The n-Category Café

Recall that one way to describe a notion of “category with structure” is by giving a 2-monad. And it’s easy to prove that for any ordinary monad $T$ on an ordinary category $C$, the category $T Alg = C^T$ of $T$-algebras inherits any limits that $C$ has (more precisely, the forgetful functor $T Alg\to C$ “creates limits”). We’d like to categorify this statement.

However, if $T$ is a 2-monad on a 2-category $C$, there is more than one choice for what one might mean by a “2-category of $T$-algebras”.

• One thing to do is to “go completely pseudo” and consider the 2-category $Ps T Alg$ of pseudo $T$-algebras and pseudo-$T$-morphisms. We can then prove a “fully bicategorical” version of the usual theorem: $Ps T Alg$ inherits any bicategorical limits (bilimits) that $C$ has. These are limits whose cones commute up to isomorphism, and which satisfy their universal property up to equivalence.

• On the other hand, we have the 2-category $T Alg_s$ of strict $T$-algebras and strict $T$-morphisms. This is the ordinary $Cat$-enriched category of algebras, and as such it inherits all strict 2-categorical limits that $C$ has. Now bicategorical limits can be modeled by strict 2-limits; a 2-category theorist says that “pseudolimits are also bilimits”. Pseudolimits also have cones commuting up to isomorphism, but satisfy their universal property up to isomorphism rather than equivalence. In particular, if $C$ is complete as a strict 2-category, so is $T Alg_s$, and both also have bicategorical limits.

  The advantage of strict algebras is that the types of “structure” on categories we consider are often not literally pseudoalgebra structures, but they are strict algebra structures. For instance, the usual notion of “monoidal category,” with a binary tensor product, unit object, and associator and unitor, describes the strict algebras for some strict 2-monad, but not the pseudo algebras for any 2-monad. (If $T$ is the 2-monad whose strict algebras are strict monoidal categories, then pseudo $T$-algebras are “unbiased” monoidal categories; these are “equivalent” to ordinary ones, but only by a nontrivial theorem.)

• On the other hand, strict monoidal functors between non-strict monoidal categories are rare, so instead of $T Alg_s$ it is often better to take a middle road: we consider the 2-category $T Alg = T Alg_p$ of strict algebras and pseudo morphisms. However, now we no longer have an easy answer to the question “what limits does $T Alg$ have?” But the answer is known: Blackwell, Kelly, and Power proved (in 2-dimensional monad theory) that $T Alg$ inherits PIE-limits from $C$. A PIE-limit is, roughly, a strict 2-limit which demands no equalities between 1-morphisms (a “non-evil strict 2-limit”). Since pseudolimits are PIE-limits, if $C$ is a complete strict 2-category (like $Cat$) then $T Alg$ has all bicategorical limits.

• Sometimes, however, even pseudo morphisms are too strict, and we need to consider lax or colax morphisms, which preserve the structure only up to a not-necessarily-invertible comparison morphism. The resulting 2-categories $T Alg_l$ and $T Alg_c$ are not generally bicategorically complete, even when $C=Cat$. So what limits do they have?

In a previous paper (Limits for lax morphisms), Steve proved that the 2-category $T Alg_l$ has colax limits (limits whose cones commute up to not-necessarily-invertible 2-cells, in one of the possible directions), and dually $T Alg_c$ has lax limits. This is already a useful thing to know. For example, Eilenberg-Moore objects (objects-of-algebras for internal monads) are a lax limit, so this implies that (for instance) the category of algebras for a colax-monoidal monad on a monoidal category inherits a monoidal structure.

(Amusingly, it also implies the lifting of 1-limits to categories of algebras. For any small class $\mathcal{X}$ of 1-limits, there is a 2-monad $T$ on $Cat$ whose algebras are categories with $\mathcal{X}$-limits. This 2-monad is colax-idempotent, which means that any functor between $T$-algebras is a colax $T$-morphism in a unique way; the colax structure is the canonical comparison map $T(\lim F) \to \lim (T F)$. Therefore, $T Alg_c$ is the 2-category of $\mathcal{X}$-complete categories and all functors between them. It follows that for any monad on an $\mathcal{X}$-complete category, the category of algebras is also $\mathcal{X}$-complete.)

