## April 28, 2014

#### Posted by Emily Riehl

Guest post by Sam van Gool

Monads provide a categorical setting for studying sets with additional structure. Similarly, 2-monads provide a 2-categorical setting for studying categories with additional structure. While there is really only one natural notion of algebra morphism in the context of monads, there are several choices of algebra morphism in the context of 2-monads. The interplay between these different kinds of morphisms is the main focus of the paper that I discuss in this post:

I will give an overview of the results and methods used in this paper. Also, especially towards the end of my post, I will also indicate some points that I think could still be clarified further by formulating some questions, which will hopefully lead to fruitful discussions below.

This post forms the 9th instalment of the series of posts written by participants of the Kan Extension Seminar, of which I’m very glad to be a part. In preparing the post I have greatly benefited from discussions with the other participants in the seminar, and of course with the seminar’s organizer, Emily Riehl. I am very grateful for the enthusiasm, encouragement and guidance that you all offered.

### 2-monads, their algebras, and their morphisms

Two-dimensional universal algebra goes beyond the $\mathbf{Cat}$-enriched setting in that it allows for non-strict morphisms. Consider the following (very) simple example.

Example. For a category $A$, let $T A$ be the category $A$ provided freely with a terminal object. This assignment can be extended to a 2-monad $T$ on $\mathbf{Cat}$. Then:

• an algebra for $T$ is (entirely determined by giving) a pair $(A,t_A)$ where $A$ is a category and $t_A$ is a designated terminal object in $A$;
• a strict morphism $(A,t_A) \to (B,t_B)$ is a functor $f$ for which $f(t_A) = t_B$;
• a pseudo morphism is a functor $f$ such that $f(t_A)$ is isomorphic to $t_B$;
• a colax morphism is just any functor from $A$ to $B$, with no additional requirement on the terminal object.

If you didn’t know them already, you will probably have guessed the general definitions of strict, pseudo and lax morphisms by now, as well as the definition of 2-cells between them. Note that, in this post, all 2-monads and algebras for them will be strict, as in [BKP].

For any 2-monad $T$, we thus get the following inclusions of 2-categories:

$T\text{-}\mathrm{Alg}_s \to T\text{-}\mathrm{Alg}_p \to T\text{-}\mathrm{Alg}_l.$

(In [BKP], the category $T\text{-}\mathrm{Alg}_p$ is denoted by $T\text{-}\mathrm{Alg}$.) Roughly the first half of the paper [BKP] is devoted to the construction of left adjoints (in the 2-categorical sense) to these inclusion functors.

Note that $T\text{-}\mathrm{Alg}_s$ is simply the Eilenberg-Moore $\mathcal{V}$-category of the $\mathcal{V}$-enriched monad $T$ in the case where $\mathcal{V} = \mathbf{Cat}$, in the sense of the second paper that we read in this seminar. The categories $T\text{-}\mathrm{Alg}_p$ and $T\text{-}\mathrm{Alg}_l$, on the other hand, are special to the $\mathbf{Cat}$-enriched setting.

### Limits in $T$-Alg$_p$

The category $T\text{-}\mathrm{Alg}_s$ has all 2-limits that the base 2-category $\mathcal{K}$ has. For $T\text{-}\mathrm{Alg}_p$, the situation is more subtle.

Example (c’t’d). In the example where $T A$ is $A$ provided freely with a terminal object, let $1 = \{\ast\}$ be the terminal category and $I$ the category with two objects $0$, $1$ and a unique isomorphism between them. There are two pseudo-morphisms $(1,\ast) \to (I,0)$, one sending $\ast$ to $0$, the other sending $\ast$ to $1$. However, if $C \to 1$ is any functor which equalizes these two morphisms, then $C$ is empty, and so it does not admit a $T$-algebra structure. Thus, the category $T\text{-}\mathrm{Alg}_p$ does not admit equalizers in general.

Assuming that the $2$-category $\mathcal{K}$ is complete, it is however possible to construct the following limits in $T\text{-}Alg_p$:

• Products,
• Inserters and iso-inserters,
• Equifiers,

and they are created by the forgetful functor $T\text{-}Alg_p \to \mathcal{K}$. As we saw in last week’s post, these PIE-limits allow for the construction of many other limits. In particular, from the results discussed last week, we see that $T\text{-}\mathrm{Alg}_p$ also has inverters and co-tensors, and hence also lax and pseudo limits.

It is also worth noting that each of the results on existence of limits “restricts to strict” (for lack of a better name), by which I mean that, for each of these limits, there exists a limiting cone such that the algebra 1-cells in the limiting cone:

1. are strict, and

2. detect strictness.

For example, for any parallel pair $f, g : B \to C$ in $T\text{-}\mathrm{Alg}_p$ there is an inserter $p : A \to B$ such that (1) $p$ is strict, and (2) if $ph$ is strict for some algebra morphism $h : D \to A$, then $h$ is strict.

### The pseudomorphism classifier

Example (c’t’d). In the example where $T A$ is $A$ provided freely with a terminal object, note that pseudo-morphisms can be mimicked using strict morphisms: for any algebra $(A,t_A)$, consider the algebra $(A',t_{A'})$, defined by adding one new object $t_{A'}$ and an isomorphism $t_{A} \cong t_{A'}$ to $A$. It is then clear that, for any algebra $(B,t_B)$, strict morphisms $(A',t_{A'}) \to (B,t_B)$ correspond to pseudo-morphisms $(A,t_A) \to (B,t_B)$. In fact, this correspondence is an isomorphism between the categories of morphisms and natural transformations between them.

The following theorem, which is arguably at the heart of the paper [BKP], says that the above phenomenon in fact occurs for any reasonably well-behaved 2-monad.

Theorem. Let $T$ be an accessible 2-monad on a 2-category $\mathcal{K}$ that is complete and cocomplete. Then the inclusion 2-functor $T\text{-}\mathrm{Alg}_s \to T\text{-}\mathrm{Alg}_p$ has a left adjoint.

Proof (Sketch). The proof of the theorem consists of three steps:

1. A general fact: in order to find a left adjoint to a 2-functor $G : \mathcal{K} \to \mathcal{L}$, it suffices to find a left adjoint to its underlying ordinary functor $G_o$, provided that $\mathcal{K}$ has cotensors with the walking arrow category $2$ and $G$ preserves them.
2. Using (1), one shows that there exists a left adjoint, $()^o$, to the inclusion functor $T\text{-}\mathrm{Alg}_s \to T/{\mathcal{K}},$ where $T/{\mathcal{K}}$ is the comma 2-category.
3. The hardest part: pseudo-morphisms out of a $T$-algebra $(A,a)$ can be mimicked by $T/\mathcal{K}$-morphisms out of a certain object $(C,c,Z)$ of $T/{\mathcal{K}}$.

