The n-Category Café

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.

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.

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).

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.

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.

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”.

The 2-category of quasi-categories certainly isn’t due to us: it’s due to André 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.

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.

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.

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.

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.

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 $T$.