The n-Category Café

I said that we will want to talk about the relationship between strict and weak morphisms — as such it will be useful to view both kinds of morphism as belonging to the same overarching structure. The right kind of structure is that of an $\mathcal{F}$-category — introduced by Steve Lack and Mike Shulman and blogged about by Mike previously — so let us begin by recalling these.

$\mathcal{F}$-categories

An $\mathcal{F}$-category $\mathbb{A}$ is a very simple thing: it is just a 2-category, whose morphisms are called loose, together with a specified subcollection of tight morphisms closed under composition and containing the identities. We typically write $A_{\lambda}$ for the whole 2-category — the 2-category of loose morphisms — and $A_{\tau}$ for the sub 2-category containing the tight morphisms together with all 2-cells between them. We write $j:A_{\tau} \to A_{\lambda}$ for the inclusion 2-functor which views tight morphisms as loose. Here are a few examples.

• Between monoidal categories are strict, strong, lax and colax monoidal functors and monoidal transformations. These can be combined into a variety of $\mathcal{F}$-categories. For our main example I’ll focus on the $\mathcal{F}$-category $\mathbb{M}onCat_{l}$ of monoidal categories, strict and lax monoidal functors — the inclusion of its 2-category of tight morphisms into its 2-category of loose ones is the 2-functor $j:MonCat_{s} \to MonCat_{l}$ which views strict monoidal functors as lax.
• But we also have the $\mathcal{F}$-categories $\mathbb{M}onCat_{p}$ of strict and strong monoidal functors, and $\mathbb{M}onCat_{c}$ of strict and colax monoidal functors and $\mathbb{M}onCat_{p,l}$ of strong and lax ones and so on.
• Likewise given a 2-monad $T$ on a 2-category $C$ (I mean a 2-monad in the strictest possible sense) we have strict $T$-algebras, strict, pseudo, lax and colax $T$-algebra morphisms and algebra transformations. These give rise to a number of different $\mathcal{F}$-categories such as the $\mathcal{F}$-category $\mathbb{T}Alg_{l}$ of strict and lax algebra morphisms. Its inclusion of tight into loose morphisms is the inclusion $j:TAlg_{s} \to TAlg_{l}$.
• Any 2-category $C$ can be viewed as an $\mathcal{F}$-category (I denote this $\mathcal{F}$-category by $C$ too) in which the tight and loose morphisms coincide. The inclusion of tight into loose morphisms is then just the identity $1:C \to C$. In this way we can view 2-categories as special $\mathcal{F}$-categories.

We also need to talk about $\mathcal{F}$-functors. An $\mathcal{F}$-functor $W:\mathbb{A} \to \mathbb{B}$ is a 2-functor which preserves tight morphisms. In other words a 2-functor $W_{\lambda}:A_{\lambda} \to B_{\lambda}$ which restricts to a 2-functor $W_{\tau}:A_{\tau} \to B_{\tau}$ as in the commuting diagram $$\array{ A_{\tau} &\stackrel{j}{\to}& A_{\lambda} \\ \downarrow^{W_{\tau}} && \downarrow^{\mathrlap{W_{\lambda}}} \\ B_{\tau} &\stackrel{j}{\to}& B_{\lambda} }$$ These are the morphisms of the category of $\mathcal{F}$-categories $\mathcal{F}$-$CAT$. Here are some examples.

• An $\mathcal{F}$-functor $W:\mathbb{A} \to B$ to a 2-category $B$ (viewed as an $\mathcal{F}$-category) just looks like a commutative triangle of 2-functors as on the left below. $$\array{ A_{\tau} &&\stackrel{j}{\to}&& A_{\lambda} &&&& MonCat_{s} &&\stackrel{j}{\to}&& MonCat_{l} \\ & {}_W_{\tau} \searrow && \swarrow_{W_{\lambda}} &&&&&& {}_V_{s} \searrow && \swarrow_{V_{l}} \\ && B &&&&&&&& Cat }$$ See how monoidal categories, strict and lax monoidal functors sit over $Cat$ for instance. Thus we have a forgetful $\mathcal{F}$-functor $V:\mathbb{M}onCat_{l} \to Cat$ with tight and loose parts $V_{s}$ and $V_{l}$ .

• Likewise given a 2-monad $T$ on a 2-category $C$ we have a forgetful $\mathcal{F}$-functor $U:\mathbb{T}Alg_{l} \to C$.

