We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense $\V$-functor $\jAE$, a $\V$-functor $r \colon D \to E$ is $j$-monadic if and only if $r$ admits a left $j$-relative adjoint and creates $j$-absolute colimits. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the ($j$-)monadicity of the composite of a $\V$-functor with a ($j$-)monadic $\V$-functor.

§ INTRODUCTION

A relative monad is a generalisation of a monad that relaxes the requirement that the monad's underlying functor be an endofunctor <cit.>. The theory of relative monads extends the theory of monads, and many aspects of the theory carry over without significant changes. In particular, every relative monad has a category of algebras, equipped with free and forgetful functors that together form a relative adjunction <cit.>.

\[\begin{tikzcd}
& {\Alg(T)} \\
A && E
\arrow[""{name=0, anchor=center, inner sep=0}, "{f_T}", from=2-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{u_T}", from=1-2, to=2-3]
\arrow["j"', from=2-1, to=2-3]
\arrow["\dashv"{anchor=center}, shift right=2, draw=none, from=0, to=1]
\end{tikzcd}\]

For a fixed functor $\jAE$, the category of $j$-relative monads is equipped with a  functor $u_{\ph} \colon \RMnd(j)\op \ffto \Cat/E$ to the category of slices over $E$, sending each relative monad $T$ to the forgetful functor $u_T \colon \Alg(T) \to E$ from its category of algebras <cit.>.
It is natural to ask whether we can characterise the essential image of $u_{({-})}$: in other words, to characterise when a functor $r \colon D \to E$ is, up to isomorphism, the forgetful functor from the category of algebras for some $j$-relative monad. Such a functor is called $j$-relatively monadic (or simply $j$-monadic); a theorem characterising $j$-monadic functors is called a relative monadicity theorem.
We shall clarify our conventions on strict versus non-strict relative monadicity in <ref>.

In this paper, we establish two relative monadicity theorems. The first (<ref>) is a characterisation in the style of beck1966untitled and pare1971absolute, establishing that it necessary and sufficient that $r$ admits a left $j$-adjoint and creates certain colimits. The second (<ref>) is a pasting law for relatively monadic functors, which may be seen as analogous to the pasting law for pullbacks, characterising the relative monadicity of one functor in terms of the relative monadicity of another.

§.§ Relative monadicity via creation of colimits

For $j$ the identity functor on a category $E$, a $j$-monad is a (non-relative) monad, and there is a well-known characterisation of $j$-monadicity: a functor $r \colon D \to E$ is monadic if and only if it admits a left adjoint and creates coequalisers of $r$-contractible pairs <cit.>; or, equivalently, if and only if it admits a left adjoint and creates absolute colimits <cit.>.

For more general functors $j$, there are also known characterisations of $j$-monadicity. However, perhaps surprisingly, these characterisations predate the modern notion of relative monad by some thirty-five years. diers1975jmonades,diers1976foncteur and lee1977relative independently established that a functor $r \colon D \to E$ is monadic relative to a dense and  functor $\jAE$ if and only if it admits a left $j$-adjoint and creates $j$-absolute colimits. Due to the relative obscurity of the sources, and because the notions of relative monad studied by  and  appear quite different to that of altenkirch2010monads ( <cit.>), these relative monadicity theorems have till now been overlooked.
A colimit in $E$ is $j$-absolute when it is preserved by the nerve functor $E(j {-}_2, {-}_1) \colon E \to [A\op, \Set]$.

While the characterisations of  and  are useful, there are two respects in which one might hope for greater generality. First, we should like to drop the assumption that the root $j$ be . Second, we should like a similar characterisation in the setting of enriched category theory, extending that for enriched monadicity <cit.>.

The purpose of <ref> is to establish such a characterisation. However, our approach is more general still. It is the philosophy of formal category theory that the various flavours of category theory – such as ordinary category theory, enriched category theory, internal category theory, and so on – should all be viewed as instances of a general theory. That is, rather than establish the fundamental theorems of category theory separately in each setting, it is valuable to instead work in a framework in which such theorems may be proven just once, and then specialised to each setting. This was the approach of <cit.>, in which we laid the foundations for a formal theory of relative monads. Herein, we follow the same approach, working in the context of a virtual equipment in the sense of cruttwell2010unified. Thus, we shall first establish a relative monadicity theorem in the context of a virtual equipment, from which an enriched relative monadicity theorem will follow as a special case ( <cit.>). However, we are able to make some simplifications for enrichment in well-behaved monoidal categories, which we describe in <ref>.

§.§ Relative monadicity via composition

Given a monadic functor $r' \colon D \to E$, it is often useful to establish when precomposing some functor $r \colon C \to D$ produces another monadic functor $(r \d r') \colon C \to E$. It is neither necessary nor sufficient that $r$ be monadic: $(r \d r')$ may admit a left adjoint even when $r$ does not; and $(r \d r')$ may not create the appropriate colimits even when $r$ does. To understand the relationship between $r$, $r'$, and $(r \d r')$, it turns out to be enlightening to consider the generalisation to the relative setting. If $r'$ is $j$-monadic for some functor $\jAE$, we may ask when $(r \d r')$ is also $j$-monadic. However, it makes no sense to ask whether $r$, too, is $j$-monadic, because $r$ and $j$ have different codomains. Instead, it is most natural to ask whether $r$ is monadic relative to $\ell' \colon A \to D$, the left $j$-adjoint of $r'$.

\[\begin{tikzcd}[sep=small]
&& C \\
&&& D \\
A &&&& E
\arrow[""{name=0, anchor=center, inner sep=0}, "r", from=1-3, to=2-4]
\arrow["j"', from=3-1, to=3-5]
\arrow[""{name=1, anchor=center, inner sep=0}, "{r'}", from=2-4, to=3-5]
\arrow[""{name=2, anchor=center, inner sep=0}, "{\ell'}"{description}, from=3-1, to=2-4]
\arrow[""{name=3, anchor=center, inner sep=0}, "\ell", from=3-1, to=1-3]
\arrow["\dashv"{anchor=center}, shift right=1, draw=none, from=2, to=1]
\arrow["\dashv"{anchor=center, rotate=18}, shift right=2, draw=none, from=3, to=0]
\end{tikzcd}\]

In fact, as we prove in <ref>, this is necessary and sufficient to ensure the $j$-monadicity of the composite $(r \d r')$. This observation may be viewed as a pasting law for relatively monadic adjunctions: supposing that the triangle on the right is relatively monadic, then the triangle on the left is relatively monadic if and only if the outer triangle is relatively monadic. This characterisation has a couple of advantages over the characterisation in terms of colimit creation: for one, it does not require that the roots $j$ or $\ell'$ be dense, and the existence of a category of algebras can be derived rather than assumed. Furthermore, this characterisation appears to be new even for $j = 1$, though we note that a similar observation appears in the work of [Theorem 1.5.5]walters1970categorical, of which our result may be seen as a refinement.
Recall that the existence of a $\V$-category of algebras for a $\V$-enriched (relative) monad requires the existence of certain limits in $\V$ ( <cit.>), and so is a nontrivial assumption for general $\V$.

§.§ Outline of the paper

In <ref>, we introduce the notions of creation of limits and of colimits in a virtual equipment, and show that the forgetful tight-cell $u_T \colon \Alg(T) \to E$ from the algebra object for a $j$-monad creates limits and $j$-absolute colimits (<ref>). In <ref>, we prove a formal relative monadicity theorem ([relative-monadicity-theorem]Theorems <ref>relative-monadicity-theorem and [relative-monadicity-theorem']<ref>relative-monadicity-theorem') and, as a consequence, show that the category of algebras for a relative monad $T$ is the category of algebras for a monad if and only if the forgetful tight-cell $u_T \colon \Alg(T) \to E$ admits a left adjoint (<ref>). In <ref>, we specialise the relative monadicity theorem to $\VCat$ and $\VCat\co$ for a well-behaved monoidal category $\V$, obtaining enriched relative monadicity and relative comonadicity theorems ([enriched-relative-monadicity-theorem]Theorems <ref>enriched-relative-monadicity-theorem and [enriched-relative-comonadicity-theorem]<ref>enriched-relative-comonadicity-theorem). To demonstrate the application of the relative monadicity theorem, we use it to prove that (1) the category of algebras for a finitary algebraic theory is (relatively) monadic over $\Set$ (<ref>); (2) that the category of algebras for a colimit-preserving monad on a free cocompletion is relatively monadic over the cocompletion (<ref>); and (3) that the category of algebras for a quantitative equational theory in the sense of mardare2016quantitative is (relatively) monadic over $\Met$ (<ref>). More generally, we show that the category of algebras for a $j$-theory in the sense of lucyshyn2023enriched is $j$-monadic (<ref>). Finally, in <ref>, we consider the interaction between the pasting law for relative adjunctions (<cit.>) and relative monadicity, and thereby deduce necessary and sufficient conditions for the composite of a tight-cell with a relatively monadic tight-cell to itself be relatively monadic (<ref>).

In addition to  and , a series of relative monadicity theorems for unenriched relative monads with  roots were also established by  <cit.>. However,  imposes a number of additional assumptions on the functor $r \colon D \to E$ that do not hold in general, and which are not appropriate for our purposes. We shall not pursue an explicit comparison to the theorems of  in this paper.

After presenting an early version of this work to the Masaryk University Algebra Seminar, the authors were informed that John Bourke, Marcelo Fiore, and Richard Garner have, in unpublished joint work, independently established an enriched relative monadicity theorem.

§.§ Notation

Following <cit.>, whose terminology and notation we adopt, we work in the context of a  $\X$. However, for readers unacquainted with <cit.>, familiarity with enriched category theory should suffice to follow the theorem statements and discussion. It will be helpful to recall that the virtual equipment $\VCat$, of categories enriched in a monoidal category $\V$, has as objects the (possibly large) $\V$-categories, as tight-cells ($\to$) the $\V$-functors, as loose-cells ($\lto$) the $\V$-distributors, and as 2-cells ($\tto$) the $\V$-natural transformations <cit.>.

One notable difference between the formal categorical setting and typical approaches to enriched category theory ( <cit.>) is that our limits and colimits are weighted by $\V$-distributors rather than $\V$-presheaves and $\V$-copresheaves: for a $\V$-distributor $p \colon X \lto Y$ and $\V$-functor $f \colon Y \to Z$, the $p$-weighted colimit of $f$, when it exists, is a $\V$-functor $p \wc f \colon X \to Z$; and for a $\V$-distributor $p \colon X \lto Y$ and $\V$-functor $g \colon X \to Z$, the $p$-weighted limit of $g$, when it exists, is a $\V$-functor $p \wl g \colon Y \to Z$.

