
$\mathrm{diag}(\mathrm{Id}_{X_{1}},\mathrm{Id}_{X_{2}})$ and its endomorphism
ring $\operatorname{End}_{\mathcal{B}(X,X)}(\mathrm{Id}_{X})$ is also diagonal
(cf. 5.1 if necessary), so that we get an isomorphism of commutative rings

$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})\cong\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X_{1}})\times\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X_{2}}).$

The latter is equivalent to a decomposition of
$1_{X}=\mathrm{id}_{\mathrm{Id}_{X}}$ as a sum $e_{1}+e_{2}$ of two orthogonal
idempotents, with $e_{i}$ corresponding to $1_{X_{i}}$. Given such a
decomposition of $1_{X}$, however, nothing guarantees that it arises as above
from a direct sum decomposition $X\simeq X_{1}\oplus X_{2}$. Whence:

######  2.10 Definition.

An additive bicategory is _weakly block complete_ if for every object $X$ and
every idempotent element $e=e^{2}$ of the ring
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$ there exists a direct sum
decomposition $X\simeq X_{1}\oplus X_{2}$ under which the 2-cells $e$ and
$1_{X}-e$ correspond to $[\,{}^{1}_{0}\,{}^{0}_{0}\,]$ and
$[\,{}^{0}_{0}\,{}^{0}_{1}\,]$, respectively, as in 2.8. As in [BD20], we say
that $\mathcal{B}$ is _block complete_ if moreover every Hom category of
$\mathcal{B}$ is idempotent complete.

######  2.11 Theorem (See [BD20, A.7.23–24]).

For every additive bicategory $\mathcal{B}$, there exists a block complete
bicategory $\mathcal{B}^{\flat}$ and a 2-fully faithful pseudofunctor
$\mathcal{B}\hookrightarrow\mathcal{B}^{\flat}$ which is 2-universal among
additive pseudofunctors to block complete bicategories.

######  2.12 Remark.

Using 2.11, we can also produce a similarly universal _weak block completion_
$\mathcal{B}^{w\flat}$ for any additive bicategory $\mathcal{B}$: It suffices
to take the smallest 2-full sub-bicategory of $\mathcal{B}^{\flat}$ which
contains (the image of) $\mathcal{B}$ and is weakly block complete. It will
have the same objects as $\mathcal{B}^{\flat}$ but may lack some of its
1-cells.

In Section 7 we will need the fact that block completeness and weak block
completeness are inherited pointwise, in the sense that bicategories of
pseudofunctors inherit these properties from their target bicategory. More
precisely:

######  2.13 Terminology.

Let $\Bbbk$ be a commutative ring. A bicategory is _$\Bbbk$ -linear_ if its
Hom categories and composition functors are $\Bbbk$-linear (i.e. enriched in
$\Bbbk$-modules). For instance, a locally additive bicategory is a
$\mathbb{Z}$-linear bicategory which also admits finite direct sums of
parallel 1-cells. A pseudofunctor $\mathcal{F}\colon\mathcal{B}\to\mathcal{C}$
between $\Bbbk$-linear bicategories is _$\Bbbk$ -linear_ if each functor
$\mathcal{F}\colon\mathcal{B}(X,Y)\to\mathcal{C}(\mathcal{F}X,\mathcal{F}Y)$
is $\Bbbk$-linear, i.e. preserves $\Bbbk$-linear combinations of 2-cells. Note
that a $\Bbbk$-linear pseudofunctor preserves whatever direct sums of 1-cells
and objects exist in its source, as the latter are equationally defined (cf.
[BD20, Prop. A.7.14]). We will denote by

$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$

the bicategory of $\Bbbk$-linear pseudofunctors $\mathcal{B}\to\mathcal{C}$,
pseudonatural transformations between them, and modifications; see e.g. [BD20,
§A.1] for details.

######  2.14 Proposition.

If $\mathcal{C}$ is (weakly) block complete then so is
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$.

###### Proof.

First note that
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$ inherits a
pointwise $\Bbbk$-linear structure. Namely, suppose
$\mathcal{F}_{1},\mathcal{F}_{2}\colon\mathcal{B}\to\mathcal{C}$ are
pseudofunctors, $t,s\colon\mathcal{F}_{1}\Rightarrow\mathcal{F}_{2}$
pseudonatural transformations, $M,M^{\prime}\colon t\Rrightarrow s$
modifications, and $\lambda\in\Bbbk$ a scalar. Then there is a modification
$M+\lambda M^{\prime}\colon t\Rrightarrow s$ whose component at each object
$X\in\mathcal{B}$ is defined by setting $(M+\lambda
M^{\prime})_{X}:=M_{X}+\lambda M^{\prime}_{X}$ in the $\Bbbk$-module
$\operatorname{Hom}_{\mathcal{B}(\mathcal{F}_{1}X,\mathcal{F}_{2}X)}(t_{X},s_{X})$.
Similarly, $\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$
inherits direct sums of 1-cells and objects from $\mathcal{C}$: The components
in $\mathcal{C}$ of the transformation $t\oplus
s\colon\mathcal{F}_{1}\Rightarrow\mathcal{F}_{2}$ are

$(t\oplus s)_{X}:=t_{X}\oplus s_{X}\colon\mathcal{F}_{1}X\to\mathcal{F}_{2}X$

for all objects $X\in\mathcal{B}$ and the diagonal

$(t\oplus s)_{F}:=t_{F}\oplus
s_{F}=\left(\,{}^{t_{F}}_{0}\,\;{}^{0}_{s_{F}}\right)\colon\mathcal{F}_{2}F\circ(t\oplus
s)_{X}\overset{\sim}{\Longrightarrow}(t\oplus s)_{Y}\circ\mathcal{F}_{1}F$

for all 1-cells $F\colon X\to Y$ of $\mathcal{B}$ (for the latter, we use that
horizontal composition in $\mathcal{C}$ preserves direct sums of 1-cells),
with the evident direct sum structure maps.

For the direct sum $\mathcal{F}_{1}\oplus\mathcal{F}_{2}$ of pseudofunctors,
set
$(\mathcal{F}_{1}\oplus\mathcal{F}_{2})(X):=\mathcal{F}_{1}X\oplus\mathcal{F}_{2}X$
on objects, with diagonal component $\Bbbk$-linear functors (at each pair
$X,Y\in\operatorname{Obj}\mathcal{B}$)

$\mathcal{F}_{1}\oplus\mathcal{F}_{2}:=\left(\,{}^{\mathcal{F}_{1}}_{0}\,\;{}^{0}_{\mathcal{F}_{2}}\right)\colon\mathcal{B}(X,Y)\longrightarrow\mathcal{C}(\mathcal{F}_{1}X\oplus\mathcal{F}_{2}X,\mathcal{F}_{1}Y\oplus\mathcal{F}_{2}Y),$

where the right-hand side is decomposed into four categories as in 2.8.

It remains to see that
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$ also inherits
the splitting of 1-cells and objects. Suppose $M=M^{2}\colon t\Rrightarrow t$
is an idempotent modification. Then each component $M_{X}\colon
t_{X}\Rightarrow t_{X}$ is an idempotent 2-cell of $\mathcal{C}$, giving rise
to a decomposition
$t_{X}\cong\operatorname{Im}(M_{X})\oplus\operatorname{Im}(\mathrm{id}_{t_{X}}-M_{X})$
as soon as that idempotent splits in
$\mathcal{C}(\mathcal{F}_{1}X,\mathcal{F}_{2}X)$. We can then assemble these
splittings into an isomorphism
$t\cong\operatorname{Im}(M)\oplus\operatorname{Im}(\mathrm{id}_{t}-M)$ in the
additive category
$\operatorname{\mathsf{PsNat}}(\mathcal{F}_{1},\mathcal{F}_{2})$ of
transformations and modifications. Thus if $\mathcal{C}$ is locally idempotent
complete so is
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$. If
$\mathcal{C}$ is weakly block complete, in the special case when
$t=\mathrm{Id}_{\mathcal{F}}$ is the identity transformation of
$\mathcal{F}:=\mathcal{F}_{1}=\mathcal{F}_{2}\colon\mathcal{B}\to\mathcal{C}$
we also have decompositions $\mathcal{F}X\simeq Y_{X}\oplus Z_{X}$ of objects
of $\mathcal{C}$ with $1_{Y_{X}}$ and $1_{Z_{X}}$ corresponding under these
equivalences to the identity 2-cells of $\operatorname{Im}(M_{X})$ and
$\operatorname{Im}(1_{\mathcal{F}X}-M_{X})$, respectively. Any choice of such
adjoint equivalences assembles into the required decomposition
$\mathrm{Id}_{\mathcal{F}}\simeq\operatorname{Im}(M)\oplus\operatorname{Im}(1_{\mathcal{F}}-M)$
in $\operatorname{\mathsf{PsNat}}(\mathcal{F}_{1},\mathcal{F}_{2})$. (If
necessary, to see why the latter assertion is true one may use the
characterizations of direct summands treated in the next section.) ∎

######  2.15 Remark.

In [DR18] a notion of “idempotent completion” for linear 2-categories is used
which in general is stronger than our block completion: It requires every Hom
category to be idempotent complete and every separable monad to split (i.e. to
arise from a separable adjunction).

##  3\. Characterizations of direct summands

We work throughout in a locally additive bicategory. Note that, when referring
to a direct sum $X_{1}\oplus X_{2}$ as in (2.4), we were implicitly assuming a
choice was made for its four projection and injection 1-cells, but not for the
2-isomorphisms implementing their relations. We can do better:

######  3.1 Proposition.

Consider any direct sum $X=X_{1}\oplus X_{2}$ as in (2.4). Then the 2-cells
implementing the direct sum relations can be chosen so that they
simultaneously form four adjunctions $P_{i}\dashv I_{i}$ and $I_{i}\dashv
P_{i}$ ($i=1,2$). More precisely, write

$\textstyle{P_{i}I_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{i}}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Id}_{X_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\eta}_{i}}$
and

for the 2-cell components; they can be chosen so that, besides satisfying the
relations

(3.2)
$\varepsilon_{i}=(\overline{\eta}_{i})^{-1}\quad\quad\quad\eta_{i}\overline{\varepsilon}_{j}=\left\\{\begin{array}[]{ll}\mathrm{id}_{I_{i}P_{i}}&\textrm{
if }i=j\\\ 0&\textrm{ if }i\neq
j\end{array}\right.\quad\quad\quad\overline{\varepsilon}_{1}\eta_{1}+\overline{\varepsilon}_{2}\eta_{2}=1_{X}$

they also partake in the adjunctions (for $i=1,2$)

$(P_{i}\dashv I_{i},\eta_{i},\varepsilon_{i})\quad\textrm{ and
}\quad(I_{i}\dashv P_{i},\overline{\eta}_{i},\overline{\varepsilon}_{i})$

as units or counits. The analogous statement holds for sums of $n\geq 2$
objects.