• An $\mathcal{F}$-functor $W:A \to \mathbb{B}$ from a 2-category is a triangle too, like left below: $$\array{ && A &&&&&&&&& A_{\tau} \\ & {}_W_{\tau}\swarrow && \searrow^{W_{\lambda}} &&&&&&& {}_1\swarrow && \searrow^{j} \\ B_{\tau} &&\stackrel{j}\to && B_{\lambda} &&&&& A_{\tau} && \stackrel{j}\to && A_{\lambda} }$$ Given any $\mathcal{F}$-category $\mathbb{A}$ we have an inclusion $\mathcal{F}$-functor $j:A_{\tau} \to \mathbb{A}$ drawn on the right above (yes, I’ve been calling it $j$ too!).

• An $\mathcal{F}$-functor between 2-categories viewed as $\mathcal{F}$-categories is just a 2-functor — in this way the 2-categorical world sits inside that of $\mathcal{F}$-categories.

Note how given an $\mathcal{F}$-functor $W:\mathbb{A} \to B$ to a 2-category $B$ the composite left below $$\array{ A_{\tau} &\stackrel{j}{\to}& \mathbb{A} &\stackrel{W}{\to}& B & & & & MonCat_{s} & \stackrel{j}{\to}& \mathbb{M}onCat_{l} &\stackrel{V}{\to}& Cat}$$ equals the 2-functor $W_{\tau}:A_{\tau} \to B$. Thus the entire $\mathcal{F}$-category $\mathbb{A}$ lies in the middle of a factorisation of a 2-functor viewed as an $\mathcal{F}$-functor. These factorisations are the key to everything I want to say, but first we need to talk about doctrinal adjunction.

Doctrinal adjunction

When people speak of doctrinal adjunction, in the context of monoidal categories, they often refer to the fact that lax monoidal structures on a right adjoint functor correspond to colax monoidal structures on its left adjoint. This is neither a 2-categorical nor an $\mathcal{F}$-categorical phenomenon in that lax and colax monoidal functors do not belong to a common 2-category or $\mathcal{F}$-category — rather, it is a double categorical phenomenon. What I want to talk about is a restricted form of doctrinal adjunction, also well known, which is $\mathcal{F}$-categorical in nature.

For monoidal categories this is the assertion that given a strong monoidal functor $F$ whose underlying functor $VF$ has a right adjoint — so that we have an adjunction $(\epsilon, V F \dashv G,\eta)$ of categories and functors — the right adjoint $G$ obtains the structure of a lax monoidal functor in such a way that $\eta$ and $\epsilon$ become monoidal transformations: which is to say that the adjunction lifts along $V$ from $Cat$ to $MonCat_{l}$. In fact the lifted adjunction is the unique such lifting, a fact not usually emphasised.

We can express this as a lifting property of an $\mathcal{F}$-functor by saying that an $\mathcal{F}$-functor $W:\mathbb{A} \to \mathbb{B}$ satisfies $l$-doctrinal adjunction if given a tight morphism $f:X \to Y \in \mathbb{A}$ and adjunction $(\epsilon, W f \dashv g,\eta)$ in $B_{\lambda}$ (ie. $g$ is allowed to be loose) then that adjunction lifts uniquely along $W$ to an adjunction $(\epsilon^{\prime},f \dashv g^{\prime},\eta^{\prime})$ in $A_{\lambda}$. Then the above observation about monoidal categories amounts to the fact that the forgetful $\mathcal{F}$-functor $V:\mathbb{M}onCat_{p,l} \to Cat$ from strong and lax monoidal functors to $Cat$ satisfies $l$-doctrinal adjunction. Since strict monoidal functors are strong it follows that $V:\mathbb{M}onCat_{l} \to Cat$ also satisfies $l$-doctrinal adjunction. This is what we want since it relates lax morphisms with the strict ones — which are those most easily understood.

Let me remark that $l$-doctrinal adjunction captures laxness, as in the orientation and non-invertibility of the comparisons $(F a)(F b) \to F(a b)$ (of tensor products) and $i^{B} \to F i^{A}$ (of units) defining a lax monoidal functor $F:A \to B$: for whilst $V:\mathbb{M}onCat_{l} \to Cat$ satisfies $l$-doctrinal adjunction neither of the forgetful $\mathcal{F}$-functors $V:\mathbb{M}onCat_{p} \to Cat$ and $V:\mathbb{M}onCat_{c} \to Cat$ do so. On the other hand there is an $\mathcal{F}$-category of monoidal categories, strict monoidal and incoherent lax monoidal functors and the forgetful $\mathcal{F}$-functor from there to $Cat$ does satisfy $l$-doctrinal adjunction. Intuitively then $l$-doctrinal adjunction captures the laxness but not the coherence axioms of a lax monoidal functor.