For tight-cells $f \colon A \to B$ and $g \colon B \to C$, we denote by $(f \d g) \colon A \to C$ or $g f \colon A \to C$ their composite. For loose-cells $p \colon X \lto Y$ and $q \colon Y \lto Z$, we denote by $q \odot p \colon X \lto Z$ and $q \odotl p \colon X \lto Z$, their composite and left-composite (<cit.>) respectively, when they exist. Every object $A$ has a loose-identity $A(1, 1) \colon A \lto A$. For every loose-cell $p \colon X \lto Y$ and tight-cells $g \colon W \to X$ and $f \colon Z \to Y$, there is a restriction loose-cell $p(f, g) \colon W \lto Z$. We denote the restriction $A(1, 1)(f, g)$ along a loose-identity by $A(f, g)$: in $\VCat$, this is given by the hom-objects of $A$.

Finally, for tight-cells $\jAE$, $\ell \colon A \to C$, and $r \colon C \to E$, we write $\ljr$ to express that $\ell$ is $j$-relatively left adjoint to $r$,  that there exists an isomorphism of loose-cells ${\sharp \colon C(\ell, 1) \iso E(j, r) \cocolon \flat}$. Every such $j$-relative adjunction induces a $j$-relative monad $T$ with carrier $t \defeq (\ell \d r) \colon A \to E$, which is equipped with an extension operator $\dag \colon E(j, t) \tto E(t, t)$ and a unit $\eta \colon j \tto t$. In this case, we say that $\ell \jadj r$ is a resolution of $T$.

§.§ Acknowledgements

The authors thank John Bourke for helpful comments.
The second author was supported by Icelandic Research Fund grant № 228684-052.

§ CREATION OF LIMITS, COLIMITS, AND ISOMORPHISMS

We start with some elementary observations on the creation of limits and certain colimits by forgetful functors from categories of algebras for relative monads.

§.§ Strict creation

We first introduce the notion of strict creation of weighted limits and colimits, which will be important in the strict relative monadicity theorem (<ref>). We then introduce creation of isomorphisms, and non-strict creation of limits and colimits in <ref>, which will be important in the non-strict relative monadicity theorem (<ref>). For the notions of weighted limit and weighted colimit in an equipment, see <cit.>.

Since there is substantial inconsistency in the literature regarding which of strict or non-strict creation (and consequently monadicity) should be taken as fundamental, we shall be explicit about strictness throughout; the unqualified terms are used only informally.

Let $p \colon Y \lto Z$ be a loose-cell, and let $f \colon Z \to W$ and $g \colon W \to X$ be tight-cells.
A colimit $(p \wc (f \d g), \lambda)$ in $X$ is strictly created by $g$ when there exists a $p$-cocone $(w, \lambda')$ for $f$, comprising a tight-cell $w \colon Y \to W$ and a 2-cell $\lambda' \colon p \tto W(f, w)$, such that

* $(w, \lambda')$ is the unique pair satisfying $p \wc (f \d g) = w \d g$ and $\lambda = \lambda' \d g$;

* $(w, \lambda')$ is the weighted colimit $p \wc f$.

In particular, strict creation of $p \wc (f \d g)$ implies preservation of $p \wc f$, in the sense of <cit.>. Our motivating example of colimit creation will be the forgetful tight-cell $u_T \colon \Alg(T) \to E$ from an algebra object for a relative monad, in the sense of <cit.>, which strictly creates every colimit that is $j$-absolute, in the sense of <cit.> (see <ref> below for a characterisation of $j$-absoluteness in the enriched context).

Let $\jAE$ be a tight-cell and let $T$ be $j$-monad admitting an algebra object $(u_T, \aop_T)$.
The tight-cell $u_T \colon \Alg(T) \to E$ strictly creates $j$-absolute colimits.

Let $p \colon Y \lto Z$ be a loose-cell and let $f \colon Z \to \Alg(T)$ be a tight-cell. Suppose that $(p \wc (f \d u_T), \lambda)$ is a $j$-absolute colimit. The tight-cell $(f \d u_T)$ forms a $T$-algebra by equipping it with the 2-cell $\aop_T(1, f) \colon E(j, (f; u_T)) \tto E(t, (f; u_T))$.

We will show first that there is a 2-cell $\aop \colon E(j, p \wc (f \d u_T)) \tto E(t, p \wc (f \d u_T))$ equipping $p \wc (f \d u_T)$ with the structure of a $T$-algebra, and then that this is the unique algebra structure for which $\lambda \colon p \tto E(u_T f, p \wc (f \d u_T))$ is a $(p)$-graded $T$-algebra morphism from $(f \d u_T)$ to $p \wc (f \d u_T)$ in the sense of <cit.>.

By definition, $j$-absoluteness of $(p \wc (f \d u_T), \lambda)$ means that the 2-cell

\[
% https://varkor.github.io/tangle/?t=W1tbNyw1LDEsOSw4XV0sW1tbMSxbMSwwLDEsMF1dLFsxLFsxLDAsMSwwXV0sWzEsWzEsMCwxLDBdXSxbMSxbMSwwLDEsMF1dXSxbWzEsWzEsMCwxLDBdXSxbMSxbMSwxLDEsMF1dLFsxLFsxLDEsMCwxXV0sWzEsWzEsMCwwLDFdXV0sW1sxLFsxLDAsMSwwXV0sWzEsWzEsMCwxLDBdXSxbMCxbXV0sWzAsW11dXV0sW1sxLDEuNSwxLFsxXV0sWzAsMi41LDEuNSxbIlxcbGFtYmRhIiwyXV0sWzEsMi41LDEsWzFdXSxbMSwxLjUsMixbMV1dLFsxLDAuNSwyLFswXV0sWzEsMC41LDEsWzBdXV0sW1swLDAsMywiaiJdLFsxLDAsMywidV9UIl0sWzIsMCwzLCJmIl0sWzMsMCwzLCJwIl0sWzAsMiwxLCJqIl0sWzEsMiwxLCJwIFxcd2MgKGYgXFxkIHVfVCkiXV1d&c=F5A3A3,F5CCA3,F5F5A3,CCF5A3,A3F5A3,A3F5CC,A3F5F5,A3CCF5,A3A3F5,CCA3F5,F5A3F5,F5A3CC
\begin{tangle}{(4,3)}[trim y]
\tgBorderA{(0,0)}{\tgColour6}{\tgColour4}{\tgColour4}{\tgColour6}
\tgBorderA{(1,0)}{\tgColour4}{\tgColour0}{\tgColour0}{\tgColour4}
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\tgBorderA{(3,0)}{\tgColour8}{\tgColour7}{\tgColour7}{\tgColour8}
\tgBorderA{(0,1)}{\tgColour6}{\tgColour4}{\tgColour4}{\tgColour6}
\tgBorderA{(1,1)}{\tgColour4}{\tgColour0}{\tgColour7}{\tgColour4}
\tgBorderA{(2,1)}{\tgColour0}{\tgColour8}{\tgColour7}{\tgColour7}
\tgBorderA{(3,1)}{\tgColour8}{\tgColour7}{\tgColour7}{\tgColour7}
\tgBorderA{(0,2)}{\tgColour6}{\tgColour4}{\tgColour4}{\tgColour6}
\tgBorderA{(1,2)}{\tgColour4}{\tgColour7}{\tgColour7}{\tgColour4}
\tgBlank{(2,2)}{\tgColour7}
\tgBlank{(3,2)}{\tgColour7}
\tgArrow{(1,0.5)}{3}
\tgCell[(2,0)]{(2,1)}{\lambda}
\tgArrow{(2,0.5)}{3}
\tgArrow{(1,1.5)}{3}
\tgArrow{(0,1.5)}{1}
\tgArrow{(0,0.5)}{1}
\tgAxisLabel{(0.5,0.75)}{south}{j}
\tgAxisLabel{(1.5,0.75)}{south}{u_T}
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\tgAxisLabel{(0.5,2.25)}{north}{j}
\tgAxisLabel{(1.5,2.25)}{north}{p \wc (f \d u_T)}
\end{tangle}
\]

is left-opcartesian, so the 2-cell on the left of the equation below factors uniquely therethrough.

(4,4)[trim y]
(1.5,3.25)northp (f ụ_T)

tangle(4,4)[trim y]
(1.5,3.25)northp (f ụ_T)

The induced 2-cell $\aop$ makes $p \wc (f \d u_T)$ into a $T$-algebra, the unit and extension operator laws following from those of $\aop_T$ by pasting the left-opcartesian 2-cell.
Moreover, the above equation is exactly the equation required for $\lambda$ to form a $(p)$-graded $T$-algebra morphism. The 2-cell $\aop$ is therefore the unique algebra structure for which $\lambda$ is such a morphism.

The universal property of the algebra object for $T$ thus induces a unique tight-cell ${(p \wc f) \colon Y \to \Alg(T)}$ and 2-cell $\lambda' \colon p \tto \Alg(T)(f, p \wc f)$ satisfying the following three equations.

\[
p \wc (f \d u_T) = (p \wc f) \d u_T
\hspace{4em}
\aop = \aop_T(1, p \wc f)
\hspace{4em}
\lambda = \lambda' \d u_T
\]

Hence the pair $(p \wc f, \lambda')$ satisfies the existence condition of (1) in <ref>. Uniqueness of this pair is immediate from uniqueness of $\aop$.

It remains to show that $(p \wc f, \lambda')$ is the colimit.
This requires us to show that we have a bijection between 2-cells $q_1, \dots, q_n \tto \Alg(T)(p \wc f, 1)$ and 2-cells $p, q_1, \dots, q_n \tto \Alg(T)(f, 1)$, given by pasting with the following 2-cell.

\[
% https://varkor.github.io/tangle/?t=W1tbOSw4LDFdXSxbW1swLFtdXSxbMSxbMSwwLDEsMF1dLFswLFtdXSxbMSxbMSwwLDEsMF1dXSxbWzEsWzAsMSwxLDBdXSxbMSxbMSwxLDAsMV1dLFsxLFswLDAsMSwxXV0sWzEsWzEsMCwxLDBdXV0sW1sxLFsxLDAsMSwwXV0sWzAsW11dLFsyLFsxXV0sWzIsWzBdXV0sW1sxLFsxLDAsMSwwXV0sWzAsW11dLFswLFtdXSxbMCxbXV1dXSxbWzAsMS41LDEuNSxbIlxcbGFtYmRhJyIsMl1dLFsxLDIuNSwyLFsxXV0sWzEsMywyLjUsWzBdXSxbMSwzLjUsMixbMF1dLFsxLDAuNSwyLFswXV0sWzEsMy41LDEsWzBdXSxbMSwwLjUsMyxbMF1dXSxbWzEsMCwzLCJwIl0sWzMsMCwzLCJwIFxcd2MgZiJdLFswLDMsMSwiZiJdXV0=&c=F5A3A3,F5CCA3,F5F5A3,CCF5A3,A3F5A3,A3F5CC,A3F5F5,A3CCF5,A3A3F5,CCA3F5,F5A3F5,F5A3CC
\begin{tangle}{(4,4)}[trim y]
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\tgAxisLabel{(3.5,0.75)}{south}{p \wc f}
\tgAxisLabel{(0.5,3.25)}{north}{f}
\end{tangle}
\]