######  3.3 Definition.

An _adjoint direct sum_ in a locally additive bicategory is the data of a
direct sum diagram together with a choice of 2-cells as in 3.1.

######  3.4 Remark.

Consider any direct sum diagram (2.4), say with 2-cell components
$\beta_{i}\colon P_{i}I_{i}\overset{\sim}{\Rightarrow}\mathrm{Id}_{X_{i}}$
whose inverse we will write $\overline{\alpha}_{i}$, and
$[\overline{\beta}_{1}\;\;\overline{\beta}_{2}]\colon I_{1}P_{1}\oplus
I_{2}P_{2}\overset{\sim}{\Rightarrow}\mathrm{Id}_{X}$ with inverse
${}^{t}[\alpha_{1}\;\;\alpha_{2}]$. These 2-cells will satisfy the direct sum
relations as in (3.2), of course. Moreover, not only are $\beta_{i}$ and
$\overline{\alpha}_{i}$ mutually inverse, but we also have

$P_{i}\overline{\beta}_{i}=(P_{i}\alpha_{i})^{-1}\colon
P_{i}\overset{\sim}{\Rightarrow}P_{i}\quad\textrm{ and
}\quad\overline{\beta}_{i}I_{i}=(\alpha_{i}I_{i})^{-1}\colon
I_{i}\overset{\sim}{\Rightarrow}I_{i}.$

Indeed, these 2-cell pairs are already mutual inverses on one side (even
before applying $P_{i}$ or $I_{i}$) by one of the defining relations. To see
that they are inverses on the other side too, it suffices to whisker the
relation $\overline{\beta}_{1}\alpha_{1}+\overline{\beta}_{2}\alpha_{2}=1_{X}$
by $P_{i}$ or $I_{i}$, respectively, and use that $P_{i}I_{j}\cong 0$ for
$i\neq j$. We will often use this observation.

###### Proof of 3.1.

This is a sharper version of [DR18, Prop. 1.1.3], and the proof is similar.
Start with a direct sum diagram with some choice of 2-cells, with notation as
in 3.4. For each $i$, consider the automorphism

$\varphi_{i}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.42406pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
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0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.57169pt\raise 6.0512pt\hbox{{}\hbox{\kern 0.0pt\raise
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of $I_{i}P_{i}$, whose inverse is
$\varphi_{i}^{-1}=(I_{i}P_{i}\overline{\beta}_{i})(I_{i}\overline{\alpha}_{i}P_{i})$
by 3.4. We claim that the following slightly corrected 2-cells

$\eta_{i}:=\varphi_{i}^{-1}\alpha_{i}\quad\quad\varepsilon_{i}:=\beta_{i}\quad\quad\overline{\eta}_{i}:=\overline{\alpha}_{i}\quad\quad\overline{\varepsilon}_{i}:=\overline{\beta}_{i}\varphi_{i}$

satisfy the required conditions. They again form a direct sum for the same
1-cells, indeed the relations (3.2) are immediately checked. It remains to see
that we have two adjunctions $(P_{i}\dashv I_{i},\eta_{i},\varepsilon_{i})$
and $(I_{i}\dashv P_{i},\overline{\eta}_{i},\overline{\varepsilon}_{i})$. For
the first adjunction, 3.4 and the commutative diagram on the left (where we
dropped the $i$’s)

|
---|---
$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P\eta}$$\scriptstyle{P\alpha}$$\textstyle{PIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{PI\overline{\alpha}P}$$\scriptstyle{\beta
P}$$\textstyle{PIPIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{PIP\overline{\beta}}$$\scriptstyle{\beta
PIP}$$\textstyle{PIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon
P}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\alpha}P}$$\textstyle{PIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P\overline{\beta}}$$\textstyle{P}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\eta}P}$$\scriptstyle{P\alpha}$$\textstyle{PIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P\overline{\varepsilon}}$$\scriptstyle{PIP\alpha}$$\textstyle{PIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\alpha}PIP}$$\scriptstyle{\beta
P}$$\textstyle{PIPIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{PI\beta
P}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\alpha}P}$$\textstyle{PIP\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P\overline{\beta}}$$\textstyle{P}$

verify the triangular identity
$(\varepsilon_{i}P_{i})(P_{i}\eta_{i})=\mathrm{id}_{P_{i}}$. This also implies
$(I_{i}\varepsilon_{i})(\eta_{i}I_{i})=(I_{i}\varepsilon_{i})(I_{i}\varepsilon_{i}P_{i}I_{i})(I_{i}P_{i}\eta_{i}I_{i})(\eta_{i}I_{i})=(I_{i}\varepsilon_{i})(\eta_{i}I_{i})(I_{i}\varepsilon_{i})(\eta_{i}I_{i})$
is an idempotent $I_{i}\Rightarrow I_{i}$. As the latter 2-cell is invertible
(by 3.4) it must be equal to $\mathrm{id}_{I_{i}}$, and we get the second
triangular identity. For the second adjunction, the above diagram on the right
displays (again thanks to 3.4) the triangular identity
$(P_{i}\overline{\varepsilon}_{i})(\overline{\eta}_{i}P_{i})=\mathrm{id}_{P_{i}}$,
and the other one follows by the precise same argument as before. ∎

######  3.5 Remark.

There is a far less elementary but perhaps more suggestive proof of 3.1, which
goes as follows. Firstly note that the 2-category $\mathsf{ADD}$ of additive
categories has a canonical choice of _adjoint_ direct sums: Just take
Cartesian products, with the projections $P_{i}$ and injections $I_{i}$ as in
2.6 and the evident 2-cells (involving solely identity and zero natural
transformations). Indeed, in this case we even have $P_{i}I_{i}=\mathrm{Id}$
and all required triangular identities are trivially true. Secondly, for any
locally additive (or just $\mathbb{Z}$-linear) bicategory $\mathcal{B}$, the
bicategory
$\operatorname{\mathsf{PsFun}}_{\mathbb{Z}}(\mathcal{B}^{\mathrm{op}},\mathsf{ADD})$
of additive pseudofunctors (2.13) inherits finite adjoint direct sums from
$\mathsf{ADD}$, by the very same level-wise construction as in the proof of
2.14. Thirdly, observe that adjoint direct sums (or even just their 2-cells)
can be transported along (adjoint) equivalences. Finally, recall (e.g. from
[BD20, Rem. A.7.25]) that for any locally additive bicategory $\mathcal{B}$
there is a Yoneda embedding
$\mathcal{B}\to\operatorname{\mathsf{PsFun}}_{\mathbb{Z}}(\mathcal{B}^{\mathrm{op}},\mathsf{ADD})$
which is an additive pseudofunctor and a biequivalence
$F\colon\mathcal{B}\overset{\sim}{\to}\mathcal{B}^{\prime}$ onto its full
image
$\mathcal{B}^{\prime}\subset\operatorname{\mathsf{PsFun}}_{\mathbb{Z}}(\mathcal{B}^{\mathrm{op}},\mathsf{ADD})$.
Using $F$ and a pseudoinverse $F^{-1}$, any direct sum diagram that exists in
$\mathcal{B}$ can therefore be corrected into an adjoint sum by the previous
remarks.

In the following, we will tacitly assume all our direct sums to be adjoint in
the sense of 3.3. Let us now study a single summand at a time.

######  3.6 Definition.

In a locally additive bicategory, a _splitting datum_ of an object $X$
consists of an object $Y$, two 1-cells $I\colon Y\to X$ and $P\colon X\to Y$
and two adjunctions

$(P\dashv I,\eta,\varepsilon)\quad\textrm{ and }\quad(I\dashv
P,\overline{\eta},\overline{\varepsilon})$

such that $\varepsilon=\overline{\eta}^{-1}$ and
$\eta\overline{\varepsilon}=\mathrm{id}_{IP}$. A splitting datum for $X$ is a
_direct summand of $X$_ if it appears as one of the ambijunctions $I_{i}\dashv
P_{i}\dashv I_{i}\colon X_{i}\leftrightarrows X$ for some decomposition into
an adjoint direct sum $X\simeq X_{1}\oplus\cdots\oplus X_{n}$ as in 3.3. A
_morphism_ of splitting data
$(Y,I,P,\eta,\varepsilon,\overline{\eta},\overline{\varepsilon})$ and
$(Y^{\prime},I^{\prime},P^{\prime},\eta^{\prime},\varepsilon^{\prime},\overline{\eta}^{\prime},\overline{\varepsilon}^{\prime})$
is a 1-cell $F\colon Y\to Y^{\prime}$ together with a 2-isomorphism
$\theta_{F}\colon FP\cong P^{\prime}$, and a _2-morphism_
$(F_{1},\theta_{1})\Rightarrow(F_{2},\theta_{2})$ is a 2-cell $\alpha\colon
F_{1}\Rightarrow F_{2}$ such that $\theta_{2}(\alpha P)=\theta_{1}$. (In other
words, forgetting some structure, here we are considering splitting data as
objects of the comma bicategory of $\mathcal{B}$ under $X$.) We thus obtain a
notion of _equivalence_ of splitting data for $X$.

In the following series of remarks, we mention several ways one can slice and
repackage the information contained in a splitting datum.

######  3.7 Remark.

An adjunction in a bicategory is a _reflection_ if its counit is invertible
and a _coreflection_ if its unit is. Thus the adjunctions $(P\dashv
I,\eta,\varepsilon)$ and $(I\dashv P,\overline{\eta},\overline{\varepsilon})$
in 3.6 display $Y$ as a reflection, respectively a coreflection, of $X$.
Indeed the counit $\varepsilon$ and the unit $\overline{\eta}$ are mutually
inverse, and we also have $\eta\overline{\varepsilon}=\mathrm{id}_{IP}$.

######  3.8 Remark.

Recall that a monad or comonad on an object $X$ in a bicategory is
_idempotent_ if its (co)multiplication is invertible. Thus writing
$E:=IP\colon X\to X$, we see that a splitting datum induces both a monad and a
comonad on $X$

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with the same underlying 1-cell $E$ which are idempotent and satisfy
$\eta\overline{\varepsilon}=\mathrm{id}_{E}$.