However, Steve also proved that $T Alg_l$ has a number of other limits besides colax ones. It has limits of all diagrams of strict morphisms (that is, the inclusion $T Alg_s \to T Alg_l$ preserves limits). It has inserters of 2-cells $f\to g$ for parallel 1-morphisms $f,g\colon A\rightrightarrows B$ if $f$ is strict. It has equifiers of parallel 2-morphisms $\alpha,\beta\colon f \rightrightarrows g \colon A \rightrightarrows B$ if $f$ is strict. It has comma objects $(f\downarrow g)$ if $f$ is strict. Moreover, all of these limits in $T Alg_l$ have a curious property: there is some specified collection of projections from the limit which are strict $T$-morphisms, and which detect strictness in the sense that a map into the limit is strict if and only if its composites with these distinguished projections are all strict. (Blackwell-Kelly-Power also proved this latter fact for PIE-limits in $T Alg$.)

Note that none of these types of limits can be expressed as purely 2-categorical properties of $T Alg_l$; they all require knowing which of the morphisms in $T Alg_l$ are strict. So the limit-structure of $T Alg_l$ becomes much richer if we enhance it by supplying the datum of the inclusion $T Alg_s \to T Alg_l$. The first main idea of the current paper is that instead of regarding this datum as a 2-functor $T Alg_s \to T Alg_l$, we can regard it as a single enriched category.

The category we are enriching over, denoted $\mathcal{F}$, has as objects categories equipped with a full subcategory, which is to say functors that are fully faithful and injective on objects. Thus an $\mathcal{F}$-category has a collection of objects, and (like a 2-category) between any two objects it has a hom-category — but also it has a specified full subcategory of that hom-category. In general, we call the morphisms in the specified subcategories tight, and the general morphisms loose; in the case of $T Alg_l$ they are of course the strict and lax $T$-morphisms, respectively.

It turns out that all the odd-looking types of limits that Steve found $T Alg_l$ to admit, including both the requirements that some of the morphisms in the diagram be strict and the additional universal property that some of the projections are strict and detect strictness, can be expressed exactly as certain weighted limits in $\mathcal{F}$-enriched category theory. I find this absolutely amazing!

Once we get over our amazement, however, we realize that all we’ve done so far is introduce a language in which to talk about the limits which $T Alg_l$ has. We still need to prove a theorem about which ones those actually are; there are plenty of $\mathcal{F}$-limits that $T Alg_l$ generally doesn’t have.

To motivate the statement of this theorem, let’s go back to pseudo morphisms and PIE-limits. Recall that I said the PIE-limits were roughly “limits that don’t involve any equalities between morphisms”. Now a given type of 2-limit is described by a weight, which consists of a functor $\Phi\colon D \to Cat$. There is a “projective” model structure on such weights, and one might guess that the PIE-weights would be the cofibrant objects therein—but that isn’t quite right. The cofibrant weights, which 2-categorists traditionally call flexible, are slightly more general than PIE-weights; splitting of idempotents is flexible, but not PIE. This makes some sense, since cofibrant objects are like projective things, which are usually retracts of free things, while PIE-weights are like free things.

(In general, $T Alg$ does not admit all flexible limits: idempotent pseudo morphisms need not split. Interestingly, though, $T Alg$ does have all flexible limits if $T$ itself is flexible as a 2-monad.)