By the universal property of the algebra object, this is equivalent to having a bijection between $T$-algebra morphisms $q_1, \dots, q_n \tto E(p \wc (f \d u_T), u_T)$ and $T$-algebra morphisms $p, q_1, \dots, q_n \tto E(f, u_T)$, given by pasting with the following 2-cell.

\[
% https://varkor.github.io/tangle/?t=W1tbOSw4LDVdXSxbW1swLFtdXSxbMSxbMSwwLDEsMF1dLFswLFtdXSxbMSxbMSwwLDEsMF1dXSxbWzEsWzAsMSwxLDBdXSxbMSxbMSwxLDAsMV1dLFsxLFswLDAsMSwxXV0sWzEsWzEsMCwxLDBdXV0sW1sxLFsxLDAsMSwwXV0sWzAsW11dLFsyLFsxXV0sWzIsWzBdXV0sW1sxLFsxLDAsMSwwXV0sWzAsW11dLFswLFtdXSxbMCxbXV1dXSxbWzAsMS41LDEuNSxbIlxcbGFtYmRhIiwyXV0sWzEsMi41LDIsWzFdXSxbMSwzLDIuNSxbMF1dLFsxLDMuNSwyLFswXV0sWzEsMC41LDIsWzBdXSxbMSwzLjUsMSxbMF1dLFsxLDAuNSwzLFswXV1dLFtbMSwwLDMsInAiXSxbMywwLDMsInAgXFx3YyAoZiBcXGQgdV9UKSJdLFswLDMsMSwiZiBcXGQgdV9UIl1dXQ==&c=F5A3A3,F5CCA3,F5F5A3,CCF5A3,A3F5A3,A3F5CC,A3F5F5,A3CCF5,A3A3F5,CCA3F5,F5A3F5,F5A3CC
\begin{tangle}{(4,4)}[trim y]
\tgBlank{(0,0)}{\tgColour8}
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\tgBorderA{(0,1)}{\tgColour8}{\tgColour8}{\tgColour4}{\tgColour8}
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\tgAxisLabel{(1.5,0.75)}{south}{p}
\tgAxisLabel{(3.5,0.75)}{south}{p \wc (f \d u_T)}
\tgAxisLabel{(0.5,3.25)}{north}{f \d u_T}
\end{tangle}
\]

Since $(p \wc (f \d u_T), \lambda)$ is a colimit, pasting induces a bijection between 2-cells.
That this bijection preserves $T$-algebra morphisms is immediate from the defining equation for $\aop$ above.

Though we shall not make use of it in our proof of relative monadicity, it is nonetheless useful to observe that forgetful tight-cells strictly create all limits.
Strict creation of limits is dual to strict creation of colimits (using that a weighted limit $p \wl f$ in an equipment $\X$ is precisely a weighted colimit $p \wc f$ in the dual equipment $\X\co$).
We spell out the definition explicitly for convenience.

Let $p \colon Y \lto Z$ be a loose-cell, and let $f \colon Y \to W$ and $g \colon W \to X$ be tight-cells.
A limit $(p \wl (f \d g), \mu)$ in $X$ is strictly created by $g$ when there exists a $p$-cone $(w, \mu')$ for $f$, comprising a tight-cell $w \colon Z \to W$ and a 2-cell $\mu' \colon p \tto W(w, f)$, such that

* $(w, \mu')$ is the unique pair satisfying $p \wl (f \d g) = w \d g$ and $\mu = \mu' \d g$;

* $(w, \mu')$ is the weighted limit $p \wl f$.

In particular, strict creation of limits implies preservation, in the sense of <cit.>.

Let $\jAE$ be a tight-cell and let $T$ be $j$-monad admitting an algebra object $(u_T, \aop_T)$.
The tight-cell $u_T \colon \Alg(T) \to E$ strictly creates limits.

Let $p \colon Y \lto Z$ be a loose-cell and let $f \colon Y \to \Alg(T)$ be a tight-cell. Suppose that $(p \wl (f \d u_T), \mu)$ is a limit in $E$.
The tight-cell $p \wl (f \d u_T)$ forms a $T$-algebra by equipping it with the following 2-cell,

\[
\aop \colon E(j, p \wl (f \d u_T))
\iso p \rx E(j, (f \d u_T))
\xtto{p \rx \aop_T(1, f)} p \rx E(t, (f \d u_T))
\iso E(t, p \wl (f \d u_T))
\]

where the isomorphisms are induced by the universal property of the limit.
That $\aop$ satisfies the $T$-algebra laws follows from the fact that $\aop_T$ does.
This extension operator $\aop$ is the unique algebra structure for which the universal 2-cell $\mu \colon p \tto E(p \wl (f \d u_T), (f \d u_T))$ is a $(p)$-graded algebra morphism from $p \wl (f \d u_T)$ to $(f \d u_T)$. Thus the $p$-cone $(p \wl (f \d u_T), \mu)$ lifts uniquely through $u_T$ as a $p$-cone $(p \wl f, \mu')$.
The proof that $(p \wl f, \mu')$ is the limit in $\Alg(T)$ is analogous to the proof that the unique lifting in <ref> is the colimit.

From <ref>, we recover [Proposition 1.2.1]manes1967triple[Corollary 5]frei1969some and <cit.> for non-relative monads in $\Cat$ and $\VCat$, and <cit.> for relative monads in $\Cat$ with  roots ( <cit.>).

Observe that, since it is not possible to formulate the notions of weighted limit and weighted colimit in an arbitrary 2-category, it is not possible to recover analogues of
<ref> for algebra objects in a 2-category in the sense of [1]street1972formal that specialise appropriately to enriched categories of algebras. In fact, <ref> appear to be new even for non-relative monads in the setting of $\VCat$.
[Theorem II.2.1 & Proposition II.4.7 & II.4.8]dubuc1970kan establishes the creation of coequalisers of $u_T$-contractible pairs, powers, conical limits, and ends, but not general weighted limits (the notion of which postdates 's work <cit.>).

§.§ Non-strict creation

Next, we consider the non-strict creation of limits and colimits, which will be important in the non-strict relative monadicity theorem (<ref>). It will be useful to observe that non-strict creation of (co)limits is implied by strict creation of (co)limits together with the creation (or, equivalently, the reflection) of isomorphisms.

A tight-cell $g \colon W \to X$ is conservative if, for each parallel pair of tight-cells $f_1, f_2 \colon Z \rightrightarrows W$ and each 2-cell $\phi \colon f_1 \tto f_2$, invertibility of $(\phi \d g)$ implies invertibility of $\phi$.

Let $\jAE$ be a tight-cell and let $T$ be $j$-monad admitting an algebra object $(u_T, \aop_T)$.
The tight-cell $u_T \colon \Alg(T) \to E$ is conservative.

Let $f_1, f_2 \colon Z \rightrightarrows \Alg(T)$ be a parallel pair of tight-cells and let $\phi \colon f_1 \tto f_2$ be a 2-cell. Suppose that $(\phi \d u_T)$ admits an inverse. Since $(\phi \d u_T)$ is an algebra morphism, so is its inverse, so that the inverse factors uniquely through $u_T$ by the universal property of $\Alg(T)$.

We shall mark with a prime ($'$) the non-strict variants of definitions and results corresponding to strict variants.

Let $p \colon Y \lto Z$ be a loose-cell, and let $f \colon Z \to W$ and $g \colon W \to X$ be tight-cells, such that the colimit $p \wc (f \d g)$ exists.
The colimit $p \wc (f \d g)$ is non-strictly created by $g$ when

* for every $p$-cocone $(w, \lambda')$ for $f$, the $p$-cocone $((w \d g), (\lambda' \d g))$ is the $p$-colimit of $(f \d g)$ if and only if $(w, \lambda')$ is the $p$-colimit of $f$;

* $W$ admits the colimit $p \wc f$.

Let $p \colon Y \lto Z$ be a loose-cell, and let $f \colon Y \to W$ and $g \colon W \to X$ be tight-cells, such that the limit $p \wl (f \d g)$ exists.
The limit $p \wl (f \d g)$ is non-strictly created by $g$ when

* for every $p$-cone $(w, \mu')$ for $f$, the $p$-cone $((w \d g), (\mu' \d g))$ is the $p$-limit of $(f \d g)$ if and only if $(w, \mu')$ is the $p$-limit of $f$;

* $W$ admits the limit $p \wl f$.

As with strict creation, non-strict creation of (co)limits implies preservation. Conversely, preservation of (co)limits implies non-strict creation assuming conservativity.

Let $p \colon Y \lto Z$ be a loose-cell and $g \colon W \to X$ be a conservative tight-cell. For any tight-cell $f \colon Z \to W$, if $W$ admits a colimit $p \wc f$ and $g$ preserves it, then $g$ non-strictly creates the colimit $p \wc (f \d g)$. Similarly, for any tight-cell $f \colon Y \to W$, if $W$ admits a limit $p \wl f$ and $g$ preserves it, then $g$ non-strictly creates the limit $p \wl (f \d g)$.

Since $g$ is conservative, a 2-cell $\phi \colon p \wc f \tto w$ is invertible if and only if $(\phi \d g) \colon (p \wc f) \d g \iso p \wc (f \d g) \tto w \d g$ is invertible. Therefore, the $p$-cocone corresponding to $\phi$ is colimiting if and only if the $p$-cocone corresponding to $(\phi \d g)$ is colimiting. The limit case follows analogously.

The non-strict analogues of <ref> follow.

Let $\jAE$ be a tight-cell and let $T$ be $j$-monad admitting an algebra object $(u_T, \aop_T)$.
The tight-cell $u_T \colon \Alg(T) \to E$ non-strictly creates limits and $j$-absolute colimits.

Follows directly from <ref>.

From <ref>, we recover <cit.> for non-relative monads in $\Cat$.

§ RELATIVE MONADICITY

In this section, we establish a relative monadicity theorem (<ref>([relative-monadicity-theorem']$'$)) for relative monads with dense roots. We start by reviewing our approach to strictness.

Our primary interest, as motivated in the introduction, is with strict relative monadicity, characterising those functors for which a canonical comparison functor is an isomorphism. This is in line with the philosophy of <cit.>, in which we are concerned with categories of relative monads, and of their resolutions, in which the appropriate notion of indistinguishability is isomorphism (rather than with 2-categories or virtual double categories, in which the appropriate notion of indistinguishability is equivalence). We therefore proceed to first establish a strict relative monadicity theorem, as it requires more care than the non-strict variant. Since a non-strict relative monadicity theorem is also of interest, we provide one in <ref>. However, as the proof strategy is identical to the strict case, up to appropriate replacement of identities with isomorphisms, our treatment of the non-strict case is succinct.

We shall now outline our proof strategy, and discuss the subtleties that arise in the formal setting. Fix a tight-cell $\jAE$. Given a tight-cell $r \colon D \to E$, we should like to characterise when it exhibits the forgetful tight-cell for an algebra object for a $j$-monad, in the sense of <cit.>. Assuming that algebra objects exist, we may reformulate this condition, using the universal property, into the following.