######  3.9 Remark.

An idempotent monad $(T\colon X\to X,\mu\colon
T^{2}\overset{\sim}{\Rightarrow}T,\iota\colon\mathrm{Id}_{X}\Rightarrow T)$ on
an object $X$ is the same thing as a _localization_ of $X$, namely a pair
$(T,\iota\colon\mathrm{Id}_{X}\Rightarrow T)$ such that $T\eta$ and $\eta
T\colon T\Rightarrow T^{2}$ are equal and invertible (their common inverse
then recovers the multiplication $\mu$). Dually, an idempotent comonad is the
same as a _colocalization_. Thus 3.8 says that a splitting datum amounts to a
pair

$(E,\eta\colon\mathrm{Id}_{X}\Rightarrow E)\quad\textrm{ and
}\quad(E,\overline{\varepsilon}\colon E\Rightarrow\mathrm{Id}_{X})$

of a localization and a colocalization of $X$, respectively, with the same
underlying 1-cell $E=IP$ and such that $\overline{\varepsilon}$ is a section
of $\eta$.

######  3.10 Remark.

The monad and comonad of 3.8 form an _idempotent special Frobenius monad_
$(T,\mu,\iota,\delta,\epsilon)$ on $X$ such that
$\iota\epsilon=\mathrm{id}_{T}$. Indeed, the latter equation is
$\eta\overline{\varepsilon}=\mathrm{id}_{E}$ again; the monad and comonad
automatically form a Frobenius monad as they originate from an ambijunction
(see e.g. [Del21, Prop. 7.4]); “idempotent” refers to multiplication $\mu$ and
comultiplication $\delta$ being invertible; and “special” refers to the
equation $\mu\delta=\mathrm{id}_{E}$, which holds since $\mu=(I\varepsilon
P)=(I\overline{\eta}P)^{-1}=\delta^{-1}$.

######  3.11 Definition.

For any splitting datum
$(Y,I,P,\eta,\varepsilon,\overline{\eta},\overline{\varepsilon})$ for $X$, the
composite 2-cell
$e:=\overline{\varepsilon}\eta\colon\mathrm{Id}_{X}\Rightarrow\mathrm{Id}_{X}$
is an idempotent element of the commutative ring
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$. We call it _the
idempotent associated_ to the given splitting datum. Note that the idempotents
$e_{i}$ associated to the direct summands of a direct sum decomposition
$X\simeq X_{1}\oplus\cdots\oplus X_{n}$ are orthogonal and their sum is
$1_{X}$.

Our last goal for this section is to show that a splitting datum is determined
up to equivalence by its associated idempotent (3.13). Let us first recall
that the “image” of a localization is similarly uniquely, essentially because
it is both the Kleisli object and the Eilenberg–Moore object of the associated
monad.

######  3.12 Lemma.

Let $I\colon Y\leftrightarrows X\\!:P$ and $I^{\prime}\colon
Y^{\prime}\leftrightarrows X\\!:P^{\prime}$ be two reflections of an object
$X$ in a bicategory $\mathcal{B}$, as in 3.7. Let $(T,\iota)$ and
$(T^{\prime},\iota^{\prime})$ be the associated localizations of $X$, as in
3.9. There is an isomorphism $T\cong T^{\prime}$ identifying $\iota$ with
$\iota^{\prime}$ if and only if there is an equivalence $Y\simeq Y^{\prime}$
identifying the two adjunctions.

###### Proof.

Write $\eta,\varepsilon$ and $\eta^{\prime},\varepsilon^{\prime}$ for the unit
and counit of $P\dashv I$ and $P^{\prime}\dashv I^{\prime}$, respectively. By
hypothesis $\varepsilon$ and $\varepsilon^{\prime}$ are invertible, and
suppose that $(T,\iota)\cong(T^{\prime},\iota^{\prime})$, i.e. that there is
an invertible 2-cell $\theta\colon IP\cong I^{\prime}P^{\prime}$ such that
$\theta\eta=\eta^{\prime}\colon\mathrm{Id}_{X}\Rightarrow
I^{\prime}P^{\prime}$. We claim that the composites $F:=P^{\prime}I\colon Y\to
Y^{\prime}$ and $G:=PI^{\prime}\colon Y^{\prime}\to Y$ are part of an adjoint
equivalence $Y\simeq Y^{\prime}$ in $\mathcal{B}$.

Indeed, from the triangular identity $(\varepsilon P)(P\eta)=\mathrm{id}_{P}$
and the invertibility of $\varepsilon$ we deduce that $P\eta\colon
P\overset{\sim}{\Rightarrow}PIP$ is invertible. From
$(P\theta)(P\eta)=P(\theta\eta)=P\eta^{\prime}$ we further deduce the
invertibility of $P\eta^{\prime}\colon P\Rightarrow PI^{\prime}P^{\prime}$,
hence of the composite

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0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 76.82068pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{PI^{\prime}P^{\prime}I=GF.}$}}}}}}}}}}}\ignorespaces$

By symmetry, we similarly obtain the isomorphism

$\beta\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.27112pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Id}_{Y^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.42235pt\raise 5.70694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.70694pt\hbox{$\scriptstyle{{\varepsilon^{\prime}}^{-1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.27112pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 34.27112pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{P^{\prime}I^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
57.95648pt\raise 6.62778pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.26668pt\hbox{$\scriptstyle{P^{\prime}\eta I^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 80.34067pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 80.34067pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{P^{\prime}IPI^{\prime}=FG.}$}}}}}}}}}}}\ignorespaces$

The commutative diagram of isomorphisms

(which uses a triangular identity of $P\dashv I$ and one of $P^{\prime}\dashv
I^{\prime}$) displays one of the triangular identities for an adjunction
$(F\dashv G,\alpha,\beta^{-1})$; the other one is similar.

We must still verify that this adjoint equivalence $F\colon Y\simeq
Y^{\prime}\\!:G$ matches the given adjunctions $P\dashv I$ and
$P^{\prime}\dashv I^{\prime}$. Indeed, we have the isomorphism
$P^{\prime}\varepsilon\colon
FP=P^{\prime}IP\overset{\sim}{\Rightarrow}P^{\prime}$, and its mate under
adjunction is an isomorphism $I^{\prime}\cong IG$. (One also checks
immediately that $F\dashv G$ is in fact an adjoint equivalence both in the
comma bicategory of $\mathcal{B}$ under $X$ and that over $X$.)

The converse implication is obvious. ∎

######  3.13 Proposition.

Let $X$ be an object of an additive bicategory. Let $I\colon Y\leftrightarrows
X\\!:P$ and $J\colon Z\leftrightarrows X\\!:Q$ be two splitting data for $X$
(3.6). Then the two are equivalent, as splitting data for $X$, if and only if
their associated idempotent elements of $\operatorname{End}(\mathrm{Id}_{X})$
are equal (3.11).

###### Proof.

Let us introduce the following notation for the units and counits of the four
adjunctions $I\dashv P\dashv I$ and $J\dashv Q\dashv J$:

$\displaystyle\eta_{Y}\colon\mathrm{Id}_{X}\Rightarrow
IP\quad\quad\varepsilon_{Y}\colon
PI\overset{\sim}{\Rightarrow}\mathrm{Id}_{Y}\quad\quad\overline{\eta}_{Y}\colon\mathrm{Id}_{Y}\overset{\sim}{\Rightarrow}PI\quad\quad\overline{\varepsilon}_{Y}\colon
IP\Rightarrow\mathrm{Id}_{X}$
$\displaystyle\eta_{Z}\colon\mathrm{Id}_{X}\Rightarrow
JQ\quad\quad\varepsilon_{Z}\colon
QJ\overset{\sim}{\Rightarrow}\mathrm{Id}_{Z}\quad\quad\overline{\eta}_{Z}\colon\mathrm{Id}_{Z}\overset{\sim}{\Rightarrow}QJ\quad\quad\overline{\varepsilon}_{Z}\colon
JQ\Rightarrow\mathrm{Id}_{X}$

Recall that $\varepsilon_{Y}=\overline{\eta}_{Y}^{-1}$,
$\varepsilon_{Z}=\overline{\eta}_{Z}^{-1}$,
$\eta_{Y}\overline{\varepsilon}_{Y}=\mathrm{id}_{IP}$ and
$\eta_{Z}\overline{\varepsilon}_{Z}=\mathrm{id}_{JQ}$ by definition. The two
associated idempotents are $e_{Y}:=\overline{\varepsilon}_{Y}\eta_{Y}$ and
$e_{Z}:=\overline{\varepsilon}_{Z}\eta_{Z}$.

Suppose first that there is an equivalence $Y\simeq Z$ of splitting data, i.e.
an equivalence $F\colon Y\overset{\sim}{\to}Z$ equipped with an isomorphism
$\theta\colon FP\overset{\sim}{\Rightarrow}Q$ in $\mathcal{B}$. We must show
that $e_{Y}=e_{Z}$. By taking mates, we easily obtain $I\cong JF$ and
therefore $IP\cong JQ$. But there is an ambiguity: We can take mates with
respect either to the left or the right adjunctions, and the resulting
isomorphism will either match $\eta_{Y}$ with $\eta_{Z}$ or
$\overline{\varepsilon}_{Y}$ with $\overline{\varepsilon}_{Z}$, accordingly,
but _a priori_ not both. Luckily, the strong bond between the left and the
right adjoint implies the two mates coincide. Let us be more precise.

The isomorphism $JF\cong I$ resulting from $I\dashv P$ amounts to the
composite

$\phi\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate}}}}\ignorespaces$

(this formally requires choosing an adjoint quasi-inverse for $F$, but then
the associated data gets canceled out.) Using $P\dashv I$ instead, we get the
isomorphism

$\psi\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}&{\immediate}{\immediate}{\immediate}\crcr}{\immediate}}}{\immediate}{\immediate}\ignorespaces{\immediate}{\immediate}\ignorespaces}\immediate\immediate}}}}\ignorespaces$

From $\eta_{Z}\overline{\varepsilon}_{Z}=\mathrm{id}$ and
$\varepsilon_{Y}=\overline{\eta}_{Y}^{-1}$, we immediately see that
$\psi\phi=\mathrm{id}_{JF}$. Hence $\phi$ and $\psi$ are mutually inverse.
(Incidentally, note that the composite $\phi\psi$ computes as $e_{Z}I\colon
I\Rightarrow I$, showing that $e_{Z}I=\mathrm{id}_{I}$.)