Now, composing (2) and (3), one associates to any $T$-algebra $(A,a)$ the $T$-algebra $(C,c,Z)^o$ and observes that this gives an ordinary (1-categorical) left adjoint to $T\text{-}\mathrm{Alg}_s \to T\text{-}\mathrm{Alg}_p$. Then, by (1) and the fact that cotensors exist in $T\text{-}\mathrm{Alg}_p$, it is also a 2-categorical left adjoint. $\qed$

The image under the left adjoint of an algebra $A$, seen as an object of $T\text{-}\mathrm{Alg}_p$, is denoted by $A'$ and called the pseudo-morphism classifier of $A$. Under the conditions of the Theorem, there is also a lax morphism classifier.

There are more conceptual proofs of these facts, using the concept of codescent objects; see, for example, this paper (which will be discussed in these series in a month or so) and Section 4 of the 2-categories companion by Stephen Lack. The latter paper, by the way, has been an indispensable source for me in preparing this post, and those who are familiar with it will probably recognize its influence throughout the post.

### Flexibility

We denote by the letters $p$ and $q$ the unit and co-unit of the adjunction

$J : T\text{-}\mathrm{Alg}_s \leftrightarrows T\text{-}\mathrm{Alg}_p : ()'$

from the above theorem. For any algebra $A$ in $T\text{-}\mathrm{Alg}_p$, the morphism $q_A : A' \to A$ is in fact always a surjective equivalence in the 2-category $T\text{-}\mathrm{Alg}_p$, but in general $q_A$ does not even need to be an equivalence in $T\text{-}\mathrm{Alg}_s$, as we will see shortly. If $q_A : A' \to A$ is an equivalence in $T\text{-}\mathrm{Alg}_s$, then $A$ is called semi-flexible, and $A$ is called flexible if $q_A$ is a surjective equivalence in $T\text{-}\mathrm{Alg}_s$. The flexible objects are the cofibrant objects in a model structure on $T\text{-}\mathrm{Alg}_s$ lifted from the model structure on $\mathcal{K}$, and the pseudomorphism classifier $A'$ is then a special cofibrant replacement of $A$ (see Section 7.3 of the 2-categories companion for more details about this).

Several equivalent characterizations of flexibility and semi-flexibility are given in Theorems 4.4 and 4.7, respectively, of [BKP]. One useful equivalent way to say that a $T$-algebra $A$ is semi-flexible is that every pseudo-morphism out of $A$ is isomorphic to a strict morphism out of $A$. With this definition, we can see that not every $T$-algebra is semi-flexible:

Example. Let $T$ be the 2-monad on $\mathbf{Cat}$ whose algebras are small categories with assigned finite limits. Let $A$ be the terminal category, with finite limits assigned in the only possible way. Let $B$ be any category with assigned finite limits in which $t_B$ is the assigned terminal object and the assigned product $t_B \times t_B$ is not equal to $t_B$ (the two objects will of course be isomorphic). Then the functor $A \to B$ which sends the unique object of $A$ to $t_B$ is a pseudo-morphism, but it is clearly not isomorphic to any strict morphism.

The following example shows that flexibility and semi-flexibility are really different concepts.

Example. Categories whose objects are functors can also often be represented as the $T$-algebras for an appropriate monad $T$ on an appropriate base 2-category $\mathcal{K}$. For instance, there is a 2-monad $T$ on $\mathbf{Cat} \times \mathbf{Cat}$, given on objects by $T(X,Y) := (X,X+Y)$, such that $T$-algebras are functors, a pseudomorphism from $f : A \to B$ to $g : C \to D$ is a diagram of the form $\begin{matrix} A & \overset{f}{\to} & B \\ u \downarrow & \overset{\alpha}{\cong} & \downarrow v \\ C & \overset{g}{\to} & D \end{matrix}$ and such a pseudomorphism is strict exactly when $\alpha$ is the identity. Now, letting $1$ denote the terminal category, it is easy to describe the pseudomorphism classifier of the $T$-algebra $a : A \to 1$: this is the inclusion functor $j : A \to \overline{A}$, where $\overline{A}$ is the indiscrete category on objects $\mathrm{ob}(A) + \{\ast\}$ (As a simple but nice exercise, you may check that, indeed, any pseudomorphism out of the algebra $a : A \to 1$ corresponds uniquely to a strict morphism out of the algebra $j : A \to \overline{A}$.) Now, letting $I$ again denote the category with two objects $0$, $1$ and a unique isomorphism between them, one may check that the algebra $I \to 1$ is equivalent in $T\text{-}\mathrm{Alg}_s$ to the algebra $1 \to 1$, which is flexible, and therefore $I \to 1$ is semi-flexible. However, $I \to 1$ is not flexible. (See example 4.11 in [BKP]).

### Biadjunctions and bicolimits in $T\text{-}\mathrm{Alg}_p$

So far, we have only considered limits, which one would expect to exist in a category of algebras. On the other hand, we wouldn’t generally expect colimits to exist in a category of algebras, but as it turns out, in the last section of [BKP], the authors prove that:

1. the category $T\text{-}\mathrm{Alg}_p$ admits bicolimits, and
2. any strict map of 2-monads $\theta : S \to T$ induces a map $T\text{-}\mathrm{Alg}_p \to S\text{-}\mathrm{Alg}_p$ that has a left biadjoint.

Both of these results are consequences of the following more technical fact:

Theorem. If $G : T\text{-}\mathrm{Alg}_p \to \mathcal{L}$ is a 2-functor so that the composite 2-functor

$\begin{matrix} T-Alg_s & \overset{J}{\to} & T-Alg_p & \overset{G}{\to} & L \end{matrix}$

has a left adjoint $H$, then $H$ maps into flexible algebras, and $J \circ H$ is left biadjoint to $G$.

From the above theorem and the relation between biadjoints and bicolimits that we discussed last week, bicolimits can now be constructed in $T\text{-}\mathrm{Alg}_p$, as claimed in (1) above. To prove (2), one first notices that 2-functor $\theta^* : T\text{-}\mathrm{Alg}_s \to S\text{-}\mathrm{Alg}_s$ extends to a 2-functor $\theta^\# : T\text{-}\mathrm{Alg}_p \to S\text{-}\mathrm{Alg}_p$ making the diagram

$\begin{matrix} T\text{-}\mathrm{Alg}_s & \overset{\theta^*}{\to} & S\text{-}\mathrm{Alg}_s \\ J\downarrow & \quad & \downarrow J \\ T\text{-}\mathrm{Alg}_p & \overset{\theta^\sharp}{\to} & S\text{-}\mathrm{Alg}_p \end{matrix}$

commute. One may then apply the Theorem in the case $G = \theta^\#$.

Above I motivated the concepts and theorems in [BKP] with some simple examples of 2-monads. The last section of [BKP] contains many more examples. About the general method for constructing such examples, the authors make the following interesting comment.