A tight-cell $r \colon D \to E$ is strictly $j$-relatively monadic (alternatively strictly monadic relative to $j$, or simply strictly $j$-monadic) if it admits a left $j$-adjoint, the induced $j$-monad $T$ admits an algebra object, and the comparison tight-cell $\unit_{\ljr} \colon D \to \Alg(T)$ is an isomorphism.

\[\begin{tikzcd}
D && {\Alg(T)} \\
& E
\arrow["r"', from=1-1, to=2-2]
\arrow["{\unit_{\ljr}}", from=1-1, to=1-3]
\arrow["{u_T}", from=1-3, to=2-2]
\end{tikzcd}\]

Explicitly, this means that, to exhibit $r$ as the forgetful tight-cell for an algebra object, we must construct a tight-cell $\Alg(T) \to D$ that is inverse to the comparison tight-cell $\unit_{\ljr}$. To do so, we shall observe that $u_T \colon \Alg(T) \to E$ can be exhibited as a certain weighted colimit (<ref>). Supposing that $r$ creates this colimit, the lifting then provides the desired inverse. Furthermore, the weighted colimit in question will be shown to be $j$-absolute, and so creation of $j$-absolute colimits provides both a necessary and sufficient condition for $j$-monadicity, by <ref>.

Note, however, that when the $j$-monad induced by $\ljr$ does not admit an algebra object, such an approach is not possible, as there is no comparison tight-cell that we may ask to be invertible.
The relative monadicity theorem we prove in this section is therefore a detection result, rather than a construction result: it characterises when a tight-cell is relatively monadic assuming that an algebra object exists. This is not a trivial assumption even for enriched (non-relative) monadicity. However, it is also not a particularly restrictive assumption, as algebra objects typically exist in settings of interest (for instance, when the enriching category $\V$ is closed and has enough limits <cit.>).

§.§ Strict relative monadicity

The first step towards a relative monadicity theorem is to identify the appropriate colimits for creation. To do so, we observe in the following proposition that, for dense $j$, a right-morphism of relative adjunctions $(\ljr) \to (\ljrp)$, exhibits $r'$ as a left extension of $r$ ( <cit.>).
In particular, this is true of morphisms of resolutions, in which the 2-cell $\rho$ below is the identity (<cit.>).

Recall that, in general, a right-morphism $(c, \rho) \colon (\ljr) \to (\ljrp)$ comprises a tight-cell $c \colon C \to C'$ commuting with the left relative adjoints, and a 2-cell $\rho \colon r \tto c \d r'$ compatible with the transposition operators <cit.>. However, when $j$ is dense, the 2-cell $\rho$ is uniquely determined by the transposition operators <cit.>. This is the case in the following proposition.

Let $\jAE$ be a dense tight-cell and let $\ljr$ be a $j$-adjunction. Consider tight-cells $c \colon C \to C'$ and $r' \colon C' \to E$, and a 2-cell $\rho \colon r \tto c \d r'$. Define $\ell' \defeq \ell \d c$.

\[\begin{tikzcd}
& {C'} \\
A & C & E
\arrow["\ell"', from=2-1, to=2-2]
\arrow["r"', from=2-2, to=2-3]
\arrow["{\ell'}", from=2-1, to=1-2]
\arrow["c"{description}, from=2-2, to=1-2]
\arrow[""{name=0, anchor=center, inner sep=0}, "{r'}", from=1-2, to=2-3]
\arrow["\rho"', shorten >=3pt, Rightarrow, from=2-2, to=0]
\end{tikzcd}\]

The following are equivalent.

* $\ljrp$, and $(c, \rho) \colon (\ljr) \to (\ljrp)$ is a right-morphism of $j$-adjunctions.

* $\rho$ exhibits $r'$ as the $j$-absolute left extension $c \plx r$.

Observe that, since $\ljr$, the loose-composite $E(j, r) \odot C'(c, 1)$ is isomorphic to $C'(\ell', 1)$:

\[C'(\ell', 1) = C'(c \ell, 1) \iso C(\ell, 1) \odot C'(c, 1) \iso E(j, r) \odot C'(c, 1)\]

(1) $\implies$ (2). Since $\ljrp$, we have

\[E(j, r') \iso C'(\ell', 1) \iso E(j, r) \odot C'(c, 1)\]

from which the result follows by <cit.> using that $j$ is dense.

(2) $\implies$ (1). We have

\[C'(\ell', 1) \iso E(j, r) \odot C'(c, 1) \iso E(j, c \plx r)\]

using $j$-absoluteness and <cit.>, so that $\ljrp$. That $(c, \rho)$ forms a right-morphism follows from the definition of $\ljrp$.

Since the left extensions of <ref> are $j$-absolute, they are created by the forgetful tight-cell $u_T \colon \Alg(T) \to E$ of any algebra object for a $j$-monad. We shall show shortly that creation of such colimits is sufficient to imply $j$-monadicity. The observation above motivates the following definition.

Let $r \colon D \to E$ be tight-cell. An $r$-extension is a left extension $c \plx r$, for some tight-cell $c$ with domain $D$.

The core technical lemma in the proof of the relative monadicity theorem is the following.

Let $\jAE$ be a dense tight-cell, and let $\ell \jadj r$ be a resolution of a $j$-monad $T$. If $r$ strictly creates $j$-absolute $r$-extensions, then every morphism of resolutions with domain $(\ell \jadj r)$ admits a retraction $c \plx 1_C$.

Let $c \colon (\ell \jadj r) \to (\ell' \jadj r')$ be a morphism of resolutions of $T$.

\[
% https://q.uiver.app/?q=WzAsNCxbMCwxLCJBIl0sWzEsMSwiQyJdLFsyLDEsIkUiXSxbMSwwLCJDJyJdLFswLDEsIlxcZWxsIiwyXSxbMSwyLCJyIiwyXSxbMCwzLCJcXGVsbCciXSxbMSwzLCJjIiwxXSxbMywyLCJyJyJdXQ==
\begin{tikzcd}
& {C'} \\
A & C & E
\arrow["\ell"', from=2-1, to=2-2]
\arrow["r"', from=2-2, to=2-3]
\arrow["{\ell'}", from=2-1, to=1-2]
\arrow["c"{description}, from=2-2, to=1-2]
\arrow["{r'}", from=1-2, to=2-3]
\end{tikzcd}
\]

By <ref>, using density of $j$, we have $r' \iso c \plx r$.
This left extension is a $j$-absolute $r$-extension, so is strictly created by $r$.
Hence there is a unique pair of a tight-cell $d \colon C' \to C$ and 2-cell
$\eta \colon 1_C \tto c \d d$ such that $d \d r = r'$ and
$\eta \d r = 1_r$. Furthermore, $\eta$ exhibits $d$ as the left extension $c \plx 1_C$.
We shall prove that $c \d d = 1_C$: this implies that $d$ is a morphism of resolutions, and hence is a retraction of $c$ in the category of resolutions of $T$, because $\ell \d c = \ell'$.

The identity 2-cell $1_r$ exhibits $r$ as the $j$-absolute $r$-extension $1_{C} \plx r$ trivially.
This left extension is strictly created by $r$, so there is a unique pair $(x, \chi)$ of a tight-cell $x \colon C \to C$ and 2-cell $\chi \colon 1_C \tto x$ such that $x \d r = r$ and $\chi \d r = 1_r$.
Clearly $(1_C, 1_{1_C})$ is such a pair, but, by the above, and the fact that $c \d r' = r$, so is $((c \d d), \eta)$.
Hence $c \d d = 1_C$ as required.

In passing, we note that, when algebra objects exist, a relative adjunction is a terminal resolution if and only if its right relative adjoint exhibits an algebra object <cit.>. However, if algebra objects are not known to exist, terminality is a weaker condition than exhibiting an algebra object. The following corollary of <ref> is useful when terminality is sufficient. One could use this fact to give a proof of the relative monadicity theorem; however, we shall give a different proof that generalises more easily to the non-strict setting.

Let $\jAE$ be a dense tight-cell, let $T$ be a $j$-monad admitting a terminal resolution $f \jadj u$, and let $\ljr$ be a resolution of $T$. If $r$ strictly creates $j$-absolute $r$-extensions, then $\ljr$ is a terminal resolution of $T$.

\[\begin{tikzcd}
D && M \\
& E
\arrow["r"', from=1-1, to=2-2]
\arrow["u", from=1-3, to=2-2]
\arrow["{\unit_{\ljr}}", from=1-1, to=1-3]
\end{tikzcd}\]

Since $(f \jadj u)$ is terminal, there is a unique morphism $(\ell \jadj r) \to (f \jadj u)$. By <ref>, this morphism is a split monomorphism, because $r$ strictly creates $j$-absolute $r$-extensions. However, every split monomorphism into a terminal object is an isomorphism, so $(\ell \jadj r) \iso (f \jadj u)$.

The relative monadicity theorem follows essentially directly from <ref>.

Let $\jAE$ be a dense tight-cell. A tight-cell $r \colon D \to E$ is strictly $j$-monadic if and only if $r$ admits a left $j$-adjoint, the induced $j$-monad admits an algebra object, and $r$ strictly creates $j$-absolute $r$-extensions.

If $r$ is strictly $j$-monadic, then it admits a left $j$-adjoint for which the induced $j$-monad has an algebra object by definition. Furthermore, $r$ is the composite of an isomorphism with $u_T$, hence strictly creates $j$-absolute $r$-extensions by <ref>. For the converse, observe that $r$ and $u_T$ both strictly create $j$-absolute $r$-extensions, so that $\unit_{\ljr}$ and $\unit_{\ljr} \plx 1_D$ exhibit $D$ and $\Alg(T)$ as retracts of one another by <ref>, and hence $\unit_{\ljr}$ is invertible.

We may additionally relax the class of colimits that $r$ need create in order to be $j$-monadic, so that the class is independent of $r$.

Let $\jAE$ be a dense tight-cell. A tight-cell $r \colon D \to E$ is strictly $j$-monadic if and only if $r$ admits a left $j$-adjoint, the induced $j$-monad admits an algebra object, and $r$ strictly creates $j$-absolute colimits.

Follows directly from <ref>, since $u_T$ strictly creates all $j$-absolute colimits by <ref>.

In practice, <ref> is often more convenient: for instance, in <ref>, we shall give a simple characterisation of the $j$-absolute colimits in $\VCat$ for a well-behaved monoidal category $\V$. However, in <ref> we shall give an example of a situation in which the sharper result of <ref> is useful.