We thus obtain the isomorphism $(J\theta)(\psi P)\colon
IP\overset{\sim}{\Rightarrow}JQ$ with inverse $(\phi P)(J\theta^{-1})$. To
prove that $e_{Z}=e_{Y}$, it therefore suffices to show that

$(J\theta)(\psi P)\eta_{Y}=\eta_{Z}\quad\textrm{ and
}\quad\overline{\varepsilon}_{Y}(\phi
P)(J\theta^{-1})=\overline{\varepsilon}_{Z}.$

After expanding the definitions of $\phi$ and $\psi$, these equations
immediately follow from one of the triangular identities for the adjunction
$P\dashv I$ or $I\dashv P$, respectively.

To prove the converse implication, suppose now that $e_{Y}=e_{Z}$ for the two
given splitting data. We must find an equivalence between the latter.

First of all, since $T:=IP\cong\operatorname{Im}(e_{Y})$ and
$S:=JQ\cong\operatorname{Im}(e_{Z})$ are two images of the same morphism in
the additive category $\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$,
there exists an isomorphism $\theta\colon T\overset{\sim}{\Rightarrow}S$ such
that $\theta\eta_{Y}=\eta_{Z}$ and
$\overline{\varepsilon}_{Z}\theta=\overline{\varepsilon}_{Y}$. In particular,
$\theta$ is an isomorphism $(T,\eta_{Y})\cong(S,\eta_{Z})$ of the
localizations of $X$ induced by the adjunctions $P\dashv I$ and $Q\dashv J$.
By 3.12, there exists an equivalence $Y\overset{\sim}{\to}Z$ preserving these
adjunctions. It is in particular an equivalence of splitting data. ∎

######  3.14 Remark.

A similar treatment of categorified direct summands can be found in [DR18]. In
particular, our uniqueness result corresponds to [DR18, Prop. 1.3.4]; neither
result implies the other, strictly speaking, but the ideas are closely
related.

##  4\. Krull–Schmidt bicategories

We are now ready to prove our main results. Indeed, it will be a very easy
task because most of the work is already done.

######  4.1 Definition.

We say that a nonzero object $X$ of a locally additive bicategory
$\mathcal{B}$ is _strongly indecomposable_ if the commutative ring
$\operatorname{End}_{\mathcal{B}(X,X)}(\mathrm{Id}_{X})$ is connected.

Let us justify the use of “strongly”:

######  4.2 Remark.

Strongly indecomposable objects are indecomposable. Indeed, nontrivial
decompositions $X\simeq X_{1}\oplus X_{2}$ produce (as in 2.8) nontrivial
decompositions $1_{X}=e_{1}+e_{2}$ in orthogonal idempotents of
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$, where $e_{i}$ is the
idempotent associated to the direct summand $X_{i}$ as in 3.11.

######  4.3 Theorem.

Let $X$ be an object of an additive bicategory $\mathcal{B}$ which admits two
direct sum decompositions

$X\simeq Y_{1}\oplus\cdots\oplus Y_{n}\quad\textrm{ and }\quad X\simeq
Z_{1}\oplus\cdots\oplus Z_{m}$

such that all $Y_{k}$ and $Z_{\ell}$ are strongly indecomposable (4.1). Then
$n=m$ and, after renumbering the factors, there are equivalences $Y_{k}\simeq
Z_{k}$ for all $k$.

###### Proof.

By construction, the identities $1_{Y_{k}}$ ($k=1,\ldots,n$) correspond under
the given equivalences to a complete family of orthogonal primitive
idempotents $e_{k}$ in the commutative ring
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$. Here each $e_{k}$ is the
idempotent associated with the direct summand $Y_{k}$. Similarly, the direct
summands $Z_{\ell}$ yield a complete family of orthogonal primitive
idempotents $f_{\ell}$. But then clearly, this being a commutative ring, each
of the $e_{k}$ must be equal to exactly one of the $f_{\ell}$. Thus $n=m$ and
we find a permutation $\sigma$ such that $e_{k}=f_{\sigma(k)}$ for all $k$. It
follows by 3.13 that $Y_{k}$ and $Z_{\sigma(k)}$ are equivalent, even as
direct summands (i.e. splitting data) of $X$. ∎

######  4.4 Definition.

We say an additive bicategory is _Krull–Schmidt_ if every nonzero object
admits a direct sum decomposition into strongly indecomposable objects.

######  4.5 Remark.

In a weakly block complete bicategory (2.10), indecomposable objects and
strongly indecomposable objects coincide. Indeed, by definition, each
nontrivial decomposition of the 2-cell $1_{X}$ as a sum of orthogonal
idempotents arises from a nontrivial direct sum decomposition of the object
$X$.

######  4.6 Corollary.

In a Krull–Schmidt bicategory (4.4), indecomposable and strongly
indecomposable objects coincide, and every object decomposes in an essentially
unique way into a direct sum of indecomposable objects.

###### Proof.

Strongly indecomposable objects are indecomposable by 4.2. By hypothesis every
object $X$ admits a decomposition $X\simeq X_{1}\oplus\cdots\oplus X_{n}$ into
strongly indecomposables. If $X$ is indecomposable, $n=1$ hence $X\simeq
X_{1}$ is strongly indecomposable. The remaining statement now follows from
4.3. ∎

######  4.7 Definition.

As in the introduction, we say a commutative ring is _semiconnected_ if it is
a direct product of finitely many connected rings.

######  4.8 Theorem.

An additive bicategory is Krull–Schmidt if and only if it is weakly block
complete and every 2-cell endomorphism ring is semiconnected.

###### Proof.

Consider an idempotent 2-cell
$e\colon\mathrm{Id}_{X}\Rightarrow\mathrm{Id}_{X}$ on an object $X$ of a
Krull–Schmidt bicategory $\mathcal{B}$. Let $X\simeq X_{1}\oplus\cdots\oplus
X_{n}$ be a decomposition in strongly indecomposable objects $X_{k}$. This
equivalence induces a decomposition $1_{X}=e_{1}+\cdots+e_{n}$ in orthogonal
primitive idempotents, where
$e_{k}\in\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$ is the idempotent
associated to $X_{k}$. It follows that $e=\sum_{k\in J}e_{k}$ for some subset
$J\subseteq\\{1,\ldots,n\\}$. By 3.13, $X\simeq\big{(}\bigoplus_{k\in
J}X_{k}\big{)}\oplus\big{(}\bigoplus_{k\not\in J}X_{k}\big{)}$ realizes the
decomposition $1_{X}=e+(1_{X}-e)$ at the level of objects. This shows that
$\mathcal{B}$ is weakly block complete. Also, each ring
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X_{k}})\cong
e_{k}\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$ is connected as
$e_{k}$ is primitive.

Now suppose that $\mathcal{B}$ is weakly block complete and that for each
object $X$ there is a decomposition
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})\cong
R_{1}\times\cdots\times R_{n}$ in finitely many connected rings $R_{k}$. Then
$1_{X}$ is a sum of $n$ primitive idempotents and, by weak block completeness,
there follows a decomposition $X\simeq X_{1}\oplus\cdots\oplus X_{n}$ with
$\operatorname{End}(\mathrm{Id}_{X_{k}})\cong R_{k}$. In particular, each
$X_{k}$ is strongly indecomposable. This shows that $\mathcal{B}$ is
Krull–Schmidt. ∎

##  5\. Notions of strong indecomposability

In this section we compare various candidates for the notion of strong
indecomposable objects in an additive bicategory (5.4). This further justifies
our choice in 4.1, if needed, and relates it to its 1-categorical analog.

######  5.1 Lemma.

Suppose $A$ and $B$ are two objects in an additive category such that the
endomorphism ring $\operatorname{End}(A\oplus B)$ of their sum is commutative.
Then $A$ and $B$ are Hom-orthogonal and therefore $\operatorname{End}(A\oplus
B)=\operatorname{End}(A)\times\operatorname{End}(B)$.

###### Proof.

For $x\in\operatorname{Hom}(A,B)$ and $y\in\operatorname{Hom}(B,A)$, consider
the elements $u=[\,{}^{1}_{0}\,{}^{0}_{0}\,]$,
$v=[\,{}^{0}_{0}\,{}^{0}_{1}\,]$, $f=[\,{}^{0}_{x}\,{}^{0}_{0}\,]$ and
$g=[\,{}^{0}_{0}\,{}^{y}_{0}\,]$ of $\operatorname{End}(A\oplus B)$. By
commutativity, the calculations $vf=f$ and $fv=0$ imply $x=0$, whereas $ug=g$
and $gu=0$ imply $y=0$. ∎

######  5.2 Proposition.

Consider the following properties of a (nonzero) additive monoidal category
$\mathcal{A}$, i.e. a category which is additive and monoidal and whose tensor
functor $\otimes\colon\mathcal{A}\times\mathcal{A}\to\mathcal{A}$ is additive
in both variables:

  1. (1)

The commutative endomorphism ring
$\operatorname{End}_{\mathcal{A}}(\mathbb{1})$ of the tensor unit $\mathbb{1}$
is connected.

  2. (2)

The tensor unit object $\mathbb{1}$ is indecomposable in $\mathcal{A}$ for the
direct sum.

  3. (3)

If a sum of two objects of $\mathcal{A}$ is tensor-invertible, one of them
must be too.

  4. (4)

If $\mathcal{A}$ is equivalent as a monoidal category to a product
$\mathcal{A}_{1}\times\mathcal{A}_{2}$ of two additive monoidal categories,
then $\mathcal{A}_{1}\simeq 0$ or $\mathcal{A}_{2}\simeq 0$.

Then (1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(4), and if $\mathcal{A}$
is idempotent complete all four are equivalent.

###### Proof.

(1)$\Rightarrow$(2): Suppose that $\mathbb{1}\cong E_{1}\oplus E_{2}$. Since
$\operatorname{End}(\mathbb{1})$ is commutative, we have
$\operatorname{End}(\mathbb{1})\cong\operatorname{End}(E_{1})\times\operatorname{End}(E_{2})$
by 5.1. Assuming $\operatorname{End}(\mathbb{1})$ is connected, we must
therefore have $\operatorname{End}(E_{i})=0$ and thus $E_{i}\cong 0$ for $i=1$
or $i=2$.