“In practice one is seldom presented with a 2-monad and invited to consider its algebras; more commonly one contemplates some structure borne by a category (…) and one concludes in certain cases that the structure is given by an action of a 2-monad (…)”

With this comment in mind, one may now construct 2-monads whose algebras are monoidal categories, symmetric monoidal closed categories (here the 2-monad is over the 2-category $\mathbf{Cat}_g$, where the 2-cells are only taken to be natural isomorphisms), and even finitary 2-monads themselves (they are the algebras for a certain 2-monad $R$ on the functor 2-category $[\mathbf{Cat}_f,\mathbf{Cat}]$, where $\mathbf{Cat}_f$ is the full subcategory of $\mathbf{Cat}$ consisting of the finitely presentable objects. This perspective was exploited in a later paper by Kelly and Power on presentations of 2-monads.

A final point of interest is that one may distinguish a special kind of 2-monad $T$, namely those for which the $T$-algebra structure on an object $A$ is unique if it exists. Such 2-monads $T$ define a property of rather than a structure on the objects of the base 2-category $\mathcal{K}$, and may thus be called property-like (as they are in this later paper by Kelly and Lack). As the authors of [BKP] remark, it “may well be a hard problem” how to distinguish the property-like 2-monads $T$ from, say, a presentation for them. A particular class of 2-monads which are ‘property-defining’ are the lax-idempotent 2-monads (which also go by the names “quasi-idempotent” and “Kock-Zöberlein” 2-monads).

### Questions

Let me finish with a (non-exhaustive) list of questions that may be interesting to discuss below.

1. Can the fact that limits in $T\text{-}\mathrm{Alg}_p$ can be chosen with a fair amount of “strictness” be understood using this account of lax / pseudo limits for morphisms between $T$-algebras using $\mathcal{F}$-enrichment?

2. The flexible algebras are exactly the strict retracts of pseudomorphism classifiers. The latter are “free algebras”, in some sense (at least in the sense that they are the images of a left adjoint). This suggests that one could think of the concept ‘flexible algebra’ as a 2-categorical version of the familiar concept ‘projective algebra’ in the 1-categorical setting. Is this a good intuition, and if so, can it be (or has it already been) made more precise?

3. In order to better understand the concept of flexible algebra and the biadjunctions in the later part of [BKP], it would probably be useful to study different examples of 2-monads, and in particular, answer the following questions in such examples:

(a) is there a concrete construction of the pseudomorphism classifier?

(b) which algebras are (semi-)flexible?

(c) (for a strict map between 2-monads) what does the biadjunction do?

Posted at April 28, 2014 9:31 PM UTC

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### Re: On Two-Dimensional Monad Theory

Very nice! A couple of slight corrections: (1) In your example of categories with a terminal object, the arbitrary functors $A\to B$ are actually the colax morphisms, whereas every lax morphism is automatically pseudo (this is because that 2-monad is colax-idempotent).

(2) You say “we wouldn’t generally expect colimits to exist in a category of algebras”, but actually in the case when the monad is accessible, we would. At least, they do exist for 1-categories of algebras and in $T\text{-}Alg_s$, by a transfinite construction. The point here is to show that, still assuming the monad to be accessible, colimits (of the bicategorical sort) also exist in $T\text{-}Alg_p$.

Posted by: Mike Shulman on April 28, 2014 10:59 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Argh! I should have caught that…

The point is that, by convention, in a lax monad morphism $f \colon (TA,a) \to (Tb,b)$, the 2-cell points down:

$\begin{matrix} TA & \overset{Tf}{\to} & TB \\ a\downarrow & \Downarrow \overline{f} & \downarrow b\\ A & \overset{f}{\to} & B \end{matrix}$

So if $T$ is the 2-monad for monoidal categories, a lax morphism is a lax monoidal functor. But if $T$ is the 2-monad for categories with a terminal object, then the 2-cell of a lax morphism points away from the terminal object in $B$; colax is the automatic direction.

I’ll fix this in the entry.

Posted by: Emily Riehl on April 28, 2014 11:10 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

The way I remember the convention is that a lax monoidal functor preserves monoids, whereas a colax monoidal functor preserves comonoids.

Thus, if $A$ and $B$ are monoidal categories, and $f:A\to B$ is a functor, and $a$ is a monoid in $A$, with unit $i\to a$ and multiplication $a\otimes a \to a$, then applying $f$ to these we obtain $f(i) \to f(a)$ and $f(a\otimes a) \to f(a)$. In order for $f(a)$ to be again a monoid, we need maps $i\to f(i)$ and $f(a) \otimes f(a) \to f(a\otimes a)$ to compose these with. So these are the “lax direction”; the other is colax.

Generalizing from this, the constraints of a lax morphism $f$ go from “the algebra-structure applied outside $f$” to “the algebra-structure applied inside $f$”. This gives you the direction of the 2-cell that Emily drew.

This convention has certain slightly odd consequences, such as the fact that colax natural transformations seem to occur more naturally (in some contexts) than lax ones, or the fact that the category $T\text{-}Alg_l$ of algebras and lax morphisms has colax limits, or that conical lax colimits are representing objects for colax natural transformations. However, it’s important to maintain a uniform convention, rather than trying to mess with the terminology to make it seem more “natural” in particular cases such as these. (I mention this because every so often someone, noticing such an example and perhaps unaware of the general convention, decides to try to switch the lax/colax convention in that case. Don’t be that person. (-: )

Posted by: Mike Shulman on April 28, 2014 11:31 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I’d like to open a thread to collect examples of “structure[s] borne by a category…[that are] given by an action of a 2-monad.”

Best would be if each contributor is either willing to describe the construction of the endofunctor and the 2-category on which it acts in some detail or can point to a place in the literature where this can be found.

Posted by: Emily Riehl on April 28, 2014 11:17 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

There are a few which are pretty easy to write down, the easiest probably being the free monoid $2$-monad on $\mathbf{Cat}$ which gives strict monoidal categories and unbiased monoidal categories. There are similar structures which can be given by a free monoid type construction - all of the ‘adjective’ strict monoidal categories, where $\text{'adjective'} \in \lbrace \text{symmetric}, \text{braided}, \text{ribbon braided}, \ldots \rbrace.$ For example, the braided strict monoidal categories $2$-monad $B \colon \mathbf{Cat} \rightarrow \mathbf{Cat}$ would be given by $B(X) = \coprod_{n \in \mathbb{N}} EBr_n \times_{Br_n} X^n,$ where $Br_n$ is the braid group on $n$ strands and $EBr_n$ is the indiscrete (or chaotic) category on $Br_n$ whose objects are the elements of $Br_n$ and for each pair of objects there is a unique isomorphism between them. (Contrast this with the strict monoidal category $2$-monad where $T(X) = \coprod X^n$.) The product over $Br_n$ is really the coequalizer of the left and right actions of $Br_n$ on the categories involved: $Br_n$ acts on the right of $EBr_n$ by group multiplication and acts on $X^n$ on the left by taking the underlying permutation of a braid and applying this to the list of things in $X$.