Now free things are also like cell complexes (while projective things are like retracts of cell complexes), and we also know another way to characterize cell complexes. Namely, in an algebraic weak factorization system, the algebraically cofibrant objects are usually closely related to cell complexes. By definition, an object is “algebraically cofibrant” when it is a coalgebra for a cofibrant replacement comonad, usually denoted $Q$. (By contrast, an object is cofibrant just when the counit $Q X \to X$ has a section, which is just the counit condition for a coalgebra; algebraicity adds the coassociativity condition.) It turns out that the PIE-weights are precisely the algebraically cofibrant objects for the model structure on weights. (Richard Garner and John Bourke also noticed this independently.)

Finally, one last important observation is that the cofibrant replacement comonad on weights has a universal property: it is the classifier for pseudo morphisms. In other words, for weights $\Phi\colon D \to Cat$ and $\Psi\colon D \to Cat$, to give a pseudo natural transformation $\Phi \to \Psi$ is precisely to give a strict natural transformation $Q\Phi\to \Psi$. A section of $Q \Psi \to \Psi$, therefore, enables us to make any pseudonatural transformation out of $\Phi$ into a strict one, and this is essentially how we can show that $T Alg$ admits PIE-limits. (This was not Blackwell-Kelly-Power’s original proof, however!)

By the way, this is in line with the general model-category philosophy that “weak maps” are maps from a cofibrant replacement to a fibrant one; in the projective model structure on weights, all objects are fibrant. There is also a dual “injective” model structure in which all objects are cofibrant and the fibrant replacement is a pseudo morphism coclassifier.

Now there is also a lax morphism classifier comonad $Q_l$, with the corresponding property that lax natural transformations $\Phi \to \Psi$ are precisely strict natural transformations $Q_l\Phi\to \Psi$. And dually there is a colax morphism classifier $Q_c$. So it is entirely reasonable to guess (with all the hindsight-inspired lead-up that I’ve given here) that the limits which $T Alg_l$ admits will have something to do with $Q_c$-coalgebras. (You might have initially said $Q_l$-coalgebras, but it turns out to be $Q_c$; recall that $T Alg_l$ has colax limits, not lax ones.)

In fact, this is a theorem: a type of 2-categorical limit lifts to $T Alg_l$, for any 2-monad $T$, if and only if its weight is a $Q_c$-coalgebra. That’s just a 2-categorical theorem, though, and as we saw above, the limit structure of $T Alg_l$ becomes much richer if we regard it as an $\mathcal{F}$-category instead. But we can mimic the above development for $\mathcal{F}$-weights as well, defining lax, colax, and pseudo $\mathcal{F}$-natural transformations and morphism classifier $\mathcal{F}$-comonads. (There are some tricky details here, but we’ll ignore them.) And we can prove that an $\mathcal{F}$-categorical limit lifts to $T Alg_l$ for all $T$ if and only if its weight is an $\mathcal{F}$-categorical $Q_c$-coalgebra… plus an extra somewhat curious condition, which roughly says that “all the projections from the limit object are generated by the tight ones.”

We call a $Q_c$-coalgebra satisfying this extra condition rigged (or more precisely “$l$-rigged”). The intuition for this is that being rigged isn’t just about “being strict”—or, in $\mathcal{F}$-categorical language, “being tight”. Rather, it’s like the rigging on a ship: the parts that should be tight are tight, but the parts that should be loose are loose, and the two interact in just the right way. To explain the actual definition of rigging would take too much space (although it’s not really that complicated), so you’ll have to read the paper. It all works out quite nicely — although I should mention that some pretty weird-looking weights can still be rigged; there are some examples in section 6 of the paper.

There is one last thing I should say, since if I don’t, someone will probably ask about it. In the current paper, Steve and I restricted ourselves to enhanced 2-categories of strict and lax (or colax) morphisms. But clearly one could combine pseudo and lax instead; or strict, pseudo, and lax; or even all four types of morphisms! We fully expect analogous theorems to hold in these cases (although it may take some extra ideas to combine lax+colax). There should even be analogous theorems for pseudoalgebras over pseudomonads and $\mathcal{F}$-enriched bicategories. It’s just that the strict+lax case was the easiest one to start with, and made the paper long enough by itself.