(2)$\Rightarrow$(3): Suppose $A\oplus B$ has a two-sided tensor inverse $C$,
with $A,B,C\in\mathcal{A}$. Then $\mathbb{1}\cong(A\oplus B)\otimes
C\cong(A\otimes C)\oplus(B\otimes C)$, hence if $\mathbb{1}$ is indecomposable
we must have $\mathbb{1}\cong A\otimes C$ or $\mathbb{1}\cong B\otimes C$.
Since $C$ is invertible, either $A$ or $B$ must be too.

(3)$\Rightarrow$(4): Given an equivalence
$\mathcal{A}\simeq\mathcal{A}_{1}\times\mathcal{A}_{2}$ of additive monoidal
categories, there follows in $\mathcal{A}$ a decomposition $\mathbb{1}\cong
E_{1}\oplus E_{2}$, where $E_{i}$ corresponds to the tensor unit of
$\mathcal{A}_{i}$. Since $\mathbb{1}$ is tensor invertible in $\mathcal{A}$,
then by (3) either $E_{1}$ or $E_{2}$ must be as well. Say $E_{1}$ is, with
inverse $F$. Since $E_{1}\otimes\mathcal{A}_{2}\simeq 0$, it follows that
$\mathcal{A}_{2}=\mathbb{1}\otimes\mathcal{A}_{2}=F\otimes
E_{1}\otimes\mathcal{A}_{2}\simeq 0$. Thus (4) holds.

Finally, we show (4)$\Rightarrow$(1) assuming that $\mathcal{A}$ is idempotent
complete. Let $\mathrm{id}_{\mathbb{1}}=e_{1}+e_{2}$ for two orthogonal
idempotents $e_{1},e_{2}\colon\mathbb{1}\to\mathbb{1}$. For every object
$A\in\mathcal{A}$, write $e_{i}\\!\\!\downharpoonright_{A}$ for the
endomorphism of $A$ corresponding to $e_{i}\otimes A$ under the structural
isomorphism $A\cong\mathbb{1}\otimes A$. Note that
$e_{1}\\!\\!\downharpoonright_{A}$ and $e_{2}\\!\\!\downharpoonright_{A}$ are
orthogonal idempotents on $A$ with
$\mathrm{id}_{A}=e_{1}\\!\\!\downharpoonright_{A}+\;e_{2}\\!\\!\downharpoonright_{A}$.
By idempotent completion, we get a decomposition $A\cong A_{1}\oplus A_{2}$
where $A_{i}=\operatorname{Im}(e_{i}\\!\\!\downharpoonright_{A})$. One easily
checks that this direct sum decomposition is functorial in $A$ and gives rise
to a decomposition $\mathcal{A}\simeq\mathcal{A}_{1}\times\mathcal{A}_{2}$ of
$\mathcal{A}$ as an additive monoidal category, where $\mathcal{A}_{i}$ is the
full subcategory $\\{A\in\mathcal{A}\mid
e_{i}\\!\\!\downharpoonright_{A}=\mathrm{id}_{A}\\}$ of $\mathcal{A}$. By (4),
we must have for either $i$ that $\mathcal{A}_{i}\simeq 0$, that is that
$\operatorname{Im}(e_{i}\\!\\!\downharpoonright_{A})\cong 0$ for all $A$ and
thus $e_{i}=0$. This shows that the ring
$\operatorname{End}_{\mathcal{A}}(\mathbb{1})$ is connected. ∎

######  5.3 Remark.

In the situation of 5.2, we could also consider the following two properties
of the category $\mathcal{A}$ which do not depend on its monoidal structure:

  1. (1)′

The commutative ring $\operatorname{End}(\mathrm{Id}_{\mathcal{A}})$ of
natural transformations
$\mathrm{Id}_{\mathcal{A}}\Rightarrow\mathrm{Id}_{\mathcal{A}}$, i.e. the
categorical “center” of $\mathcal{A}$, is connected.

  2. (4)′

If $\mathcal{A}\simeq\mathcal{A}_{1}\times\mathcal{A}_{2}$ as an additive
category then $\mathcal{A}_{1}\simeq 0$ or $\mathcal{A}_{2}\simeq 0$.

One can verify that (1)′$\Rightarrow$(1), (4)′$\Rightarrow$(4) and
(1)′$\Rightarrow$(4)′ in general, and that (1)′$\Leftrightarrow$(4)′ if
$\mathcal{A}$ is idempotent complete. But (1)′ and (4)′ do not seem otherwise
relevant here.

The endomorphism category of any object in a locally additive bicategory can
be viewed as an additive monoidal category, with tensor product given by
horizontal composition. In this case, 5.2 translates as follows:

######  5.4 Corollary.

For any (nonzero) object $X$ of a locally additive bicategory $\mathcal{B}$,
we have implications (1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(4)
between the following properties, and all four are in fact equivalent if the
additive category $\operatorname{End}_{\mathcal{B}}(X)$ is idempotent
complete:

  1. (1)

The commutative endomorphism ring
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$ is connected.

  2. (2)

The identity 1-cell $\mathrm{Id}_{X}$ is indecomposable in the additive
category $\operatorname{End}_{\mathcal{B}}(X)$.

  3. (3)

If a sum of two 1-cells $X\to X$ is an equivalence, then one of them must be
too.

  4. (4)

$\operatorname{End}_{\mathcal{B}}(X)$ is indecomposable as an additive
monoidal category. ∎

######  5.5 Remark.

Property (3) is the evident 2-categorical analog of being a _strongly
indecomposable_ object $A$ of an additive (1-)category $\mathcal{A}$. Recall
the latter means that $\operatorname{End}_{\mathcal{A}}(A)$ is a local ring,
that is: If a sum $f+g$ of two morphisms $f,g\colon A\to A$ is invertible then
either $f$ or $g$ is invertible. Since we do not wish to assume idempotent
completion of our Hom categories, we are left with the choice of at least four
reasonable 2-categorical analogs of strong indecomposability. In 4.1, we chose
the strongest and most useful one. It is also the only one which always
implies the usual indecomposability and which is stable under (weak) block
completion.

##  6\. Direct comparison of direct summands

In this section we explain how to categorify the usual proof of the uniqueness
of decompositions in strongly indecomposables. This does not use the
characterization of direct summands via idempotents (3.13), but instead
requires us to understand how to “cut down” a direct summand by another (6.2).
It is a considerably harder task than its 1-categorical analog, which is a
triviality.

######  6.1 Lemma.

Suppose that $\mathcal{B}$ is weakly block complete, and that a 1-cell
$E\colon X\to X$ is part of an idempotent monad $(E,\mu,\iota)$ and of an
idempotent comonad $(E,\delta,\epsilon)$ such that
$\iota\epsilon=\mathrm{id}_{E}$. Then there is a decomposition $X\simeq
Y\oplus Z$ such that the direct summand $Y$ induces the given monad and
comonad as in 3.8.

###### Proof.

Since $\iota\epsilon=\mathrm{id}_{E}$, the 2-cell
$e:=\epsilon\iota\colon\mathrm{Id}_{X}\Rightarrow E\Rightarrow\mathrm{Id}_{X}$
is an idempotent element of the ring
$\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$, and so is $f:=1_{X}-e$.
By weak block completion, we deduce a direct sum decomposition $X\simeq
Y\oplus Z$ such that $\mathrm{Id}_{Y}$ and $\mathrm{Id}_{Z}$ correspond to
$\operatorname{Im}(e)$ and $\operatorname{Im}(f)$ respectively. Moreover,
since $\mathrm{Id}_{X}\cong\operatorname{Im}(e)\oplus\operatorname{Im}(f)$ and
since $\epsilon$ is a split mono by hypothesis, we have
$\operatorname{Im}(e)=\operatorname{Im}(\iota)\cong E$ in the category
$\operatorname{End}_{\mathcal{B}}(X)$, compatibly with the respective
structure 2-cells. We can now easily conclude. ∎

######  6.2 Proposition.

Let $Y$ and $Z$ be two direct summands (for two different decompositions) of
an object $X$ in a weakly block complete additive bicategory $\mathcal{B}$,
with structural 1-cells denoted respectively by

$Y\overset{I}{\longrightarrow}X\overset{P}{\longrightarrow}Y\quad\textrm{ and
}\quad Z\overset{J}{\longrightarrow}X\overset{Q}{\longrightarrow}Z.$

Suppose that $PJQI$ is an equivalence $Y\overset{\sim}{\to}Y$. Then $QI\colon
Y\to Z$ and $PJ\colon Z\to Y$ are part of a direct sum decomposition $Z\simeq
Y\oplus Y^{\prime}$.

###### Proof.

By 6.1, it suffices to show that $E:=(QI)(PJ)$ is both part of an idempotent
monad $(E,\mu,\iota)$ and an idempotent comonad $(E,\delta,\epsilon)$ such
that $\iota\epsilon=\mathrm{id}_{E}$. Suppose $P,I$ and $Q,J$ are part of two
splitting data for $X$ as usual, and write the units and counits of the four
adjunctions $I\dashv P\dashv I$ and $J\dashv Q\dashv J$ as follows:

$\displaystyle\eta_{Y}\colon\mathrm{Id}_{X}\Rightarrow
IP\quad\quad\varepsilon_{Y}\colon
PI\overset{\sim}{\Rightarrow}\mathrm{Id}_{Y}\quad\quad\overline{\eta}_{Y}\colon\mathrm{Id}_{Y}\overset{\sim}{\Rightarrow}PI\quad\quad\overline{\varepsilon}_{Y}\colon
IP\Rightarrow\mathrm{Id}_{X}$
$\displaystyle\eta_{Z}\colon\mathrm{Id}_{X}\Rightarrow
JQ\quad\quad\varepsilon_{Z}\colon
QJ\overset{\sim}{\Rightarrow}\mathrm{Id}_{Z}\quad\quad\overline{\eta}_{Z}\colon\mathrm{Id}_{Z}\overset{\sim}{\Rightarrow}QJ\quad\quad\overline{\varepsilon}_{Z}\colon
JQ\Rightarrow\mathrm{Id}_{X}$

Thus we have $\varepsilon_{Y}=(\overline{\eta}_{Y})^{-1}$ and
$\eta_{Y}\overline{\varepsilon}_{Y}=\mathrm{id}_{IP}$, and similarly for $Z$.
Then the two composite adjunctions $PJ\dashv QI\dashv PJ$ have the following
units and counits:

$\displaystyle\eta\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{QJ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.8864pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 78.10748pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{QIPJ}$}}}}}}}}}}}\ignorespaces\quad\quad\quad\quad\varepsilon\colon\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
16.70103pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
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0.0pt\hbox{$\textstyle{PJQI\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.31303pt\raise 6.52222pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15555pt\hbox{$\scriptstyle{P\,\overline{\varepsilon}_{Z}I}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.70103pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 40.70103pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{PI\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
65.45294pt\raise 5.19028pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\varepsilon_{Y}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
65.96837pt\raise-4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.6906pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 83.6906pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Id}_{Y}}$}}}}}}}}}}}\ignorespaces$
$\displaystyle\overline{\eta}\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.83112pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
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0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15555pt\hbox{$\scriptstyle{\overline{\eta}_{Y}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.10889pt\raise-4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.83112pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 33.83112pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{PI\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
53.19484pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\raise-1.70833pt\hbox{$\scriptstyle{P\,\eta_{Z}I}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 76.82068pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{PJQI}$}}}}}}}}}}}\ignorespaces\quad\quad\quad\quad\overline{\varepsilon}\colon\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
16.70103pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
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0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.70103pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{QJ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
67.39128pt\raise-4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Id}_{Z}}$}}}}}}}}}}}\ignorespaces$

The monad $(E,\mu,\iota)$ and the comonad $(E,\delta,\epsilon)$ on $Z$ are
those induced by $PJ\dashv QI$ and $QI\dashv PJ$, respectively. In particular
$\iota=\eta$ and $\epsilon=\overline{\varepsilon}$, so that the required
relation $\iota\epsilon=\eta\overline{\varepsilon}=\mathrm{id}_{QIPJ}$
immediately follows from $\varepsilon_{Z}=(\overline{\eta}_{Z})^{-1}$ and
$\eta_{Y}\overline{\varepsilon}_{Y}=\mathrm{id}_{IP}$.

It remains to verify that this monad and this comonad are idempotent. For the
monad, recall that by definition its multiplication is
$\mu=(QI)\varepsilon(PJ)$, that is

$\mu\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 69.027pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 99.027pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{QIPIPJ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
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156.59026pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
166.6964pt\raise-4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1.5}\lx<EMAIL_ADDRESS>0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 199.41862pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{QIPJ=E}$}}}}}}}}}}}\ignorespaces,$

hence we must show that $QIP\,\overline{\varepsilon}_{Z}IPJ$ is invertible.

In fact, we can show that $P\,\overline{\varepsilon}_{Z}I$ is already
invertible. Indeed, the composite $PJQI$ is an equivalence by hypothesis. It
follows that the composite adjunction $PJQI\dashv PJQI$ is an adjoint
equivalence. In particular, its counit

$\alpha\colon\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
0.0pt\hbox{\ignorespaces\immediate\immediate\immediate{\ignorespaces{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}{\immediate}\hbox{\vtop{{\immediate}\kern
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is an isomorphism. Thus $P\,\overline{\varepsilon}_{Z}I$ is a split epi, since
$(PJ\varepsilon_{Z}QI)(PJQ\,\overline{\varepsilon}_{Y}JQI)\alpha^{-1}\varepsilon_{Y}$
provides a section for it. But $P\,\overline{\varepsilon}_{Z}I$ is also a
split mono, because $\overline{\varepsilon}_{Z}$ is a split mono by
construction. Therefore $P\,\overline{\varepsilon}_{Z}I$ is invertible as
claimed.

The proof that the comonad is idempotent is similar and is left to the reader.
∎

###### Alternative proof of 4.3.

We start by replacing $\mathcal{B}$ with its block completion
$\mathcal{B}^{\flat}$ as in 2.11. Note that the embedding
$\mathcal{B}\hookrightarrow\mathcal{B}^{\flat}$ preserves direct sums (because
it is an additive pseudofunctor) and strongly indecomposable objects (because
it is 2-fully faithful). Also by being 2-fully faithful, it reflects whether
two given 1-cells of $\mathcal{B}$ form (adjoint) equivalences, as we are
setting out to establish. Thus we may and will assume that $\mathcal{B}$ is
block complete.

For all $k$ and $\ell$, let

$Y_{k}\overset{I_{k}}{\longrightarrow}X\overset{P_{k}}{\longrightarrow}Y_{k}\quad\textrm{
and }\quad
Z_{\ell}\overset{J_{\ell}}{\longrightarrow}X\overset{Q_{\ell}}{\longrightarrow}Z_{\ell}$

denote the inclusion and projection 1-cells of the two given direct sum
decompositions of $X$, and let $E_{k}:=I_{k}P_{k}$ and
$F_{\ell}:=J_{\ell}Q_{\ell}$ be the corresponding localizations of $X$. The
isomorphism

$\mathrm{Id}_{X}\cong F_{1}\oplus\ldots\oplus F_{s}$

in $\operatorname{End}_{\mathcal{B}}(X)$ induces an isomorphism

$\mathrm{Id}_{Y_{1}}\cong P_{1}I_{1}\cong P_{1}(F_{1}\oplus\ldots\oplus
F_{s})I_{1}\cong P_{1}F_{1}I_{1}\oplus P_{1}F_{2}I_{1}\oplus\ldots\oplus
P_{1}F_{s}I_{1}$

in $\operatorname{End}_{\mathcal{B}}(Y_{1})$. Since $Y_{1}$ is strongly
indecomposable, by 5.4 (3) we can find an $\ell\in\\{1,\ldots,s\\}$ such that
$P_{1}F_{\ell}I_{1}$ is an equivalence $Y_{1}\overset{\sim}{\to}Y_{1}$. By
renumbering the $Z_{\ell}$’s, we may assume that $\ell=1$. Thus

(6.3) $P_{1}F_{1}I_{1}\cong(P_{1}J_{1})(Q_{1}I_{1})$

is an equivalence. By 6.2 and the (weak) block completeness of $\mathcal{B}$,
the two composites $Q_{1}I_{1}\colon Y_{1}\to Z_{1}$ and $P_{1}J_{1}\colon
Z_{1}\to Y_{1}$ are part of a direct sum decomposition $Z_{1}\simeq
Y_{1}\oplus Y_{1}^{\prime}$. As $Z_{1}$ is strongly indecomposable it is also
indecomposable (4.2), hence we must have $Y_{1}^{\prime}\simeq 0$, that is
$Q_{1}I_{1}$ and $P_{1}J_{1}$ are in fact an adjoint equivalence $Y_{1}\simeq
Z_{1}$ of direct summands of $X$.

Note that, by construction, the latter equivalence is the component $Y_{1}\to
Z_{1}$ of the given composite equivalence (whose components are precisely the
$Q_{\ell}I_{k}$)

(6.4) $Y_{1}\oplus\underbrace{Y_{2}\oplus\ldots\oplus
Y_{n}}_{=:\,Y^{\prime}}\overset{\sim}{\to}X\overset{\sim}{\to}Z_{1}\oplus\underbrace{Z_{2}\oplus\ldots\oplus
Z_{m}}_{=:\,Z^{\prime}}\,.$

We claim that (6.4) similarly restricts to an equivalence $Y^{\prime}\simeq
Z^{\prime}$, after which the conclusion of the theorem will follow by a
routine inductive argument. To prove the claim, note that the equivalence
$Y_{1}\overset{\sim}{\to}Z_{1}$ identifies (up to isomorphism) $I_{1}$ with
$J_{1}$ and $P_{1}$ with $Q_{1}$. In particular we have $Q_{\ell}I_{1}\cong 0$
and $P_{1}J_{\ell}\cong 0$ for $\ell\neq 1$, which means that (6.4) has a
diagonal $2\times 2$ matrix form with respect to the sum decompositions
$Y_{1}\oplus Y^{\prime}$ and $Z_{1}\oplus Z^{\prime}$. As (6.4) is an
equivalence, it follows that the second diagonal component $Y^{\prime}\to
Z^{\prime}$ is also an equivalence, as claimed. ∎

##  7\. Examples

We finally consider some examples of Krull–Schmidt bicategories, concentrating
on various kinds of “2-representation theory”. We begin with a few
generalities:

######  7.1 Proposition.

Suppose $\mathcal{B}$ is an additive bicategory where the 2-cell endomorphism
ring $\operatorname{End}_{\mathcal{B}}(\mathrm{Id}_{X})$ of every object $X$
is semiconnected (4.7). Then both its block completion $\mathcal{B}^{\flat}$
(2.11) and its weak block completion $\mathcal{B}^{w\flat}$ (2.12) are
Krull–Schmidt bicategories.

###### Proof.

The block completion and weak block completion are both weakly block complete
bicategories. Moreover, the canonical embeddings
$\mathcal{B}\hookrightarrow\mathcal{B}^{\flat}$ and
$\mathcal{B}\hookrightarrow\mathcal{B}^{w\flat}$ are 2-fully faithful
pseudofunctors, and by construction the 2-cell endomorphism ring of every
object in either completion is a direct factor of that of some object of
$\mathcal{B}$ (cf. 2.8). Since direct factors of semiconnected rings are
semiconnected, we conclude with the characterization of 2.11. ∎

Clearly not every commutative ring is semiconnected (consider
$R=\prod_{\mathbb{N}}\mathbb{Z}$), hence we need some criteria. The following
two lemmas already go a long way:

######  7.2 Lemma.

Every noetherian commutative ring is semiconnected.

###### Proof.

This can be deduced from the fact that the spectrum of a noetherian ring is a
finite union of irreducible closed subsets ([Har77, Prop. I.1.5 and Ex.
II.2.13]): The latter being connected, there are only finitely many connected
components. ∎

######  7.3 Lemma.

Unital subrings of semiconnected rings are semiconnected.

###### Proof.

This boils down to injective morphisms of commutative rings inducing maps with
dense image between Zariski spectra ([AM69, Ch. 1 Ex. 21(v)]): In particular,
they induce surjective maps between sets of connected components. ∎

The following criterion follows immediately from 7.2 and 4.8.