We can similarly do this with the symmetric groups and the ribbon braid groups (and any other suitable family of groups!). It’s also possible to extend these constructions to give a $2$-monad whose algebras are the appropriate functors between monoidal categories. So in the case of strict monoidal categories, a strict $T$-algebra would be a strict monoidal functor and a $T$-pseudoalgebra would be an unbiased monoidal functor - though it should be noted that (as far as I can tell) you can’t get unbiased lax monoidal functors from these.

I may come back and tidy up some points tomorrow! I’ll run through some of the functor examples as well - there’s even one that gives isomorphisms of categories as strict algebras, and equivalences of categories as pseudoalgebras.

Posted by: Alex Corner on April 28, 2014 11:55 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

An example I like is that there is a 2-monad on the 2-category of “$Cat$-enriched graphs” whose algebras are (strict) 2-categories, whose strict/pseudo/lax morphisms are strict/pseudo/lax functors, and whose 2-cells are “icons”. This example is especially interesting because its pseudo algebras are (unbiased) bicategories, and thus a general strictification theorem for pseudoalgebras over 2-monads implies automatically that any bicategory is equivalent to a strict 2-category.

One dimension up, there is a 2-monad on the 2-category of “$Cat$-enriched 2-graphs” whose algebras are strict 3-categories, and whose pseudoalgebras are a kind of semistrict tricategory (weaker than Gray-categories). Its pseudo morphisms are a fairly strict sort of functor, and its 2-cells are called “ico-icons”. This 2-monad violates the strictification theorem for pseudoalgebras, because not every tricategory is equivalent to a strict 3-category.

These 2-monads are constructed fairly explicitly in arXiv:1005.1520, as well as in other places (some of which are cited therein).

Posted by: Mike Shulman on April 29, 2014 6:39 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Steve Lack’s A 2-categories companion section 5 describes how to construct 2-monads via a presentation. Let me try and work out how to use these ideas to get a 2-monad for categories equipped with a functorial factorization.

The 2-monad I’m trying to build will live on the 2-category $CAT$ but everything I’ll say works in other locally finitely presentable 2-categories.

Write

• $Mnd_f(CAT)$ for the 2-category of finitary 2-monads on $CAT$

• $End_f(CAT)$ for the 2-category of finitary endofunctors

• $[ob CAT_f, CAT]$ for the functor 2-category whose domain is the set of finitely presentable categories.

Steve begins by observing that the forgetful functors

$Mnd_f(CAT) \to End_f(CAT) \qquad \mathrm{and} \qquad Mnd_f(CAT) \to [ob CAT_f, CAT]$

are monadic. In particular, we can define free monads on finitary endofunctors or on certain collections of categories. It also follows that $Mnd_f(CAT)$ is complete and cocomplete (because it’s locally finitely presentable).

We’ll start with the latter. Regard a functor $X : ob CAT_f \to CAT$ as an assignment, to an arity $c \in CAT_f$ of a type $Xc \in CAT$. The example I’ll consider has only a single non-empty arity, the walking arrow category $2$, to which it assigns the type $3$, the (ordinal) category indexing a pair of composable morphisms. The point is an algebra for the free monad $F_X$ on $X$ is a category $A$ together with a functor $A^2 \to A^3$.

This functor provides the data of a functorial factorization if and only if it is a section of the canonical composition functor $A^3 \to A^2$, which is induced by precomposition with the map of ordinals that I’ll call $d^1 \colon 2 \to 3$. We’ll impose this condition by defining a coequalizer diagram in $Mnd_f(CAT)$ of a pair of strict morphisms of monads with codomain $F_X$. The domain of this pair of maps will be free on a finitary endofunctor $E$, so by adjunction it will suffice to define a pair of natural transformations $E \Rightarrow U_X F_X$, where I’m writing $U_X F_X$ for the endofunctor bit of the free 2-monad on $X$. Taking mates, it suffices to define a pair of natural transformations $E U_X \Rightarrow U_X \colon F_X\text{-}Alg_s \to CAT$ and this is what I’ll do.

Let $E \colon CAT \to CAT$ be the functor $A \mapsto A^2 \times 2$. There’s a natural transformation from this to the identity given by evaluation — this is one of the desired pair. The other only exists a map $E U_X \Rightarrow U_X$. Given an algebra $(A, A^2 \to A^3)$ we can define a functor

$A^2 \times 2 \overset{1 \times d^1}{\to} A^2 \times 3 \to A$

where the second factor is a transpose of the algebra structure map. This is natural in strict algebra maps.

In this way, we get a pair of strict monad maps $H_E \to F_X$, where I’ve written $H_E$ for the free monad on the finitary endofunctor $E$. I claim that the coequalizer is the 2-monad whose algebras are categories with a functorial factorization.

Posted by: Emily Riehl on April 30, 2014 12:21 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

This is a great exposition, Sam. I think it’s going to be a great reference for me to look things up instead of trying to remember where I read them in the paper!

Posted by: Alex Corner on April 28, 2014 11:57 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

There’s a certain parametericity in category theory which really starts to confuse me when we talk about strict things. But it starts before that.

• When I talk about the abelian group $\mathbb{Z}$, everything I say is covariant in isomorphisms. If you have an isomorphic copy of $\mathbb{Z}$ in mind, everything I say about my copy of $\mathbb{Z}$ transports across an isomorphism between our two copies in a truth-preserving way.

• When I talk about the category of abelian groups $\mathbf{Ab}$, everything I say is covariant in equivalences. If your copy of $\mathbf{Ab}$ is equivalent to mine, we can transport across this equivalence in a truth-preserving way. This is so even if my copy of $\mathbf{Ab}$ is skeletal and yours is not, because we’ve learned not to talk about the number of objects in our categories.

• When I talk about the 2-category $\mathbf{Cat}$ of categories, everything I say should be covariant in 2-equivalences. Perhaps my 2-category has just one representative of each equivalence class of categories and yours has many; perhaps my functor categories are skeletal and yours are not (actually – is this right?) – it shouldn’t matter.

Now, when we say that monoidal categories are the strict algebras for a strict 2-monad $T$ on $\mathbf{Cat}$, we identify the 2-category $T\text{-}\mathrm{Alg}$ of strict monoidal categories and pseudo monoidal functors up to isomorphism. But this is only once we’ve specified our 2-category $\mathbf{Cat}$ up to isomorphism, which we typically don’t actually do!

So how does one deal with this? Is it important to pick strict conventions and then stick with them all the way through?

It leads to questions like this:

• Considering $2\text{-}\mathbf{Cat}$ as a strict 2-category, strict 2-monads are monads in $2\text{-}\mathbf{Cat}$ and an equivalence in $2\text{-}\mathbf{Cat}$ would, I suppose, be called a “strict 2-equivalence”. If I’m not mistaken, monads “transport” along equivalences in any 2-category (if $T$ is a monad and $F\dashv G$ is an adjoint equivalence, then $T' = G T F$ with appropriate 2-cells is the transported monad); in particular, strict 2-monads transport along strict 2-equivalences. Do the various algebra constructions transport along the same 2-equivalence? That is, do we have a strict 2-equivalence between $T'\text{-}\mathrm{Alg}_w$ and $T\text{-}\mathrm{Alg}_w$ for $w = s,p,l,c$?