The assumption that $j$ be dense in the statement of <ref> is necessary. For instance, denoting by $0$ the empty category, every functor $r \colon D \to E$ (for arbitrary categories $D$ and $E$) is right adjoint to the unique functor $[]_D \colon 0 \to D$ relative to the unique functor $[]_E \colon 0 \to E$, which is dense only when $E$ is indiscrete.

\[
% https://q.uiver.app/?q=WzAsMyxbMCwxLCIwIl0sWzEsMCwiRCJdLFsyLDEsIkUiXSxbMCwxLCJbXV9EIl0sWzEsMiwiciJdLFswLDIsIltdX0UiLDJdLFszLDQsIiIsMCx7Im9mZnNldCI6MiwibGV2ZWwiOjEsInN0eWxlIjp7Im5hbWUiOiJhZGp1bmN0aW9uIn19XV0=
\begin{tikzcd}
& D \\
0 && E
\arrow[""{name=0, anchor=center, inner sep=0}, "{[]_D}", from=2-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "r", from=1-2, to=2-3]
\arrow["{[]_E}"', from=2-1, to=2-3]
\arrow["\dashv"{anchor=center}, shift right=2, draw=none, from=0, to=1]
\end{tikzcd}
\]

The trivial $[]_E$-monad is the unique $[]_E$-monad. Consequently, $r \colon D \to E$ is $[]_E$-monadic if and only if it is an isomorphism <cit.>. However, every colimit in $E$ is $[]_E$-absolute (by <ref> below), and so any non-invertible functor $r \colon D \to E$ that strictly creates colimits is a counterexample to the statement of <ref>. We leave as an open question whether there are sufficient conditions for relative monadicity in the absence of density.

Algebra objects for monads in a 2-category (which are weaker than algebra objects for monads in an equipment <cit.>) can be characterised representably in terms of algebra objects for monads in $\Cat$ <cit.>. Consequently, monadicity for monads in a 2-category $\K$ may be characterised representably in terms of monadicity in the 2-category $\Cat$ <cit.>. [Proposition 22]wood1985proarrows makes use of this characterisation to give a formal monadicity theorem, which is, in essence, an objectwise version of 's monadicity theorem <cit.>. 's monadicity theorem is thus of an entirely different nature to <ref>, which is a characterisation internal to the equipment $\X$.
A consequence is that <ref>, in contrast to 's monadicity theorem, directly specialises to the monadicity theorem for enriched monads.

A particularly useful consequence of the relative monadicity theorem is the following proposition, which relates algebra objects for relative monads that have different roots. Observe that the proof follows easily from <ref>, whereas a proof based on <ref> is less straightforward.

Let $\jAE$ and $\jEI$ be tight-cells, and let $T$ be a $(j \d j')$-monad admitting an algebra object. Suppose that $j'$ is dense. Then $u_T \colon \Alg(T) \to E'$ is strictly $j'$-monadic if and only if it admits a left $j'$-adjoint and the induced $j'$-monad admits an algebra object.

The only if direction is trivial. For the other direction, observe that every $j'$-absolute $r$-extension is a $(j \d j')$-absolute $r$-extension, since every right-morphism of $j'$-adjunctions induces a right-morphism of $(j \d j')$-adjunctions by precomposing $j$ <cit.>. Therefore, since $u_T$ strictly creates $(j \d j')$-absolute $r$-extensions, it strictly creates, in particular, $j'$-absolute $r$-extensions, from which the result follows by <ref>.

In particular, if every monad admits an algebra object, we obtain that the algebra object for a relative monad $T$ is the algebra object for a monad whenever the forgetful tight-cell $u_T \colon \Alg(T) \to E$ admits a left adjoint.

§.§ Non-strict relative monadicity

We now briefly discuss the case of non-strict relative monadicity. While strict relative monadicity is often the appropriate notion, there are situations in which it is convenient to consider a property that is invariant under equivalence. This motivates the following definition.

Let $\jAE$ be a tight-cell. A tight-cell $r \colon D \to E$ is non-strictly $j$-relatively monadic (alternatively non-strictly monadic relative to $j$, or simply non-strictly $j$-monadic) if it admits a left $j$-adjoint, the induced $j$-monad $T$ admits an algebra object, and the comparison tight-cell $\unit_{\ljr} \colon D \to \Alg(T)$ is an equivalence.

Informally, a characterisation of non-strict relative monadicity is obtained by relaxing <ref> through the replacement of the equality of tight-cells by isomorphism, and thereby asking for non-strict creation of colimits rather than strict creation. With respect to the proof strategy, there is one notable difference to the strict case. While, just as in the strict setting, the comparison tight-cell $\unit_{\ljr} \colon D \to \Alg(T)$ is a morphism of resolutions, the tight-cell $\Alg(T) \to D$ constructed via the non-strict creation of colimits only commutes with the left and right adjoints up to isomorphism. However, once this subtlety is taken into account, the proof then proceeds in essentially the same way as the strict case.

Below, our proof follows the structure of the strict case. Each statement is numbered according to its non-strict pair and marked with a prime ($'$) to denote non-strictness.

A pseudo-morphism of resolutions of $T$ from $\ljr$ to $\ljrp$ is a tight-cell $c \colon C \to C'$ between the apices rendering the following diagram pseudo-commutative.

\[\begin{tikzcd}[sep=small]
& C \\
A && E \\
& {C'}
\arrow["\ell", from=2-1, to=1-2]
\arrow["r", from=1-2, to=2-3]
\arrow["{\ell'}"', from=2-1, to=3-2]
\arrow["{r'}"', from=3-2, to=2-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "c"{description}, from=1-2, to=3-2]
\arrow["\iso"{description}, draw=none, from=2-1, to=0]
\arrow["\iso"{description}, draw=none, from=0, to=2-3]
\end{tikzcd}\]

Let $\jAE$ be a dense tight-cell, and let $\ell \jadj r$ be a resolution of a $j$-monad $T$. If $r$ non-strictly creates $j$-absolute $r$-extensions, then every pseudo-morphism of resolutions with domain $(\ell \jadj r)$ admits a pseudo-retraction $c \plx 1_C$.

Note that, unlike a morphism of resolutions, a pseudo-morphism of resolutions $c \colon (\ljr) \to (\ljrp)$ is not necessary a right-morphism. However, since $j$ is dense, it induces a unique right-morphism $(c, \rho) \colon (\ljr) \to ((\ell \d c) \jadj r')$ ( <ref>). Therefore, by <ref>, $\rho$ exhibits $r' \iso c \plx r$ as a $j$-absolute $r$-extension.
The proof of <ref> then carries through with respect to non-strict creation of colimits after replacing the equalities of tight-cells with isomorphisms.

Let $\jAE$ be a dense tight-cell. A tight-cell $r \colon D \to E$ is non-strictly $j$-monadic if and only if $r$ admits a left $j$-adjoint, the induced $j$-monad admits an algebra object, and $r$ non-strictly creates $j$-absolute $r$-extensions.

If $r$ is non-strictly $j$-monadic, then it admits a left $j$-adjoint for which the induced $j$-monad has an algebra object by definition. Furthermore, $r$ is the composite of an equivalence with $u_T$, hence non-strictly creates $j$-absolute $r$-extensions by <ref>. For the converse, observe that $r$ and $u_T$ both non-strictly create $j$-absolute $r$-extensions, so that $\unit_{\ljr}$ and $\unit_{\ljr} \plx 1_D$ exhibit $D$ and $\Alg(T)$ as pseudo-retracts of one another by <ref>, and hence that $\unit_{\ljr}$ is an equivalence.

Let $\jAE$ be a dense tight-cell. A tight-cell $r \colon D \to E$ is non-strictly $j$-monadic if and only if $r$ admits a left $j$-adjoint, the induced $j$-monad admits an algebra object, and $r$ non-strictly creates $j$-absolute colimits.

Follows directly from <ref> together with <ref>.

Equivalently, in the statements of <ref>, rather than ask that $r$ non-strictly create the requisite colimits, we could ask for $r$ to be conservative and preserve the requisite colimits (using <ref> and that conservative tight-cells are closed under equivalences and composition), which is often more convenient in practice.

Let $\jAE$ and $\jEI$ be tight-cells and suppose that $j'$ is dense. A non-strictly $(j \d j')$-monadic tight-cell $r \colon D \to E'$ is non-strictly $j'$-monadic if and only if it admits a left $j'$-adjoint and the induced $j'$-monad admits an algebra object.

The proof of <ref> carries through with respect to non-strict creation of colimits.

§ ENRICHED RELATIVE MONADICITY

Instantiating <ref>([relative-monadicity-theorem']$'$) and its corollaries in the equipment $\VCat$ of categories enriched in a monoidal category $\V$ (<cit.>), we immediately obtain a relative monadicity theorem for enriched relative monads. Under some additional assumptions on $\V$, we may make some simplifications.
We first make note of the following characterisation of $j$-absolute colimits, which is a special case of <cit.>.
Below, when they exist, we write $\P A$ for the $\V$-category of presheaves on a $\V$-category $A$, and write $n_j \colon E \to \P A$ for the nerve of a $\V$-functor $\jAE$, which is defined by $n_j e \defeq E(j{-}, e)$. Dually, we write $\cl Q Z$ for the $\V$-category of copresheaves on a $\V$-category $Z$, and write $m_i \colon U \to \cl Q Z$ for the co-nerve of a $\V$-functor $i \colon Z \to U$, which is defined by $m_i u \defeq U(u, i{-})$.

Let $\V$ be a complete left- and right-closed monoidal category and let $\jAE$ be a $\V$-functor with small domain.
Given a $\V$-distributor $p \colon X \lto Y$ and $\V$-functor $f \colon Y \to E$, a colimit $p \wc f$ in $E$ is $j$-absolute exactly when it is preserved by the nerve $n_j \colon E \to \P A$.

Since $A$ is small, $\P A$ exists by <cit.>, and the result follows by <cit.>.

We spell out <ref>([relative-monadicity-theorem-absolute']$'$) in the equipments $\VCat$ and $\VCat\co$, where $\V$ is a well-behaved monoidal category, using the characterisation of $j$-absolute colimits above.

Let $\V$ be a complete left- and right-closed monoidal category and let $\jAE$ be a dense $\V$-functor with small domain.
A $\V$-functor $r \colon D \to E$ is (non)strictly $j$-monadic if and only if $r$ admits a left $j$-adjoint and (non)strictly creates those colimits that are preserved by the nerve $n_j \colon E \to \P A$.

Since $A$ is small, $\V$-categories of algebras for $j$-monads exist by <cit.>.
The result then follows directly from <ref>([relative-monadicity-theorem-absolute']$'$) together with <ref>.

<ref>enriched-relative-monadicity-theorem$\co$[Enriched relative comonadicity]

Let $\V$ be a complete left- and right-closed monoidal category and let $i \colon Z \to U$ be a codense $\V$-functor with small domain.
A $\V$-functor $\ell \colon W \to U$ is (non)strictly $i$-comonadic if and only if $\ell$ admits a right $i$-coadjoint and (non)strictly creates those limits that are preserved by the co-nerve $m_i \colon U \to \cl Q Z$.