######  7.4 Corollary.

Suppose $\mathcal{B}$ is a weakly block complete bicategory which is
$\Bbbk$-linear over some noetherian commutative ring $\Bbbk$ and such that
every 2-cell Hom space is finitely generated as a $\Bbbk$-module. Then
$\mathcal{B}$ is a Krull–Schmidt bicategory. ∎

We now turn to more concrete situations, beginning with a non-example:

######  7.5 Example.

Recall from 2.6 the additive 2-category $\mathsf{ADD}$ of additive categories.
Consider its 2- and 1-full additive sub-bicategory
$\mathsf{ADD}^{\mathsf{ic}}$ of idempotent complete categories: It is block
complete (see [BD20, A.7.18]) but not Krull–Schmidt, because it contains
$\mathcal{A}=\operatorname{Mod}R$ for rings $R$ whose center
$\mathrm{Z}(R)\cong\operatorname{End}(\mathrm{Id}_{\mathcal{A}})$ is not
semiconnected. Here, and again below, we use the well-known identification of
the center of a ring with the center
$\operatorname{End}(\mathrm{Id}_{\operatorname{Mod}R})$ of its module
category.

######  7.6 Example.

Consider the full 2-subcategory $\mathcal{B}$ of $\mathsf{ADD}^{\mathsf{ic}}$
whose objects are the idempotent complete additive categories $\mathcal{A}$
such that $\operatorname{End}(\mathrm{Id}_{\mathcal{A}})$ is semiconnected.
Then $\mathcal{B}$ is Krull–Schmidt. Indeed, if
$1_{\mathcal{A}}=e_{1}+e_{2}\in\operatorname{End}(\mathrm{Id}_{\mathcal{A}})$
for such a category $\mathcal{A}$ with $e_{1},e_{2}$ two orthogonal
idempotents, we get a corresponding decomposition
$\mathcal{A}\simeq\mathcal{A}_{1}\oplus\mathcal{A}_{2}$ in
$\mathsf{ADD}^{\mathsf{ic}}$, as the latter is weakly block complete. But then
$\operatorname{End}(\mathrm{Id}_{\mathcal{A}})=R_{1}\times R_{2}$ with
$R_{k}\cong\operatorname{End}(\mathrm{Id}_{\mathcal{A}_{k}})$, hence the
categories $\mathcal{A}_{k}$ are again in $\mathcal{B}$ since direct factors
of semiconnected rings are semiconnected. Thus $\mathcal{B}$ is weakly block
complete and we conclude with 4.8.

######  7.7 Example.

Many variations of the bicategory $\mathcal{B}$ in 7.6 are possible. For
instance, we may consider the full subbicategory
$\mathcal{B}^{\prime}\subset\mathcal{B}$ of module categories
$\mathcal{A}=\operatorname{Mod}R$ over rings $R$ whose center is
semiconnected. (Indeed, if
$\operatorname{Mod}R\cong\mathcal{A}_{1}\oplus\mathcal{A}_{2}$ in
$\mathsf{ADD}^{\mathsf{ic}}$ for idempotents
$e_{1},e_{2}\in\operatorname{End}(\mathrm{Id}_{\operatorname{Mod}R})\cong\mathrm{Z}(R)$,
we have $\mathcal{A}_{k}\simeq\operatorname{Mod}R_{k}$ for the rings
$R_{k}:=e_{k}Re_{k}=Re_{k}$ ($k=1,2$). Moreover, each $\mathrm{Z}(R_{k})$ is
semiconnected since $\mathrm{Z}(R)$ is and
$\mathrm{Z}(R)\cong\mathrm{Z}(R_{1})\times\mathrm{Z}(R_{2})$.) Similarly, we
may further restrict to rings $R$ with noetherian center (by 7.2) and/or
replace $\operatorname{Mod}R$ with the category of finitely
presented/generated modules, or projective modules, finitely generated
projectives, etc., indeed any option for which the previous splitting argument
still goes through. We may also take module categories over finite dimensional
algebras $R$ over a fixed field $\Bbbk$ and $\Bbbk$-linear functors between
them, since the center of a finite dimensional algebra is again finite
dimensional hence noetherian.

######  7.8 Example.

Consider the bicategory of rings, bimodules and bimodules maps, and its sub-
bicategory $\mathcal{B}^{\prime\prime}$ of rings whose center is
semiconnected. Then $\mathcal{B}^{\prime\prime}$ is Krull–Schmidt. To see
this, just recall that by the Eilenberg-Watts theorem we may identify
$\mathcal{B}^{\prime\prime}$ with the 2-full sub-bicategory of
$\mathcal{B}^{\prime}\subset\mathsf{ADD}^{\mathsf{ic}}$ as in 7.7 having the
same objects but where the only 1-cells are the colimit-preserving functors.
Since all functors involved in direct sum decompositions are colimit-
preserving (they all have two-sided adjoints), the same object decompositions
of $\mathcal{B}^{\prime}$ equally work in $\mathcal{B}^{\prime\prime}$, hence
$\mathcal{B}^{\prime\prime}$ is also weakly block complete. As before, we may
replace “semiconnected” with “noetherian”, or we may work over a fixed
$\Bbbk$.

Our original motivation for this article is the following application:

######  7.9 Corollary.

Let $\Bbbk$ be any noetherian commutative ring. Every bicategory of
$\Bbbk$-linear _Mackey 2-motives_ in the sense of [BD20] is Krull–Schmidt.
Similarly, every bicategory of _cohomological_ Mackey 2-motives as in [BD21]
is Krull–Schmidt. By 4.3, we deduce the uniqueness of the motivic
decomposition of every finite group in each of these bicategories.

###### Proof.

These bicategories are constructed as block completions, hence in particular
they are weakly block complete. Moreover their 2-cell Hom spaces are easily
seen to be finitely generated free $\Bbbk$-modules. We may thus apply 7.1 or
7.4. (The cohomological case is also a variant of 7.8.) ∎

$*\;*\;*$

From now on we look at various flavors of linear 2-representation theory,
starting with some commonly used target 2-categories for said
2-representations:

######  7.10 Example.

Let $\Bbbk$ be any field. Consider the 2-category $\mathsf{2FVect}_{\Bbbk}$ of
_finite dimensional 2-vector spaces_ (over $\Bbbk$) in the sense of
Kapranov–Voevodsky [KV94] and Neuchl [Neu]. Recall that
$\mathsf{2FVect}_{\Bbbk}$ can be defined as the $\Bbbk$-linear 2-category of
finite semisimple $\Bbbk$-linear categories, $\Bbbk$-linear functors and
natural transformations. Here a $\Bbbk$-linear category is _finite semisimple_
if it is abelian and there is a finite set of absolutely simple objects
(objects with one-dimensional endomorphism algebra) such that every other
object is a direct sum of these in an essentially unique way. (Each finite
semisimple category is in fact equivalent to $\mathrm{vect}_{\Bbbk}^{n}$ for
some $n$.) One easily checks that $\mathsf{2FVect}_{\Bbbk}$ is block complete
and all its 2-cell Hom $\Bbbk$-vector spaces are finite dimensional. In
particular it is Krull–Schmidt by 7.4.

######  7.11 Example.

Consider the 2-category 2FHilb of _finite-dimensional 2-Hilbert spaces_ in the
sense of Baez [Bae97] (see also [HV19]). Recall that a 2-Hilbert space is a
dagger category which is enriched on finite dimensional Hilbert spaces
(compatibly with adjoints and inner products) and has finite direct sums and
split idempotents; it is finite dimensional if it admits a finite set of
simple objects generating all objects under finite direct sums. Then 2FHilb is
Krull–Schmidt. Indeed, there is a 2-functor
$\textsf{2FHilb}\overset{\sim}{\to}\textsf{2FVect}_{\mathbb{C}}$ forgetting
daggers and inner products which is a biequivalence of bicategories (this is
analogous to the equivalence between the 1-categories of finite dimensional
Hilbert spaces and finite dimensional complex vector spaces).

######  7.12 Example.

Let $\Bbbk$ be a field. A $\Bbbk$-linear category is _finite_ if it is
equivalent to that of finite dimensional representations of some finite
dimensional associative unital $\Bbbk$-algebra. The 2-category
$\mathsf{CAT}_{\Bbbk}^{\mathsf{fin}}$ of finite $\Bbbk$-linear categories,
$\Bbbk$-linear functors and natural transformations is Krull–Schmidt; it is
indeed a variation of 7.7.

######  7.13 Example.

For a field $\Bbbk$, consider the 2-category
$\mathsf{TRI}_{\Bbbk}^{\mathsf{fin}}$ whose objects are triangulated
categories equivalent to the bounded derived category of some finite
$\Bbbk$-linear category (7.12), triangulated functors and triangulated natural
transformations. We leave it as an exercise to verify that
$\mathsf{TRI}_{\Bbbk}^{\mathsf{fin}}$ is Krull–Schmidt.

The next result will essentially take care of all our remaining examples:

######  7.14 Theorem.

Let $\Bbbk$ be any commutative ring (for instance $\Bbbk=\mathbb{Z}$ or a
field). Consider the $\Bbbk$-linear bicategory
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$ of
$\Bbbk$-linear pseudofunctors between two $\Bbbk$-linear bicategories
$\mathcal{B}$ and $\mathcal{C}$ (2.13). Then
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$ is
Krull–Schmidt if $\mathcal{C}$ is Krull–Schmidt and $\mathcal{B}$ has finitely
many equivalence classes of objects.

###### Proof.

By replacing $\mathcal{B}$ with a biequivalent sub-bicategory if necessary, we
may assume it has finitely many objects. If $\mathcal{C}$ is weakly block
complete then so is
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$ by 2.14. Now
recall that for (non-necessarily $\Bbbk$-linear) pseudofunctors
$\mathcal{F}_{1},\mathcal{F}_{2}\colon\mathcal{B}\to\mathcal{C}$ and
transformations $t,s\colon\mathcal{F}_{1}\Rightarrow\mathcal{F}_{2}$, a
modification $M\colon t\Rrightarrow s$ consists of a family $\\{M_{X}\\}_{X}$
of 2-cells $M_{X}\colon t_{X}\Rightarrow s_{X}$ of $\mathcal{C}$ indexed by
the objects $X$ of $\mathcal{B}$, satisfying some relations. In particular,
for every $\Bbbk$-linear pseudofunctor
$\mathcal{F}\colon\mathcal{B}\to\mathcal{C}$ the ring
$\operatorname{End}(\mathrm{Id}_{\mathcal{F}})$ of self-modifications of the
identity transformation is a unital subring of the product
$\prod_{X\in\operatorname{Obj}\mathcal{B}}\operatorname{End}_{\mathcal{C}(\mathcal{F}X,\mathcal{F}X)}(\mathrm{Id}_{\mathcal{F}X})$.
Being a product of finitely many semiconnected rings, the latter is
semiconnected. Hence so is $\operatorname{End}(\mathrm{Id}_{\mathcal{F}})$ by
7.3. Once again, we conclude with 4.8. ∎

######  7.15 Remark.

If $\mathcal{B}$ and $\mathcal{C}$ are 2-categories with $\mathcal{C}$
Krull–Schmidt, the same proof (including that of 2.14) shows that the
2-category
$\operatorname{\mathsf{2Fun}}Fun^{\mathsf{str}}_{\Bbbk}(\mathcal{B},\mathcal{C})$
of _strict_ 2-functors, _strict_ transformations (i.e. 2-natural
transformations) and modifications is also Krull–Schmidt. In fact, if the
2-category $\mathcal{B}$ is small (which we had better assume anyway!), it
should be possible to show that the 2-full inclusion
$\operatorname{\mathsf{2Fun}}Fun^{\mathsf{str}}_{\Bbbk}(\mathcal{B},\mathcal{C})\subset\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathcal{B},\mathcal{C})$
is a biequivalence by adapting to the $\Bbbk$-linear case the coherence
results for 2-monads due to Power and Lack; see [Pow89] [Lac02] [Lac07].