• By transporting along a strict 2-equivalence, we can replace a 2-category with its skeleton, I believe (where there is only one object in each equivalence class – or perhaps we can only get down to isomorphism classes?). But perhaps we want to do something more radical. For instance, the 2-category of skeletal categories must be biequivalent to the 2-category of categories (whatever the latter means). But I suspect that strict 2-monads can’t be transported from the latter to the former. Are there other strict details about my 2-category of categories that I need to be careful about?

• There are biequivalent copies of $\mathbf{Cat}$ that I actually care about. Let $\mathbf{Cat}$ be the usual 2-category of categories internal to $\mathbf{Set}$, and let $\mathbf{Cat'}$ be the 2-category of categories internal to $\mathbf{Set'}$, where $\mathbf{Set'}$ is a skeleton of $\mathbf{Set}$. Do strict 2-monads on $\mathbf{Cat}$ transport to strict 2-monads on $\mathbf{Cat'}$? How about to $\mathbf{Cat''}$, the 2-category of categories enriched in $\mathbf{Set'}$?

All of this is without even pausing to think about size issues.

Posted by: Tim Campion on April 29, 2014 4:46 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I may have more to say later, but perhaps the following remarks from the beginning of the 2-categories companion are worth quoting:

In $Cat$-[enriched] category theory one deals with higher-dimensional versions of the usual notions of functor, limit, monad, and so on, without any “weakening”. The passage from category theory to $Cat$-category theory is well understood; unfortunately $Cat$-category theory is generally not what one wants to do — it is too strict, and fails to deal with the notions that arise in practice.

In bicategory theory all of these notions are weakened. One never says that arrows are equal, only isomorphic, or even sometimes only that there is a comparison 2-cell between them. If one wishes to generalize a result about categories to bicategories, it is generally clear in principle what should be done, but the details can be technically very difficult.

2-category theory is a “middle way” between $Cat$-category theory and bicategory theory. It uses enriched category theory, but not in the simple minded way of $Cat$-category theory; and it cuts through some of the technical nightmares of bicategories.

Posted by: Mike Shulman on April 29, 2014 6:43 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I suppose my concern is this: it seems like the question of whether e.g. a 2-monad is strict can depend on the choice of what set theory underlies our constructions. For example, the 2-category of categories enriched in a skeleton of $\mathbf{Set}$ or the 2-category of categories internal to a skeleton of $\mathbf{Set}$ might not admit the same strict 2-monads as the category of categories in a non-skeletal $\mathbf{Set}$. Is this accurate? Does this mean that we have to be careful about our foundational assumptions in order to do 2-category theory in Lack’s sense? If not, why not?

Posted by: Tim Campion on April 29, 2014 7:26 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

It doesn’t depend on your foundations as long as your foundations are set theory. (If your foundations are homotopy type theory, on the other hand, then things may be different.)

You can indeed replace any 2-category by a 2-equivalent one which is skeletal in the $Cat$-enriched sense of having only one object in each isomorphism class. Strict 2-monads do transport along 2-equivalences, as do all the algebra constructions.

If you want something skeletal in the bicategorical sense of having only one object in each equivalence class, then what you’ve got is only biequivalent to what you started with, and a strict 2-monad doesn’t transport — but then you’re no longer doing strict 2-category theory, you’re doing bicategory theory.

Your third example is actually a 2-equivalence, not just a biequivalence. In general, if $C$ and $C'$ are equivalent 1-categories with pullbacks, then $Cat(C)$ and $Cat(C')$ are 2-equivalent 2-categories, and if $V$ and $V'$ are equivalent monoidal 1-categories, then $V\text{-}Cat$ and $V'\text{-}Cat$ are also 2-equivalent 2-categories.

Posted by: Mike Shulman on April 29, 2014 11:12 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

It somehow surprises me that an equivalence of categories with pullbacks induces a strict 2-equivalence of 2-categories of internal categories, and an equivalence of monoidal categories induces a strict 2-equivalence of 2-categories of enriched categories. It feels like a fully weak equivalence is giving rise to a not-fully-weak equivalence. But of course, in this generality there is no 3-dimensional structure available to make things weaker. Anyway, this allows a small sigh of relief. But I still feel as though I need to be on my guard in case a dependence on set theory crops up in some other part of the theory.

Posted by: Tim Campion on April 30, 2014 4:12 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I couldn’t have quoted either of these results

In general, if $C$ and $C'$ are equivalent 1-categories with pullbacks, then Cat($C$) and Cat($C'$) are 2-equivalent 2-categories, and if $V$ and $V'$ are equivalent monoidal 1-categories, then $V$-Cat and $V'$-Cat are also 2-equivalent 2-categories.

but they don’t surprise me. All of these notions of equivalence start with a map of objects that is bijective on isomorphism classes and then stipulate that all other categorical structures correspond up to the strongest possible notion of isomorphism.

It’s interesting that you describe the concept of equivalence of categories as “fully weak.” I suppose that’s true but instead it feels like the notion is “just right” in the sense that it perfectly accommodates my inability to decide whether I’m thinking about multiple objects in any isomorphism class or not.

Posted by: Emily Riehl on April 30, 2014 7:57 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

It feels like a fully weak equivalence is giving rise to a not-fully-weak equivalence.

It is indeed! But that’s because the categorical dimension is going up (we start with a 1-category $C$ and produce a 2-category $Cat(C)$), so the meaning of “fully weak” is changing. If we count our notions of equivalence “from the bottom rather than from the top”, we can see that an equivalence of one sort (an equivalence of (Set-enriched) categories) is giving rise to an equivalence of the same sort (an equivalence of Cat-enriched categories). It’s just that in the codomain we happen to also have “more room” at the top to be able to talk about weaker kinds of equivalence.

This sort of thing also happens one dimension down. For instance, consider the second construction in this email: from a finite group $G$ we define the category whose objects are the orbits $G/H$, considered as $G$-sets, and whose morphisms are the $G$-maps between them. Isomorphisms between groups lead to isomorphisms (not just equivalences) between the corresponding categories of orbits.

It’s right to be somewhat suspicious of 2-category theory. But just as for all other areas of mathematics, the evidence suggests that it is not dependent on the details of the underlying set theory.

Posted by: Mike Shulman on May 1, 2014 6:21 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

It’s right to be somewhat suspicious of 2-category theory.

What does this mean?

Posted by: Tom Ellis on May 1, 2014 8:34 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

What I mean is that, like classical algebraic topology and model category theory, 2-category theory is not naturally invariant under the most appropriate notion of sameness for its objects of study. As you’ve pointed out, a pair of biequivalent 2-categories are morally the same structure — but they can be very different to the eyes of 2-category theory. One of my favorite examples of this is the biequivalence between the 2-category of internal categories in $C$ and the 2-category of fibrations over $C$ that are representable by some such internal category (see e.g. the remarks after B2.3.3 in the Elephant).