In writing the paper I found a refinement of the intuitive notion of $l$-doctrinal adjunction to be what was really needed and called $\mathcal{F}$-functors satisfying this refinement $l$-doctrinal. For our purposes it doesn’t matter what precisely an $l$-doctrinal $\mathcal{F}$-functor is: it suffices to say that any $\mathcal{F}$-functor that satisfies $l$-doctrinal adjunction, is faithful on 2-cells and reflects identity 2-cells is $l$-doctrinal. For example $V:\mathbb{M}onCat_{l} \to Cat$ satisfies these two additional conditions concerning 2-cells because monoidal transformations are just natural transformations with properties — thus $V$ is $l$-doctrinal. I call the class of $l$-doctrinal $\mathcal{F}$-functors $l-doct$.

Colax limits of loose morphisms

Mike and Steve introduced $\mathcal{F}$-categories to explain the behaviour of limits of weak homomorphisms in 2-dimensional universal algebra. Curious things happen in this world and I barely want to touch on them here but one kind of $\mathcal{F}$-categorical limit is crucial: the colax limit of a loose morphism. Recall that given a functor $F:A \to B$ between categories we can form the comma category $B/F$ — the one with objects like $(\alpha:b \to F a,a)$. This is a kind of 2-categorical limit, the so-called colax limit of $F$, and has a colax cone which looks like $$\array{ && B/F \\ & {}_P\swarrow & \Downarrow^{\lambda} & &\searrow_{Q} \\ A && \underset{F}{\to} &&& B }$$ Here $P$ and $Q$ are projection functors — $P(\alpha:b \to F a,a) = a$ and $Q(\alpha:b \to F a,a) = b$ — and the natural transformation $\lambda:Q \to FP$ has component at $(\alpha:b \to F a,a)$ given by $\alpha$ itself. The limit property of $B/F$ is that this is the universal such cone.

Now if $F:A \to B$ is a lax monoidal functor it turns out that $B/F$ obtains a unique monoidal structure such that the projections $P$ and $Q$ become strict monoidal and $\lambda$ a monoidal natural transformation — thus the colax cone lifts to the 2-category $MonCat_{l}$. In addition to this the projections $P$ and $Q$ jointly detect the property of a lax monoidal functor in $B/F$ being strict monoidal and the lifted cone has the same universal property in $MonCat_{l}$. All of these properties of the lifted cone can be expressed $\mathcal{F}$-categorically: in $\mathbb{M}onCat_{l}$ they assert precisely that the lifted cone is the colax limit of the loose morphism $F$.

What is slightly magical about this is that if you work through this claim — or look in the paper — you’ll see that it uses exactly the coherence axioms for a lax monoidal functor. Whilst $l$-doctrinal adjunction captures laxness somehow colax limits of loose morphism capture the coherence axioms.

Incidentally I don’t think anyone (at least me!) has a satisfactory conceptual explanation as to why colax limits of loose morphisms are so important — it seems they are though.

Pinning down lax morphisms and monadicity

Now for the main theorem - I’ll only talk about the lax case. In it I mention tight pullbacks which I haven’t defined. Don’t worry about them — they are just pullbacks of tight morphisms which have their universal property with respect to the loose ones too, like pullbacks of strict monoidal functors.

Theorem. Consider an $\mathcal{F}$-functor $W:\mathbb{A} \to B$ to a 2-category and suppose that $\mathbb{A}$ has colax limits of loose morphisms and tight pullbacks, and that $W$ is $l$-doctrinal. Then the decomposition in $\mathcal{F}$-$CAT$ $$\array{A_{\tau} &\stackrel{j}{\to}& \mathbb{A} &\stackrel{W}{\to}& B}$$ of the 2-functor $W_{\tau}:A_{\tau} \to B$ is an orthogonal $(^{\bot}l-doct,l-doct)$-decomposition.

In other words the inclusion $j:A_{\tau} \to \mathbb{A}$ is orthogonal to each $l$-doctrinal $\mathcal{F}$-functor. It follows from the theorem that the decomposition $$\array{MonCat_{s} &\stackrel{j}{\to}& \mathbb{M}onCat_{l} &\stackrel{V}{\to}& Cat}$$ of the forgetful 2-functor $V_{s}:MonCat_{s} \to Cat$ is an orthogonal decomposition. Likewise for any 2-monad $T$ on $Cat$ the decomposition $$\array{TAlg_{s} &\stackrel{j}{\to}& \mathbb{T}Alg_{l} &\stackrel{U}{\to}& Cat}$$ is an orthogonal decomposition.