Since $A$ is small, copresheaf $\V$-categories exist, as do $\V$-categories of coalgebras for $i$-comonads by <cit.>. The result then follows as for <ref>.

When $A$ is large, presheaf $\V$-categories can no longer be assumed to exist, and so $j$-absoluteness cannot be characterised in terms of the nerve without a change of enrichment base. In this case <ref>([relative-monadicity-theorem']$'$) and <ref>([relative-monadicity-theorem-absolute']$'$) may be used directly.

From <ref>, taking $j = 1$, we recover the unenriched monadicity theorem of  [Theorem 3.5]pare1969absoluteness[Theorem]pare1969absolute[Theorem 7.3]pare1971absolute, which is a reformulation of the unenriched monadicity theorem of beck1966untitled in terms of absolute colimits; as well as an analogous characterisation of enriched monadicity ( [Theorem 2.11]bunge1969relative[Theorem II.2.1]dubuc1970kan). Taking $j$ to be dense and , we recover the unenriched relative monadicity theorems of  [Théoremè 2.5]diers1975jmonades[Theoreme 5.1]diers1976foncteur and [Corollary 2.8]lee1977relative.

It does not appear to be possible to obtain a characterisation of relative monadicity in terms of contractible coequalisers without much stronger assumptions on the root $j$, which seldom hold in examples of interest other than $j = 1$; we do not pursue such a characterisation.

§.§ Examples

We now present several prototypical situations in which the relative monadicity theorem may be applied; our intention is to demonstrate some typical situations in which the theorem is useful.

In the first situation we present, one has a notion of theory and an accompanying notion of algebra. For instance, we may show that the category of algebras for a finitary algebraic theory is monadic relative to the inclusion of finite sets into small sets ( <cit.>).

[Algebraic theories induce monads]

Denote by $\F$ the free category with strict finite coproducts on a single object $1$. The inclusion $j \colon \F \equiv \FinSet \ffto \Set$ of finite ordinals into small sets exhibits a cocompletion under sifted colimits, and is hence dense and .

Now consider a finitary algebraic theory $k \colon \F \to K$ in the sense of <cit.>,  an  functor preserving finite coproducts. The category of algebras for $k$ is, up to equivalence, the category $\Cart[K\op, \Set]$ of finite-product-preserving functors from $K\op$ to $\Set$. The forgetful functor, which we denote by $u_k \colon \Cart[K\op, \Set] \to \Set$, is the composite of $\Cart[k\op, \Set]$ with the equivalence $\Cart[\F\op, \Set] \equiv \Set$. Since $K$ has finite coproducts, $u_k$ is a continuous and sifted-cocontinuous functor between locally strongly finitely presentable categories <cit.>, and consequently admits a left adjoint $f_k \colon \Set \to \Cart[K\op, \Set]$. Denoting by $\sigma_L \colon K \ffto \Cart[K\op, \Set]$ the Yoneda embedding, we therefore have the following diagram in $\CAT$, in which the rightmost square is a pseudopullback.

\[\begin{tikzcd}
K & {\Cart[K\op, \Set]} & {[K\op, \Set]} \\
\F & \Set & {[\F\op, \Set]}
\arrow[""{name=0, anchor=center, inner sep=0}, "{u_k}", curve={height=-12pt}, from=1-2, to=2-2]
\arrow["{n_{\sigma_K}}", hook, from=1-2, to=1-3]
\arrow["{n_j}"', hook, from=2-2, to=2-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{[k\op, \Set]}", from=1-3, to=2-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "k", from=2-1, to=1-1]
\arrow[""{name=3, anchor=center, inner sep=0}, "{f_k}", curve={height=-12pt}, from=2-2, to=1-2]
\arrow["{\sigma_K}", hook, from=1-1, to=1-2]
\arrow["j"', hook, from=2-1, to=2-2]
\arrow["\dashv"{anchor=center}, draw=none, from=3, to=0]
\arrow["\iso"{description}, draw=none, from=2, to=3]
\arrow["{\it{ps.\;pb}}"{description}, draw=none, from=0, to=1]
\end{tikzcd}\]

Since $k$ is identity-on-objects, $[k\op, \Set]$ is an amnestic isofibration and strictly creates colimits. By the former property, the pseudopullback is equivalent to the strict pullback <cit.>.
The forgetful functor $u_k$ therefore non-strictly creates those colimits that $n_j$ preserves <cit.>, which are precisely the $j$-absolute colimits by <ref>. By <cit.>, we have $(j \d f_k) \jadj u_k$ and so $u_k$ is $j$-monadic by <ref>. Furthermore, by <ref>, it is additionally non-strictly monadic. Every finitary algebraic theory therefore induces both a $j$-monad, and a sifted-cocontinuous monad on $\Set$ – the former obtained by precomposing the latter by $j$ <cit.> – whose categories of algebras are concretely equivalent.

More generally, the methods of <ref> may be used to show that, for an arity class $\kappa$ in the sense of <cit.> (for instance, any regular cardinal), the category of algebras for a $\kappa$-ary algebraic theory is monadic relative to $\Set_\kappa \ffto \Set$, the full subcategory inclusion of the $\kappa$-small sets. When $\kappa = \N$, we recover finitary algebraic theories; and when $\kappa$ is the cardinality of the universe, we recover infinitary algebraic theories. Another useful case is given by taking $\kappa = \{ 1 \}$, as in the following example.

A unary algebraic theory is an  functor with domain $1$, and is understood syntactically to be a theory presented solely by unary operations. Concretely, each unary algebraic theory is equivalent to a monoid $M$, viewed as a one-object category, and its category of algebras is the presheaf category $\Set^{M\op}$ of right-actions of $M$ (sometimes called right $M$-sets). Consequently, a functor is $(1 \ffto \Set)$-monadic precisely when its domain is equivalent to a presheaf category on a single object, and its action is given by evaluation at that unique object.

Alternatively, we may give a characterisation via the relative monadicity theorem. Observe that the nerve of $1 \ffto \Set$ is isomorphic to the identity functor on $\Set$. Hence every colimit is $(1 \ffto \Set)$-absolute. A functor $u \colon D \to \Set$ is therefore $(1 \ffto \Set)$-monadic if and only if it creates colimits and admits a left $(1 \ffto \Set)$-adjoint. This latter condition holds exactly when $u$ is corepresentable,  when there exists an object $d \in D$ such that $u \iso D(d, {-})$.

<Ref> demonstrate the typical relationship between monadicity relative to different functors. Suppose that we have functors $\jAE$ and $\jEI$. In general, $(j \d j')$-monadicity is neither stronger nor weaker than $j'$-monadicity: $(j \d j')$-adjointness is a weaker property than $j'$-adjointness, but creation of the requisite $(j \d j')$-absolute colimits is a stronger property than creation of the requisite $j'$-absolute colimits. However, in many cases of interest, a $(j \d j')$-monadic functor will admit a left $j'$-adjoint, in which case <ref>([monadic-iff-left-adjoint']$'$) will apply.

Consider the inclusions $j \colon 1 \ffto \F$ and $j' \colon \F \ffto \Set$, so that $(j \d j')$-monadicity is the $(1 \ffto \Set)$-monadicity of <ref> and $j'$-monadicity is the $(\F \ffto \Set)$-monadicity of <ref>. By the adjoint functor theorem, the forgetful functor from the category of algebras for any unary algebraic theory admits a left adjoint, and is thus monadic, and hence also $j'$-monadic. Thus $(j \d j')$-monadicity implies $j'$-monadicity. The converse is not true in general: although the forgetful functor from the category of algebras for a finitary algebraic theory admits a left adjoint, and thus also a $(j \d j')$-adjoint, it will not create all colimits in general. For instance, the underlying set of a coproduct of monoids is not the coproduct of their underlying sets.

More generally, for arity classes $\kappa \subseteq \kappa'$, the methods above may be used to show that the category of algebras for a $\kappa$-ary algebraic theory is always $(\Set_{\kappa'} \ffto \Set)$-monadic. Taking $\kappa'$ to be the cardinality of the universe, every $\kappa$-ary algebraic theory is seen to induce a monad on $\Set$ of rank $\kappa$.

In the second situation we present, one has a class of weights $\Phi$, and a monad $T$ on a free $\Phi$-cocompletion $\Phi A$, for which $T$ preserves $\Phi$-weighted colimits. In this case, the category of algebras for $T$ is monadic relative to the cocompletion $A \to \Phi A$.

Let $\Phi$ be a class of weights, let $A$ be a small $\V$-category admitting a free cocompletion $A \to \Phi A$ under $\Phi$-weighted colimits, and let $T$ be a $\Phi$-cocontinuous monad on $\Phi A$. Since $T$ is $\Phi$-cocontinuous, it preserves $(A \to \Phi A)$-absolute colimits <cit.>. Hence, the forgetful functor $u_T \colon \Alg(T) \to \Phi A$ strictly creates $(A \to \Phi A)$-absolute colimits ( <cit.>). Consequently, since every cocompletion is dense, and $u_T$ admits a left $(A \to \Phi A)$-adjoint <cit.>, <ref> implies that the forgetful functor $u_T$ is strictly $(A \to \Phi A)$-monadic. For instance, the category of algebras for a finitary monad $T$ on a locally finitely presentable category $E$ is the category of algebras for the $(E_\tx{fp} \ffto E)$-monad induced by precomposing $T$ with the inclusion of the full subcategory $E_\tx{fp}$ of finitely presentable objects in $E$ <cit.>.

<Ref> gives an alternative method to <ref> for proving that finitary algebraic theories induce $(\F \ffto \Set)$-relative monads, at least supposing that one already knows that every algebraic theory induces a sifted-cocontinuous monad on $\Set$ (which is the cocompletion of $\F$ under sifted colimits).

In the third situation, one has a presentation of a structure by operations and equations, and consequently an induced notion of algebra for the presentation. For instance, we may show that the category of algebras for any quantitative equational theory in the sense of <cit.> is relatively monadic. We give the specific example of quantitative semigroups below; the general case follows analogously.

Denote by $\Met$ the category of extended metric spaces and nonexpanding maps, and by $j \colon \FinMet \ffto \Met$ the full subcategory inclusion of the finite metric spaces. For a metric space $X$, denote by $d_X \colon X \times X \to [0, \infty]$ its distance function. Since $\Met$ is a cartesian-monoidal category, we may consider the category $\Semigrp(\Met)$ of semigroups internal to $\Met$,  metric spaces $X$ equipped with an associative function $\ph \monprod \ph \colon X \times X \to X$ such that $\max\{d_X(x_1, x_1'), d_X(x_2, x_2')\} \geq d_X((x_1 \monprod x_2), (x_1' \monprod x_2'))$.
The forgetful functor $u \colon \Semigrp(\Met) \to \Met$ admits a left adjoint (an explicit construction is given in <cit.>), and hence a left $j$-adjoint <cit.>.
Moreover, we may prove that $u$ strictly creates every $j$-absolute colimit $C \defeq \colim_i (u(D_i))$, where $D$ is a diagram in $\Semigrp(\Met)$.