So the kinds of questions you’re asking are important: you do have to worry about when and whether you need to transport a construction along an “equivalence” that’s weaker than is allowed. In general, however, this doesn’t happen very often. One place in my experience where it does tend to happen is if you start trying to do 2-category theory with “very large” 2-categories, whose objects are large 1-categories. Of course, technically there’s no relationship between universe-level and category-dimension, but small categories often tend to be defined up to isomorphism rather than equivalence (or more precisely, to be strict categories), so that (large) 2-categories of small categories tend to be defined up to 2-equivalence rather than biequivalence — while large categories have objects that are built out of sets and only defined up to isomorphism, hence themselves are defined only up to equivalence, so that “very large” 2-categories of such will be defined only up to biequivalence. Of course, that’s not very precise, but maybe you get the idea.

Posted by: Mike Shulman on May 1, 2014 4:38 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

(I think you probably assumed I was Tim Campion, but I’m just an observer asking out of interest.)

So if we are to be suspicious of 2-category theory because “2-category theory is not naturally invariant under the most appropriate notion of sameness” what are we to be not suspicious of? Would a treatment under Homotopy Type Theory with Univalence ensure the right sort of invariance? Or is that too glib an assumption?

Posted by: Tom Ellis on May 1, 2014 7:18 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Haha, oops. Sorry Tom!

There’s no reason to be suspicious of classical bicategory theory, where categories are always treated up to equivalence (and bicategories up to biequivalence). Or, at least, no reason to be any more suspicious of it than we are of classical category theory, in which sets are treated up to isomorphism and categories are treated up to equivalence. A univalent treatment would indeed make us absolutely sure that everything is invariant, but even in classical foundations, if you work in a theory whose basic notions are invariant, it’s usually pretty easy not to slip up.

By the way, by using the word “suspicious” I didn’t mean to imply that anything is necessarily wrong with 2-category theory. I just mean that I think it requires an extra degree of carefulness, beyond what is required by category theory or bicategory theory, to make sure that we are strict or non-strict at the appropriate places.

Posted by: Mike Shulman on May 2, 2014 7:25 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Tim, let me make a general comment that does not get to the technical heart of your questions, but is perhaps something worth considering. You write

When I talk about the abelian group $\mathbb{Z}$, everything I say is covariant in isomorphisms. If you have an isomorphic copy of $\mathbb{Z}$ in mind, everything I say about my copy of $\mathbb{Z}$ transports across an isomorphism between our two copies in a truth-preserving way.

Although this is of course true if you restrict the things you want to say to be statements about $\mathbb{Z}$ itself, namely first-order sentences involving $(+,0)$, it is not, I think, correct to infer from this fact (i.e. truth preservation under isomorphism) that you must never distinguish between isomorphic copies of $\mathbb{Z}$, or isomorphic copies of any other group for that matter. Whether or not you should distinguish isomorphic groups (or equivalent categories, biequivalent bicategories etc.) depends on the context. For example, there is often a need to distinguish between isomorphic Sylow $p$-subgroups, if we are viewing them in the context of a bigger group. Or, even more simply, although we might not want to distinguish between $\mathbb{Z}/2$ and any other $2$-element group $G$ when studying them in a context-free fashion, we certainly want to be able to distinguish between two distinct order $2$ elements of a given group, even if the order $2$ subgroups they each generate are isomorphic.

So I’m wondering whether the case of studying $T$-Alg for a particular monad $T$ on, say, Cat is relevantly different from studying Cat as a free-floating $2$-category in just the above sense - and therefore that your questions demand more of the situation that they ought to. Namely, the “context” of studying $T$-Alg for a $2$-monad $T$ on Cat may require us to distinguish between equivalent copies of Cat in just the same way that the “context” of studying the Sylow $p$-subgroups of a given group $G$ may require us to distinguish between isomorphic such $p$-subgroups (e.g. for the purpose of counting them.) So the situation is not really that different from the cases which you call “covariant” like the case of abelian groups.

In summary: Distinguishing between equivalent (=equivalent under the “correct” notion of equivalence for that particular type of structure) structures should be expected in certain contexts, and the fact that we have to do so with equivalent copies of Cat in the context of studying $2$-monads on them is not a loss of “covariance”.

Posted by: Dimitris on April 29, 2014 7:50 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

From a categorical point of view, a subgroup of a group $G$ is not just a group that happens to be a subgroup of $G$, but a group together with an inclusion into $G$. There is, of course, a forgetful functor from the poset of subgroups of $G$ to the category of groups. But distinct subgroups that happen to have isomorphic underlying groups are still not isomorphic subgroups. You just have to keep track of what category your isomorphisms live in.

The question of distinguishing between isomorphic copies of $Cat$, when considering monads on it, is different: it’s more analogous to considering distinct isomorphic copies of the ambient group $G$. And it certainly doesn’t matter whether you and I are using different isomorphic copies of $G$ when we’re talking about Sylow $p$-subgroups of $G$.

Posted by: Mike Shulman on April 29, 2014 11:04 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Similarly, given a 2-monad $T$ on $CAT$, $T$-Alg is defined up to isomorphism relative to the 2-category $CAT$.

Posted by: Emily Riehl on April 30, 2014 7:58 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Right - but relative here means “up to isomorphism” right? Namely, $T-Alg$ is fixed up to isomorphism only if Cat is fixed up to isomorphism. If I fix Cat up to $2$/bi-equivalence then $T-Alg$ is only fixed up to $2$/bi-equivalence right (in the sense that $T-Alg$ is $2$/bi-equivalent to $T'-Alg$ where $T'$ is $T$ composed with the given equivalence)?

(I apologize if the answer to this is contained in the discussion above, but I probably wasn’t able to parse it.)

Posted by: Dimitris on May 4, 2014 12:51 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

[I]t’s more analogous to considering distinct isomorphic copies of the ambient group $G$. And it certainly doesn’t matter whether you and I are using different isomorphic copies of $G$ when we’re talking about Sylow $p$-subgroups of $G$.

In the analogy I had in mind the $p$-groups were playing the role of Cat (and thus you were forced to distinguish between them more finely than their preferred criterion of sameness.) But I’m not sure exactly what the ‘ambient group’ would be in the way I was thinking about it. Maybe something along these lines: In considering $2$-monads on Cat, i.e. when we are in some $3$-category with objects a pair consisting of a $2$-category and a $2$-monad on it, we are forced to distinguish between copies of Cat more finely than when we consider Cat as an object in a the $3$-category $2$-Cat (no matter how weak we stipulate the arrows in $2$-Cat to be.)

Posted by: Dimitris on May 4, 2014 12:52 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

My point was that you’re not distinguishing the $p$-subgroups any more finely than their preferred criterion of sameness. The preferred criterion of sameness for subgroups is an isomorphism that respects the inclusions into the ambient group.