Now orthogonal decompositions are unique up to isomorphism. Consequently we can interpret the theorem as saying: given a 2-category $A_{\tau}$ sitting over $B$ then $A_{\tau}$ can be extended to an $\mathcal{F}$-category $\mathbb{A}$ over $B$ in at most one way such that the loose morphisms of $\mathbb{A}$ behave like lax morphisms (ie. satisfy the hypotheses of the theorem).

Anyway that’s not very snappy — lets see how the theorem relates to monadicity. The starting point here is that monadicity for strict morphisms is easily understood: we can use Beck’s theorem in the enriched setting to show that $V_{s}:MonCat_{s} \to Cat$ is strictly monadic — which is to say that we have an isomorphism of 2-categories $E$ over $Cat$ as below $$\array{ MonCat_{s} &&\stackrel{E}{\to}&& TAlg_{s} \\ & {}_V_{s} \searrow && \swarrow_{U_{s}} \\ && Cat }$$ So strict monoidal functors correspond to strict $T$-algebra maps. What remains is to show that the lax monoidal functors correspond to the lax $T$-algebra maps.

Now the isomorphism $E$ over $Cat$ just asserts the commutativity of the outside of the diagram

$$\array{ MonCat_{s} &\stackrel{j}{\to}& \mathbb{M}onCat_{l} &\stackrel{V}{\to}& Cat \\ \downarrow^{E} && \downarrow^{\mathrlap{E_{l}}} && \downarrow^{\mathrlap{1}} \\ TAlg_{s} &\stackrel{j}{\to}& \mathbb{T}Alg_{l} &\stackrel{U}{\to}& Cat }$$

Since both horizontal rows are orthogonal decompositions and because both vertical 2-functors ($E$ and $1$) are isomorphisms we obtain a unique isomorphism of $\mathcal{F}$-categories $E_{l}:\mathbb{M}onCat_{l} \to \mathbb{T}Alg_{l}$ as in the middle — thus drawing the desired correspondence between lax monoidal functors and lax $T$-algebra maps.

Thats the idea of the main result on monadicity, Theorem 21, of the paper — and there are also entirely similar variants treating pseudo and colax morphisms. Let me make a few final points about what its all good for.

• Firstly I should point out that it is well known that monoidal categories and their morphisms (and many other structures in 2-dimensional universal algebra) can be described by 2-monads and their algebra maps — that’s why 2-dimensional monad theory was developed! The standard approach to showing this is via colimit presentations — here you start with the algebraic structure you have in mind and translate it into a presentation of a 2-monad $T$ as a colimit of free ones — see Steve’s A 2-categories companion for a good exposition of this. You then show that each kind of $T$-algebra map, 2-cell between, and their various compositions match those intended by a series of lengthy calculations backtracking through the construction of $T$. These calculations tend not to be very illuminating — so although the monadicity theorems yield nothing really new in this context they give an alternative approach which has the advantage that the calculations involved are illuminating, in that they involve natural concepts such as doctrinal adjunction and the behaviour of limits. Of course colimit presentations are very useful in other contexts as well: in particular they make the connection between flexible algebraic structure and pie colimits transparent.
• For genuinely new applications we can look to situations where colimit presentations do not apply — here is one such example. Recall that if $C$ is a monoidal category then if the forgetful functor from monoids in $C$ has a left adjoint it is automatically monadic (by Beck’s theorem). Likewise if $C$ is a monoidal 2-category and the forgetful 2-functor from monoids (or pseudomonoids) and strict homomorphisms has a left 2-adjoint then it too is monadic (by the enriched version of Beck’s theorem). This tell us that the strict monoid maps correspond to the strict $T$-algebra maps for the induced 2-monad $T$. Now our monadicity theorem allows us to show — assuming $C$ has some finite limits — that the pseudo, lax and colax monoid maps correspond to the pseudo, lax and colax $T$-algebra maps too.
• Another thing worth mentioning is that the main theorem, on orthogonal decompositions, is nothing to do with 2-monads but is purely a result about $\mathcal{F}$-categories. Whilst this can be easily used to prove monadicity theorems it is not bound by the formalism of 2-monads, and so can equally be used to give recognition theorems for weak morphisms relative to other abstract frameworks — such as two dimensional Lawvere theories.