A _$\Bbbk$ -linear tensor category_ is a monoidal categories whose tensor
product is a $\Bbbk$-linear functor of both variables. For a fixed such tensor
category $\mathcal{A}$, we may consider the 2-category
$\operatorname{\mathsf{PsMod}}\mathcal{A}$ whose objects are _( $\Bbbk$-linear
left) module categories over $\mathcal{A}$_, whose 1-morphisms are _module
functors_ and 2-morphisms _module natural transformations_ ; see e.g. [Gre10]
for detailed definitions. (These notions specialize those of pseudomonoids and
their left pseudomodules [DS97] [McC00] to the symmetric monoidal 2-category
$\mathsf{CAT}_{\Bbbk}$ of $\Bbbk$-linear categories, $\Bbbk$-linear functors
and natural transformations.) In fact:

######  7.16 Lemma.

For any $\Bbbk$-linear monoidal category $\mathcal{A}$, let
$\mathrm{B}\mathcal{A}$ denote its delooping, i.e. the bicategory with a
single object having $\mathcal{A}$ as monoidal endo-category. Then there is a
canonical biequivalence (actually an isomorphism) of $\Bbbk$-linear
2-categories between $\operatorname{\mathsf{PsMod}}\mathcal{A}$ as defined
above and
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathrm{B}\mathcal{A},\mathsf{CAT}_{\Bbbk})$
(2.13).

###### Proof.

The exponential law yields a bijection between objects of the 2-categories,
with a (coherently associative and unital) pseudoaction
$-\odot-\colon\mathcal{A}\otimes_{\Bbbk}\mathcal{M}\to\mathcal{M}$
corresponding to a strong monoidal functor
$\mathcal{A}\to\operatorname{End}_{\Bbbk}(\mathcal{M})$, that is to a
pseudofunctor $\mathrm{B}\mathcal{A}\to\mathsf{CAT}_{\Bbbk}$ sending the
unique object of $\mathrm{B}\mathcal{A}$ to $\mathcal{M}$. It is immediate
from the definitions that module functors
$F\colon\mathcal{M}_{1}\to\mathcal{M}_{2}$ and module natural transformations
$\varphi\colon F\Rightarrow F^{\prime}$ correspond to pseudonatural
transformations and modifications. ∎

######  7.17 Remark.

By replacing $\mathsf{CAT}_{\Bbbk}$ with a full 2-subcategory
$\mathsf{ADD}_{\Bbbk}$, $\mathsf{ADD}^{\mathsf{ic}}_{\Bbbk}$,
$\mathsf{2FVect}_{\Bbbk}$ etc. in the above, we can restrict attention to
those tensor categories and their module categories which are additive,
idempotent complete etc., as appropriate.

We conclude by specializing 7.14 to some examples of interest.

######  7.18 Corollary.

Let $\mathcal{A}$ be any finite (multi-) tensor category in the sense of
Etingof–Ostrik [Ost03] [EO04], over an algebraically closed field $\Bbbk$.
Then the 2-category of finite module categories over $\mathcal{A}$ is
Krull–Schmidt, as well as its 2-subcategory of exact module categories.

###### Proof.

A finite tensor (or “multi-tensor”) category $\mathcal{A}$ is in particular a
$\Bbbk$-linear tensor category in our sense above; it is one which is _finite_
as in 7.12. Since the 2-subcategory
$\mathsf{CAT}_{\Bbbk}^{\mathsf{fin}}\subset\mathsf{CAT}_{\Bbbk}$ of finite
$\Bbbk$-linear categories is Krull–Schmidt, so is
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathrm{B}\mathcal{A},\mathsf{CAT}_{\Bbbk}^{\mathsf{fin}})$
by 7.14. We conclude with 7.16 and 7.17 that finite $\mathcal{A}$-module
categories form a Krull–Schmidt 2-category.

By definition, a module category $\mathcal{M}$ is _exact_ if $P\odot M$ is a
projective object in $\mathcal{M}$ for every $M\in\mathcal{M}$ and every
projective $P\in\mathcal{A}$. Over a finite $\mathcal{A}$, exact module
categories are finite ([Ost03, Lemma 3.4]) and are clearly closed under taking
direct summands, hence they form a full 2-subcategory which is also
Krull–Schmidt. ∎

######  7.19 Corollary.

Every finite semisimple 2-category in the sense of Douglas–Reutter [DR18] is
Krull–Schmidt, and its indecomposable and simple objects coincide.

###### Proof.

As in [DR18], here we work over an algebraically closed field $\Bbbk$ of
characteristic zero. Then by [DR18, Thms. 1.4.8-9], the 2-category of finite
(left) semisimple module categories over a multifusion category $\mathcal{A}$
is an example of a finite semisimple 2-category, and the latter are all of
this form up to biequivalence. (The cited result is formulated for right
module categories, but it holds equally well for left ones because multifusion
categories are stable under reversing their tensor product). By definition, a
multifusion category is just a finite semisimple tensor category, thus in
particular a $\Bbbk$-linear tensor category $\mathcal{A}$ as above. By 7.16,
with $\mathsf{CAT}_{\Bbbk}$ replaced by $\mathsf{2FVect}_{\Bbbk}$ (7.17), we
see that the 2-category of finite semisimple module categories over
$\mathcal{A}$ is biequivalent to
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\mathrm{B}\mathcal{A},\mathsf{2FVect}_{\Bbbk})$.
Since $\mathsf{2FVect}_{\Bbbk}$ is Krull–Schmidt (7.10), we can therefore
conclude with 7.14.

It is easily checked that simples and indecomposables agree. ∎

######  7.20 Remark.

We found it satisfying to derive 7.19 from 7.14, but this is overkill: More
directly, and more generally, it follows from the results in [DR18] that all
_locally finite_ semisimple 2-categories are also Krull–Schmidt.

Recall now that a _2-groupoid_ is a bicategory in which every 1-cell is an
equivalence and every 2-cell an isomorphism. It is a _2-group_ if it has a
single object, i.e. if it is the delooping of a monoidal groupoid whose
objects are tensor-invertible.

######  7.21 Corollary.

Let $\Bbbk$ be any field. Let $\mathcal{G}$ be a 2-groupoid with only finitely
many equivalence classes of objects (e.g. a 2-group). Then the bicategory

$\mathsf{2FRep}_{\Bbbk}\mathcal{(}\mathcal{G}):=\operatorname{\mathsf{PsFun}}(\mathcal{G},\mathsf{2FVect}_{\Bbbk})$

of finite dimensional $\Bbbk$-linear 2-representations of $\mathcal{G}$ is
Krull–Schmidt.

###### Proof.

Consider the (easily constructed) free $\Bbbk$-linearization of $\mathcal{G}$,
which is a $\Bbbk$-linear bicategory $\Bbbk\mathcal{G}$ with the same objects
as $\mathcal{G}$ and equipped with a pseudofunctor
$\mathcal{G}\to\Bbbk\mathcal{G}$ inducing a biequivalence
$\operatorname{\mathsf{PsFun}}_{\Bbbk}(\Bbbk\mathcal{G},\mathcal{C})\overset{\sim}{\to}\operatorname{\mathsf{PsFun}}(\mathcal{G},\mathcal{C})$
for every $\Bbbk$-linear bicategory $\mathcal{C}$. In particular, for
$\mathcal{C}=\mathsf{2FVect}_{\Bbbk}$ we obtain a biequivalence between
$\mathsf{2FRep}_{\Bbbk}\mathcal{(}\mathcal{G})$ and a Krull–Schmidt bicategory
as in 7.14. ∎

######  7.22 Remark.

By varying the 2-groupoid $\mathcal{G}$, we can specialize
$\mathsf{2FRep}_{\Bbbk}(\mathcal{G})$ to an impressive variety of families of
interesting 2-categories already over an algebraically closed field of
characteristic zero, i.e. when we are at the semisimple intersection of 7.21
and 7.19. See the detailed bestiary in [DR18, §1.4.5].

######  7.23 Remark.

By replacing $\mathsf{2FVect}_{\Bbbk}$ with $\mathsf{2FHilb}$ (7.11) we obtain
the Krull–Schmidt property for _unitary_ finite dimensional representations of
2-groups.

######  7.24 Corollary.

Let $\mathcal{B}$ be any $\Bbbk$-finitary 2-category in the sense of
Mazorchuk–Miemietz [MM11], over an algebraically closed field $\Bbbk$. Then
the 2-category $\mathcal{B}\mathfrak{-mod}$ of $\Bbbk$-linear
2-representations of $\mathcal{B}$ is Krull–Schmidt.

###### Proof.

By definition
$\mathcal{B}\mathfrak{-mod}:=\operatorname{\mathsf{2Fun}}Fun^{\mathsf{str}}_{\Bbbk}(\mathcal{B},\mathsf{CAT}^{\mathsf{fin}}_{\Bbbk})$
as in 7.15 and $\mathcal{B}$ has finitely many objects, hence we may once
again apply 7.14. ∎

Many further variations are possible, for instance one may wish to study
finite “derived” rather than abelian or semisimple 2-representations by using
the Krull–Schmidt 2-category $\mathsf{TRI}^{\mathsf{fin}}_{\Bbbk}$ as target
(7.13). As sampled above, most directions of 2-representation theory found in
the literature impose strong finiteness conditions (often motivated by
combinatorial goals) which ensure the Krull–Schmidt property. We trust
interested readers to adapt our proofs to any such situations.

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