It is true that we have to consider $Cat$ up to different notions of sameness, due to the issues raised above: when doing 2-category theory (such as working with strict 2-monads), we have to consider $Cat$ up to 2-equivalence, whereas in other cases we might want to consider it only up to biequivalence. But I don’t think it’s analogous to the $p$-subgroups situation.

Posted by: Mike Shulman on May 4, 2014 7:57 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I have a few questions, more like reference requests if anything, some of whose answers might be trivial, but which I have no way of verifying at the moment. I guess this is the best forum to air them, and I would greatly appreciate any help.

Firstly, there are several papers that are referred to as “in preparation” whose fate remains unclear to me:

1. Did the paper “On finitary enriched monads and their presentations” referenced as [27] in [BKP] become the paper “Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads”, Kelly-Power (1993)?
2. What happened to the paper “Essentially algebraic structures in the enriched context, II” referenced as [6] in Dubuc-Kelly “A Presentation of Topoi as Algebraic relative to Categories of Graphs”? (I’m guessing it was supposed to be a sequel to “Structures defined by finite limits in the enriched context, I” (1982) ?)
3. Kelly-Power in their (1993) above refer in the introduction (the second sentence) to “a paper in preparation to which the present paper is preliminary” - did this ever appear?

Secondly, I’m wondering if anyone has studied/answered any of the following situations/questions:

1. Is the $2$-category of $(\infty, 1)$-categories $2$-monadic over Cat? This question is intentionally vague, hopefully in order to convey the general query. One precisification would be to take $(\infty,1)$-categories to mean quasi-categories and the relevant $2$-category is the one defined by Riehl-Verity here. Is it reasonable to expect monadicity in this case? Is the horn-filling condition clearly a property and not a structure? Can we infer non-monadicity in this case by the lack of certain limits in $qCat_2$, or otherwise? In general, is it reasonable to expect some definition of a $2$-category of $\infty$-categories to be $2$-monadic over Cat?
2. Has anyone ever studied $2$-monads over the $2$-category of (Grothendieck) toposes with geometric morphisms between them? It seems to have all the necessary $2$-limits as to not make it unsuitable to such study - but perhaps there is some other obstruction? For example, questions such as: Are coherent Grothendieck toposes monadic over Grothendieck toposes? Or are these crazy questions?
Posted by: Dimitris on April 29, 2014 8:38 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

The 2-category of quasi-categories certainly isn’t due to us: it’s due to Andr&eacute Joyal. Good evidence that it is “the 2-category of $(\infty,1)$-categories” is provided by this paper of Zhen Lin Low.

A quick review: $qCat_2$ is the 2-category of quasi-categories, “functors” (maps of simplicial sets), and “natural transformations”. Given functors $f,g : A \to B$, a natural transformation $f \Rightarrow g$ is an equivalence class of 1-simplices in the quasi-category $B^A$ from $f$ to $g$. Here the equivalence relation is the usual one: two 1-simplices with the same source and target are equivalent if and only if there is a 2-simplex bounded by these and the appropriate degeneracy.

Now the usual adjunction $h \colon qCat_2 \leftrightarrows Cat_2 \colon N$ is a 2-adjunction; moreover $hN \cong 1$ identifies $Cat_2$ as a reflective sub 2-category. So $Cat_2$ is 2-monadic over $qCat_2$ via the idempotent 2-monad $Nh$.

I don’t believe that there is another adjunction which makes $qCat_2$ monadic or comonadic over $Cat_2$, but it’s an interesting question.

Posted by: Emily Riehl on April 29, 2014 9:52 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I believe your guess in (1) is correct, but you might get better answers to your first trio of questions from the categories mailing list.

The “homotopy category” functor $qCat_2 \to Cat$ is not conservative; therefore, it is not monadic or comonadic. I can’t think offhand of any other functor that it would be natural to consider.

There are some interesting 2-monads (or, perhaps, pseudomonads — I’m not sure) on the 2-category of Grothendieck toposes. A few of them are discussed in B4.4 and B4.5 of Sketches of an Elephant.

Posted by: Mike Shulman on April 29, 2014 11:00 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Thanks a lot for this.

The “homotopy category” functor $qCat_2 \rightarrow$ Cat is not conservative; therefore, it is not monadic or comonadic. I can’t think offhand of any other functor that it would be natural to consider.

Right, thanks. I don’t know exactly what the right setting is for my question, but the basic idea is this: “Is it reasonable to think of an $(\infty,1)$-category as structure on a category?” Ideally this would be answered in the positive by some $2$-monad on Cat. But are there any other precise ways to answer ‘yes’ to the above question, even forgetting about $2$-categorical structure?

Posted by: Dimitris on May 4, 2014 12:53 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I would say it’s not reasonable to try to think of an (∞,1)-category as a category with extra structure, but you can certainly think of it as a category with extra stuff.

Posted by: Mike Shulman on May 4, 2014 6:14 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I have a question.

I think I remember that there is a 2-monad on the 2-category of 2-categories, 2-functors, and 2-natural transformations whose algebras are 2-categories equipped with a 2-monad.

But were it not for this vague recollection I might have guessed that some sort of 3-monad (or Gray-monad?) would be necessary to describe structures borne by 2-categories.

How can I gauge the dimension needed to encode higher categorical structures monadically? For example, motivated by this mathOverflow question, is there a 2-monad whose algebras are 2-categories with certain 2-(co)limits?

Posted by: Emily Riehl on May 1, 2014 10:43 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Yes. More generally given $V$ as usual, and a small class of weights $\Phi$, there is a 2-monad on $V-Cat$ whose algebras are small $V$-categories with chosen $\Phi$-colimits (or limits).

This is explained in Kelly and Lack’s paper “On the monadicity of categories with chosen colimits.”

Since each 2-monad has an underlying 1-monad indeed there is a (1)-monad $T$ whose algebras are 2-categories with chosen colimits. But you need to look at the pseudomorphisms of $T$-algebras to get the functors preserving colimits only up to isomorphism.

Posted by: John on May 2, 2014 1:44 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Thanks for the reference.

Posted by: Emily Riehl on May 2, 2014 11:23 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

there is a 2-monad on the 2-category of 2-categories, 2-functors, and 2-natural transformations whose algebras are 2-categories equipped with a 2-monad.

Right. Indeed, for any nice $V$ there is a 2-monad on the 2-category of $V$-categories, $V$-functors, and $V$-natural transformations whose algebras are $V$-categories equipped with a $V$-monad.

In general, if a 2-categorical statement is the case $V=Cat$ of a $V$-categorical statement, then there’s no need to bring in any 3-dimensional structure. It’s when you start wanting to do something that only makes sense in the case $V=Cat$ that you generally need 3-categories, Gray-categories, or tricategories if you want to put it in a general context.

For instance, the EM-object of this $V$-monad consists of $V$-categories equipped with a $V$-monad and $V$-functors that strictly preserve the $V$-monads. Similarly, as John said (John who?), the EM-object of the $V$-monad for $V$-categories with certain $V$-colimits consists of $V$-categories with those chosen $V$-colimits and $V$-functors that strictly preserve the chosen $V$-colimits.