To do so, we shall show that each such $C$ admits a unique semigroup structure such that the coprojections $\copi_i$ are semigroup homomorphisms.
Since the nerve of $j$ preserves $C$ by assumption, every nonexpansive map $X \to C$ with finite domain factors through some coprojection $\copi_i \colon D_i \to C$.
In particular, every finite subobject of $C$ factors thus, and consequently every finite subset of $C$ is equal to the image $\copi_i(S)$, for some $i \in \ob D$, of a subset $S \subseteq D_i$, such that $d_{D_i}(x, x') = d_C(\copi_i(x), \copi_i(x'))$ for each pair $x, x' \in S$.

Uniqueness of the binary operation $\monprod$ on $C$ follows by considering two-element subsets, and associativity by considering three-element subsets.
Nonexpansiveness of $\monprod$ follows by considering a four-element subset, and using the condition on $d_{D_i}$ above.

It thereby follows that $u$ is strictly $(\FinMet \ffto \Met)$-monadic.

<Ref> gives another alternative method to <ref>, since every finitary algebraic theory admits a presentation <cit.>.

In fact, the three situations we have described – theories, cocontinuous monads, and presentations – are all facets of a more general relationship. To illustrate this, we give one final example: the relative monadicity of categories of algebras for a $j$-theory in the sense of [Definition 3.1.13]arkor2022monadic[Definition 3.1]lucyshyn2023enriched.

Let $\V$ be a locally small closed symmetric monoidal category with equalisers, embedding into a complete closed symmetric monoidal category $\V'$ <cit.>. Let $\jAE$ be a dense and  $\V$-functor, and let $\ell \colon A \to B$ be a $j$-theory,  an  $\V$-functor admitting a right $j$-adjoint $r \colon B \to E$. The category of (concrete) algebras for $\ell$ is defined to be the following pullback in $\V'\h\Cat$ <cit.>; we denote by $\yo_B$ the Yoneda embedding.

\[\begin{tikzcd}
B & \ell\h\Alg & {[B\op, \V]} \\
& E & {[A\op, \V]}
\arrow[""{name=0, anchor=center, inner sep=0}, "{u_\ell}"{description}, from=1-2, to=2-2]
\arrow["n", hook, from=1-2, to=1-3]
\arrow["{n_j}"', hook, from=2-2, to=2-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{[\ell\op, \V]}", from=1-3, to=2-3]
\arrow["{\yo_B}", curve={height=-30pt}, hook, from=1-1, to=1-3]
\arrow["r"', curve={height=12pt}, from=1-1, to=2-2]
\arrow["i"{description}, dashed, hook, from=1-1, to=1-2]
\arrow["{\it{pb}}"{description}, draw=none, from=0, to=1]
\end{tikzcd}\]

Just as in <ref>, it follows that the pullback is equivalent to the pseudopullback, and that $u_\ell$ strictly creates $j$-absolute colimits. The projection $\V'$-functor $n \colon \ell\h\Alg \to [B\op, \V]$ is , since $j$ is dense and  functors are closed under pullback. The relative adjunction $B(\ell, 1) \iso E(j, r)$ consequently induces a mediating $\V'$-functor $i \colon B \to \ell\h\Alg$, which is , since $i \d n \iso \yo_B$ and $n$ and $\yo_B$ are . By <cit.>, we therefore have $n \iso n_i$. Pseudocommutativity of the pseudopullback square thus asserts that $(\ell \d i) \colon A \to \ell\h\Alg$ is left- $j$-adjoint to $u_\ell$.
Consequently, by <ref>, the category of algebras for $\ell$ is strictly $j$-monadic in $\V'\h\Cat$. When $u_\ell$ furthermore admits a left adjoint ( $\ell$ is admissible in the sense of <cit.>), it is strictly monadic in $\V'\h\Cat$ by <ref>, recovering <cit.>.

<Ref> turns out to subsume the previous examples. However, explicating this connection is beyond the scope of the present paper. In future work, we shall develop this connection fully, establishing a formal correspondence between these notions ( [Chapters 3 & 7]arkor2022monadic[]arkor2022relative[5.1]lucyshyn2023enriched[]lucyshyn2022diagrammatic).

§ A PASTING LAW FOR RELATIVELY MONADIC ADJUNCTIONS

In this section, we recall the pasting law for relative adjunctions and show that it respects relative monadicity in an appropriate sense. As a consequence, we derive necessary and sufficient conditions for the composite $(r \d r')$ of a tight-cell $r$ with a $j$-monadic tight-cell $r'$ to be $j$-monadic. We continue to work in the context of a virtual equipment $\X$.

Consider the following diagram.

\[\begin{tikzcd}[sep=small]
&& C \\
&&& D \\
A &&&& E
\arrow["r", from=1-3, to=2-4]
\arrow["j"', from=3-1, to=3-5]
\arrow[""{name=0, anchor=center, inner sep=0}, "{r'}", from=2-4, to=3-5]
\arrow[""{name=1, anchor=center, inner sep=0}, "{\ell'}"{description}, from=3-1, to=2-4]
\arrow["\ell", from=3-1, to=1-3]
\arrow["\dashv"{anchor=center}, shift right=1, draw=none, from=1, to=0]
\end{tikzcd}\]

The left triangle is a relative adjunction ($\ell \radj{\ell'} r$) if and only if the outer triangle is a relative adjunction ($\ell \jadj r \d r'$).

We shall specialise <ref> to the case in which $r'$ is strictly $j$-monadic, and study the algebras for the relative monads induced by the corresponding relative adjunctions.

Consider the following situation, and let $T$ be the $f_{T'}$-monad induced by $\ell \radj{f_{T'}} r$.

\[\begin{tikzcd}[sep=small]
&& C \\
&&& {\Alg(T')} \\
A &&&& E
\arrow[""{name=0, anchor=center, inner sep=0}, "r", from=1-3, to=2-4]
\arrow["j"', from=3-1, to=3-5]
\arrow[""{name=1, anchor=center, inner sep=0}, "{u_{T'}}", from=2-4, to=3-5]
\arrow[""{name=2, anchor=center, inner sep=0}, "{f_{T'}}"{description}, from=3-1, to=2-4]
\arrow[""{name=3, anchor=center, inner sep=0}, "\ell", from=3-1, to=1-3]
\arrow["\dashv"{anchor=center, pos=.6}, draw=none, from=2, to=1]
\arrow["\dashv"{anchor=center, rotate=19}, shift right=2, draw=none, from=3, to=0]
\end{tikzcd}\]

Postcomposition by $u_{T'}$ induces a bijection between $T$-algebras and $(T \d u_{T'})$-algebras, and their graded morphisms.

We denote the carrier of the $f_{T'}$-monad $T$ by $t = \ell \d r$, the unit by $\eta \colon f_{T'} \tto t$, and the extension operator by $\dagger \colon \Alg(T')(f_{T'}, t) \tto \Alg(T')(t, t)$;
and denote the transposition operators of the relative adjunction $f_{T'} \jadj u_{T'}$ by
\sharp' \colon \Alg(T')(f_{T'}, 1) \iso E(j, r) \cocolon \flat'
Recall that every $T$-algebra (morphism) induces a $(T \d u_{T'})$-algebra (morphism) by postcomposing $u_{T'}$ (<cit.>), and that every $(T \d u_{T'})$-algebra (morphism) induces a $T'$-algebra (morphism) via the left-morphism $(r, \eta)$.

Given a $(T \d u_{T'})$-algebra $(e, \aop')$, we shall construct a $T$-algebra. The carrier $x \colon X \to \Alg(T')$ is induced by the universal property of $\Alg(T')$ from the $T'$-algebra induced by $(r, \eta)$. For the algebra structure, observe that the 2-cell $\aop' \colon E(j, e) \tto E(u_{T'} t, e)$ forms an $E(j, e)$-graded $T'$-algebra morphism in the sense of <cit.>, from $(T \d u_{T'})$ to the induced $T'$-algebra on $e$, following <cit.>. Hence, the universal property of $\Alg(T')$ induces a 2-cell $\unit_{\aop'} \colon E(j, e) \tto \Alg(T')(t, x)$ such that postcomposition by $u_{T'}$ yields $\aop'$. Let $\aop$ be as follows.

\[\aop \colon \Alg(T')(f_{T'}, x) \xtto{\sharp'(1, x)} E(j, u_{T'} x) = E(j, e) \xtto{\unit_{\aop'}} \Alg(T')(t, x)\]

The pair $(x, \aop)$ forms a $T$-algebra. The unit law follows: 1.1 by the $\sharp'$–$\flat'$ isomorphism; 1.2 by bending $u_{T'}$; 1.3 by the definition of $\unit_{\aop'}$; 1.4 by the unit law for $\aop'$; and 1.5 by the $\sharp'$–$\flat'$ isomorphism. To show the extension law, it suffices to show the equality of the corresponding 2-cells under postcomposition by $u_{T'}$, using the universal property of $\Alg(T')$. This equality follows: 2.1 by the definition of $\unit_{\aop'}$; 2.2 by the $\sharp'$–$\flat'$ isomorphism; 2.3 by the extension operator law for $\aop'$; and 2.4 by the definition of $\unit_{\aop'}$ twice.

(3,7)[trim y]

tangle(3,7)[trim y]

tangle(4,7)[trim y]

(4,7)[trim y]

tangle(3,7)[trim y]

tangle(2,7)[trim y]

(5,5)[trim y]

tangle(5,5)[trim y]

(6,6)[trim y]

tangle(6,6)[trim y]

(6,7)[trim y]

Every $(p_1, \ldots, p_n)$-graded $(T \d u_{T'})$-algebra morphism $\epsilon \colon (e_1, \aop_1') \to (e_2, \aop_2')$, is also a $(p_1, \ldots, p_n)$-graded $T'$-algebra morphism between the induced $T'$-algebras. By the universal property of $\Alg(T')$, $\epsilon$ thus induces a 2-cell $\unit_\epsilon \colon \Alg(T')(1, x_1), p_1, \ldots, p_n \tto \Alg(T')(1, x_2)$. This is a $(p_1, \ldots, p_n)$-graded $T$-algebra morphism $(x_1, \aop_1) \to (x_2, \aop_2)$ since, by postcomposing $u_{T'}$, we have:

(6,5)[trim y]
at (5,2.9) $\cdots$;

tangle(6,5)[trim y]
at (5,2.9) $\cdots$;

(6,5)[trim y]
at (5,1.9) $\cdots$;

tangle(6,5)[trim y]
at (5,0.9) $\cdots$;

3.1 by the definitions of $\unit_{\aop_1'}$ and $\unit_\epsilon$; 3.2 by the compatibility law for $\epsilon$; and 3.3 by the definitions of $\unit_{\aop_2'}$ and $\unit_\epsilon$.

We must now show that these assignments are inverse to one another. Suppose we have a $T$-algebra $(x, \aop)$, inducing a $(T \d u_{T'})$-algebra $((e \d u_{T'}), \aop')$. We must show that the $T'$-algebra induced by $x$ coincides with that induced by the $(T \d u_{T'})$-algebra. By definition $(x \d u_{T'})$ is the carrier of both $T'$-algebras. Using the unit law for $\aop$, the latter $T'$-algebra structure is given by

(4,5)[trim y]

tangle(4,5)[trim y]

which is the former $T'$-algebra structure.
That the $T$-algebra structure induced by $\aop'$ coincides with $\aop$ follows from the universal property of $\Alg(T')$ with respect to graded algebra morphisms.

Suppose we have a $(p_1, \ldots, p_n)$-graded $T$-algebra morphism $\chi \colon (x_1, \aop_1) \to (x_2, \aop_2)$. The $T'$-algebra morphism induced directly by $\chi$, and the $T'$-algebra morphism induced from the $(T \d u_{T'})$-algebra morphism are both given by $(\chi \d u_{T'})$ by definition. Thus the universal property of $\Alg(T')$ with respect to graded algebra morphisms implies that the induced $T$-algebra morphism is $\chi$.

The other direction follows immediately from the universal property of $\Alg(T')$ (with respect both to algebras and graded algebra morphisms).

A relative adjunction $\ljr$ inducing a relative monad $T$ is strictly $j$-relatively monadic (alternatively strictly monadic relative to $j$, or simply strictly $j$-monadic) if $T$ admits an algebra object and $\ljr$ is isomorphic to $f_T \jadj u_T$ in the category of resolutions of $T$.

Note that a tight-cell $r$ is strictly $j$-monadic if and only if it admits a left $j$-adjoint and the induced $j$-adjunction is strictly $j$-monadic, since both properties exhibit $r$ as being the right $j$-adjoint of a terminal resolution ( <cit.>).

It follows from <ref> that the pasting law for relative adjunctions (<ref>) respects relative monadicity.

Consider the following diagram.

\[\begin{tikzcd}[sep=small]
&& C \\
&&& D \\
A &&&& E
\arrow["r", from=1-3, to=2-4]
\arrow["j"', from=3-1, to=3-5]
\arrow[""{name=0, anchor=center, inner sep=0}, "{r'}", from=2-4, to=3-5]
\arrow[""{name=1, anchor=center, inner sep=0}, "{\ell'}"{description}, from=3-1, to=2-4]
\arrow["\ell", from=3-1, to=1-3]
\arrow["\dashv"{anchor=center}, shift right=1, draw=none, from=1, to=0]
\end{tikzcd}\]

Suppose that $\ell' \jadj r'$ is strictly $j$-monadic. Then the left triangle exhibits a strictly $\ell'$-monadic relative adjunction if and only if the outer triangle exhibits a strictly $j$-monadic relative adjunction.

Follows directly from <ref>, the latter exhibiting the universal properties of $\Alg(T)$ and $\Alg(T \d r')$ as being equivalent.

We rephrase <ref> to concern only the tight-cells $r$ and $r'$, rather than relative adjunctions, which gives necessary and sufficient conditions for the composite $(r \d r')$ to be relatively monadic.

Let $\jAE$ and $r' \colon D \to E$ be tight-cells, and assume that $r'$ is (non)strictly $j$-monadic, with left $j$-adjoint $\ell'$.
A tight-cell $r \colon C \to D$ is (non)strictly $\ell'$-monadic if and only if $(r \d r') \colon C \to E$ is (non)strictly $j$-monadic.

Let $T'$ be the $j$-monad induced by $\ell' \jadj r'$.
Without loss of generality, we may assume $\ell' \jadj r'$ is strictly $j$-monadic: the following diagram commutes, so that non-strict $\ell'$-monadicity of $r$ is equivalently non-strict $f_{T'}$-monadicity of $(r \d {\equiv})$, and non-strict $j$-monadicity of $(r \d r')$ is equivalently non-strict $j$-monadicity of $(r \d {\equiv} \d u_{T'})$.

\[\begin{tikzcd}
& D \\
& {\Alg(T')} \\
A && E
\arrow["j"', from=3-1, to=3-3]
\arrow[""{name=0, anchor=center, inner sep=0}, "{u_{T'}}"{description}, from=2-2, to=3-3]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_{T'}}"{description}, from=3-1, to=2-2]
\arrow["\equiv"{description}, from=1-2, to=2-2]
\arrow["{r'}", curve={height=-12pt}, from=1-2, to=3-3]
\arrow["{\ell'}", curve={height=-12pt}, from=3-1, to=1-2]
\arrow["\dashv"{anchor=center}, shift right=1, draw=none, from=1, to=0]
\end{tikzcd}\]

The strict case then follows directly from <ref>, considering the following triangle, where $\ell \colon A \to C$ is the left relative adjoint of either $r$ or of $(r \d u_{T'})$.

\[\begin{tikzcd}[sep=small]
&& C \\
&&& {\Alg(T')} \\
A &&&& E
\arrow["r", from=1-3, to=2-4]
\arrow["j"', from=3-1, to=3-5]
\arrow[""{name=0, anchor=center, inner sep=0}, "{u_{T'}}", from=2-4, to=3-5]
\arrow[""{name=1, anchor=center, inner sep=0}, "{f_{T'}}"{description}, from=3-1, to=2-4]
\arrow["\ell", from=3-1, to=1-3]
\arrow["\dashv"{anchor=center}, draw=none, from=1, to=0]
\end{tikzcd}\]

The non-strict case follows therefrom by precomposing $r$ by equivalences.

<Ref> may be seen as a significant generalisation of classical sufficiency results for the monadicity of a composite $(r \d r')$ functor ( <cit.>). When $r$ is , it may alternatively be seen as a weakening of 's HSP theorem <cit.>, giving necessary and sufficient conditions for a subcategory of a category of algebras to itself be a category of algebras (the HSP theorem further asks that, in this case, the induced monad morphism be a regular epimorphism, which expresses that the subcategory is formed by imposing additional axioms on the algebras).
While we expect that the HSP theorem may be derived from <ref>, we shall not do so here.

To conclude, we mention two useful consequences of <ref>. The first is the following classical cancellability result for monadic adjunctions ( <cit.>).

Consider the following situation, in which the monad induced by $\ell \adj r$ admits an algebra object.

\[\begin{tikzcd}
C & D & E
\arrow[""{name=0, anchor=center, inner sep=0}, "r", shift left=2, from=1-1, to=1-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{r'}", shift left=2, from=1-2, to=1-3]
\arrow[""{name=2, anchor=center, inner sep=0}, "{\ell'}", shift left=2, from=1-3, to=1-2]
\arrow[""{name=3, anchor=center, inner sep=0}, "\ell", shift left=2, from=1-2, to=1-1]
\arrow["\dashv"{anchor=center, rotate=90}, draw=none, from=3, to=0]
\arrow["\dashv"{anchor=center, rotate=90}, draw=none, from=2, to=1]
\end{tikzcd}\]

If $r'$ and $(r \d r')$ are (non)strictly monadic, then so is $r$.

From <ref>, we have that $r'$ is (non)strictly $\ell'$-monadic. The result then follows from <ref>([monadic-iff-left-adjoint']$'$), since the monad induced by $\ell \adj r$ admits an algebra object.

The second establishes that algebraic tight-cells – that is, those tight-cells between algebra objects that are induced by relative monad morphisms – create limits and certain colimits, and are themselves monadic as soon as they admit left adjoints.

Let $\jAE$ be a tight-cell, and consider a commutative triangle as follows, where $r$ and $r'$ are (non)strictly $j$-monadic.

\[\begin{tikzcd}
{D'} && D \\
& E
\arrow["i", from=1-1, to=1-3]
\arrow["{r'}"', from=1-1, to=2-2]
\arrow["r", from=1-3, to=2-2]
\end{tikzcd}\]

* $i$ (non)strictly creates limits and those colimits in $D$ that are sent by $r$ to $j$-absolute colimits.

* $i$ is (non)strictly monadic if and only if it admits a left adjoint.

Denote by $\ell$ the left $j$-adjoint of $r$.
By <ref>, $i$ is (non)strictly $\ell$-monadic.

For (1), $i$ (non)strictly creates limits and $\ell$-absolute colimits by <ref>. In particular, the (non)strict lifting of any $j$-absolute colimit through $r$ is $\ell$-absolute: for each loose-cell $p \colon Y \lto Z$ and tight-cell $f \colon Z \to D$ admitting a $j$-absolute colimit $p \wc (f \d r)$, we have

\begin{align*}
D(\ell, f) \odotl p & \iso E(j, r f) \odotl p \tag{$\ell \jadj r$} \\
& \iso E(j, p \wc (f \d r)) \tag{$p \wc (f \d r)$ is $j$-absolute} \\
& \iso E(j, r (p \wc f)) \tag{\cref{u_T-creates-j-absolute-colimits,u_T-nonstrict-creation}} \\
& \iso D(\ell, p \wc f) \tag{$\ell \jadj r$}
\end{align*}

with canonicity of this isomorphism following from that for $j$-absoluteness.
(Alternatively, we could establish (1) as a consequence of <ref> together with simple observations about the creation of limits and colimits, but find the approach above to be particularly satisfying.)

(2) follows immediately from $\ell$-monadicity of $i$ by <ref>([monadic-iff-left-adjoint']$'$).

The preservation of limits and certain colimits in <ref> is often sufficient, via an adjointness theorem, to imply that, for sufficiently nice roots $j$, every algebraic tight-cell admits a right adjoint, and hence is monadic.
We end with an example of this situation.

From <ref>, we have that the category of algebras $\Cart[L\op, \Set]$ of a finitary algebraic theory $\ell \colon \F \to L$ is $(j \colon \F \ffto \Set)$-monadic. Since $u_\ell$ and $n_j$ preserve sifted colimits, sifted colimits in $\Cart[L\op, \Set]$ are lifted non-strictly by $u_\ell$ via <ref>. Consequently, every concrete functor $i$ between categories of algebras for finitary algebraic theories preserves limits and sifted colimits.

\[\begin{tikzcd}
{\Cart[L'\op, \Set]} && {\Cart[L\op, \Set]} \\
& \Set
\arrow[""{name=0, anchor=center, inner sep=0}, "i"', shift right=2, from=1-1, to=1-3]
\arrow["{u_{\ell'}}"', from=1-1, to=2-2]
\arrow["{u_\ell}", from=1-3, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, shift right=2, dashed, from=1-3, to=1-1]
\arrow["\dashv"{anchor=center, rotate=-90}, draw=none, from=1, to=0]
\end{tikzcd}\]

Since categories of algebras for algebraic theories are locally strongly finitely presentable, and every continuous and sifted-cocontinuous functor therebetween is a right adjoint, every such $i$ is a right adjoint, and hence is non-strictly monadic ( [Theorem IV.2.1]lawvere1963functorial[Corollary 1.5.2]manes1967triple).

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

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