If you want pseudo or lax preservation (or if you want pseudo algebras instead of strict algebras), then you need $T\text{-}Alg_p$ or $T\text{-}Alg_l$ rather than $T\text{-}Alg_s$, and those don’t make sense for a general $V$, only for $V=Cat$. In particular, they don’t have a universal property analogous to that of EM-objects in the 2-category of 2-categories. Teasing out exactly where they have a universal property is a bit subtle however.

Given a 2-monad $T$ on a 2-category $K$, we’re asking for a universal functor $U: A\to K$ on which $T$ acts. There are three questions we have to ask that determine where this universal object can live.

1. What sort of a functor is it? If it’s a pseudofunctor, then we’re forced all the way up to tricategories. But this isn’t usually necessary; the direct construction of the 2-categories of algebras produces a forgetful functor that’s a strict 2-functor. So we can hope to remain in the world of 3-categories or Gray-categories.

2. Is the action a strict action or a pseudo action? E.g. are the two composites $T T U \to T U \to U$ strictly equal, or only coherently isomorphic? This determines whether the objects of our “category of algebras” will be strict algebras or pseudo algebras. But it doesn’t actually restrict where our universal property can live: we’re talking about equality or isomorphism of natural transformations (2-cells), which can be discussed in either a strict 3-category or a Gray-category.

3. Is the action map $T U \to U$ a strict 2-natural transformation? If so, then we can be in a 3-category; whereas if it’s only pseudo, we need to be in a Gray-category. It may not be immediately obvious, but this is what determines whether the morphisms in our category of algebras are strict or pseudo.

Thus, although strict 2-monads do live in the 2-category or 3-category of 2-categories, in order to give a universal property to the category $T\text{-}Alg_p$, we need to be living in the Gray-category of 2-categories. (And if we want a universal property for $T\text{-}Alg_l$, then we need to be in a “lax Gray-category”, defined like a Gray-category but using a lax form of the Gray tensor product.)

Finally, we can also go up another level. If $V=Cat$, then we can ask for a functor between categories with certain 2-colimits to preserve those colimits only up to equivalence rather than up to isomorphism. For this we would need to lift the 2-monad for such 2-colimits to a Gray-monad.

Posted by: Mike Shulman on May 2, 2014 7:49 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Mike, thanks. This is very helpful.

And I know which John (and so do you; if you guessed you’d be right), but perhaps it’s impolite to out him?

Posted by: Emily Riehl on May 2, 2014 11:21 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Thank you Sam!

I think some there has been some in-seminar discussion about pseudomonads, or the actual lack thereof in concrete examples. So I bring it up here: can one actually name concrete examples of situations where a 2-monad arising naturally is not associative on the nose, but only up to an invertible 2-cell?

Further, are there examples when this 2-cell is not even invertible (lax/oplax 2-monads?), but is required to satisfy some conditions? On a related note, one can imagine algebraic structures on categories which are lax or oplax: as an example, take the notion of (representable) pseudotensor categories of Beilinson and Drinfeld in their book Chiral Algebras (myself, I think they rather should be called lax monoidal categories). Thus I wonder what sort of a 2-monad classifies such things; probably a strict one again.

Best, Eduard

Posted by: Eduard Balzin on May 2, 2014 11:36 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

I’m not used to working with any concrete examples of pseudomonads but I’ll try and point to some places which mention specific examples.

First, in the paper ‘Pseudo-distributive laws’ of Cheng, Hyland, and Power, there are a number of examples given. I think my favourites are the ones to do with free cocompletions of categories under certain classes of colimits. These yield pseudomonads on $\mathbf{Cat}$. Example $2.5$ in the paper.

The example that follows talks about the $2$-monad for small symmetric monoidal categories being lifted to a pseudomonad on $\mathbf{Prof}$. This is also a focus in Richard Garner’s paper ‘Polycategories via pseudo-distributive laws’. He shows that you can lift this $2$-monad in two different ways in order to get a pseudomonad and a pseudocomonad which (pseudo)distribute over each other.

Posted by: Alex Corner on May 2, 2014 12:25 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Indeed, in the next paper we’ll even see an example of a pseudo-distributive law where colimit completions distribute over limit completions!

A very natural description of any free cocompletion monad comes by considering it as a submonad of $[(-)^\mathrm{op}, \mathbf{Set}]$. In this description the monad is definitely pseudo, I think. But I can imagine strictifying it by freely adding all the colimits in question in a “syntactic” way.

Near the beginning of “Coherence Theorems for Lax Algebras and for Distributive Laws” (unfortunately hidden behind a paywall, but you can find this in the preview), Kelly discusses the idea that strict distributive laws between 2-monads are rare in nature. I haven’t really understood the reason, but it suggests that if you take the composite pseudomonad from two 2-monads with a pseudodistributive law, maybe you’ll get an example of a pseudomonad that can’t be strictified? So, for example, categories with certain limits and certain colimits (and perhaps some compatibility condition?) might be an example?

Posted by: Tim Campion on May 2, 2014 10:00 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

As for why strict distributive laws are rare, I would guess it’s for the same sort of reason that strictly symmetric monoidal structures are rare, since a distributive law is a sort of “commutativity”.

Every (accessible) pseudomonad can however be “strictified” in the following sense: for any pseudomonad $T$, there is a 2-monad $T'$ whose strict algebras are the same as the pseudo $T$-algebras. This is an instance of the general fact that the inclusion of strict algebras into pseudo algebras has a left adjoint (which is itself a generalization of the pseudomorphism classifier from BKP), applied again to the 2-monad for 2-monads.

Posted by: Mike Shulman on May 4, 2014 6:09 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Am I right to think that the 2-monad $T'$ and the pseudomonad $T$ might live on different 2-categories?

Posted by: Emily Riehl on May 6, 2014 9:08 PM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

Nope, they live on the same 2-category (which is required to be well-behaved, e.g. locally presentable).

Posted by: Mike Shulman on May 7, 2014 12:51 AM | Permalink | Reply to this

### Re: On Two-Dimensional Monad Theory

myself, I think they rather should be called lax monoidal categories

Actually, they should be called colax monoidal categories. They can be defined either as the colax algebras for the 2-monad $T$ whose strict algebras are strict monoidal categories, or as the strict algebras for $T^\diamond$, the “colax algebra classifier” monad of $T$ (a sort of colax cofibrant replacement).

The way to tell whether something is a lax algebra or a colax algebra is that $T$-algebra structures on an object $A$ are equivalent to monad morphisms $T \to \langle A,A\rangle$, where $\langle A,A\rangle$ is the “endomorphism monad” of $A$ (the right Kan extension of $A :1\to K$ along itself). Now recall that 2-monads are the algebras for a 2-monad, so we can talk about lax and colax morphisms of monads $T \to \langle A,A\rangle$; those are lax and colax algebras, respectively, for $T$.

Posted by: Mike Shulman on May 2, 2014 7:55 PM | Permalink | Reply to this

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