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January 11, 2013

Two Dimensional Monadicity

Posted by Mike Shulman

(guest post by John Bourke)

This blog is about my recent preprint Two dimensional monadicity which is indeed about two dimensional monadicity. However, the monadicity theorems are an application, and the paper is really about how the weaker kinds of homomorphism that arise in 2-dimensional universal algebra — like strong, lax or colax monoidal functors between monoidal categories — are unique in satisfying certain properties. These properties relate the weak homomorphisms with the more easily understood strict homomorphisms and so are \mathcal{F}-categorical, rather than 2-categorical, in nature. If you want to understand what I mean then read on.

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 𝔸\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 λA_{\lambda} for the whole 2-category — the 2-category of loose morphisms — and A τA_{\tau} for the sub 2-category containing the tight morphisms together with all 2-cells between them. We write j:A τA λj: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 𝕄onCat l\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 sMonCat lj:MonCat_{s} \to MonCat_{l} which views strict monoidal functors as lax.
  • But we also have the \mathcal{F}-categories 𝕄onCat p\mathbb{M}onCat_{p} of strict and strong monoidal functors, and 𝕄onCat c\mathbb{M}onCat_{c} of strict and colax monoidal functors and 𝕄onCat p,l\mathbb{M}onCat_{p,l} of strong and lax ones and so on.
  • Likewise given a 2-monad TT on a 2-category CC (I mean a 2-monad in the strictest possible sense) we have strict TT-algebras, strict, pseudo, lax and colax TT-algebra morphisms and algebra transformations. These give rise to a number of different \mathcal{F}-categories such as the \mathcal{F}-category 𝕋Alg l\mathbb{T}Alg_{l} of strict and lax algebra morphisms. Its inclusion of tight into loose morphisms is the inclusion j:TAlg sTAlg lj:TAlg_{s} \to TAlg_{l}.
  • Any 2-category CC can be viewed as an \mathcal{F}-category (I denote this \mathcal{F}-category by CC too) in which the tight and loose morphisms coincide. The inclusion of tight into loose morphisms is then just the identity 1:CC1: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:𝔸𝔹W:\mathbb{A} \to \mathbb{B} is a 2-functor which preserves tight morphisms. In other words a 2-functor W λ:A λB λW_{\lambda}:A_{\lambda} \to B_{\lambda} which restricts to a 2-functor W τ:A τB τW_{\tau}:A_{\tau} \to B_{\tau} as in the commuting diagram A τ j A λ W τ W λ B τ j B λ\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}-CATCAT. Here are some examples.

  • An \mathcal{F}-functor W:𝔸BW:\mathbb{A} \to B to a 2-category BB (viewed as an \mathcal{F}-category) just looks like a commutative triangle of 2-functors as on the left below. A τ j A λ MonCat s j MonCat l Wτ W λ Vs V l B Cat \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 CatCat for instance. Thus we have a forgetful \mathcal{F}-functor V:𝕄onCat lCatV:\mathbb{M}onCat_{l} \to Cat with tight and loose parts V sV_{s} and V lV_{l} .

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

  • An \mathcal{F}-functor W:A𝔹W:A \to \mathbb{B} from a 2-category is a triangle too, like left below: A A τ Wτ W λ 1 j B τ j B λ A τ j A λ \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 𝔸\mathbb{A} we have an inclusion \mathcal{F}-functor j:A τ𝔸j:A_{\tau} \to \mathbb{A} drawn on the right above (yes, I’ve been calling it jj 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:𝔸BW:\mathbb{A} \to B to a 2-category BB the composite left below A τ j 𝔸 W B MonCat s j 𝕄onCat l V Cat\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 τ:A τBW_{\tau}:A_{\tau} \to B. Thus the entire \mathcal{F}-category 𝔸\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 FF whose underlying functor VFVF has a right adjoint — so that we have an adjunction (ϵ,VFG,η)(\epsilon, V F \dashv G,\eta) of categories and functors — the right adjoint GG obtains the structure of a lax monoidal functor in such a way that η\eta and ϵ\epsilon become monoidal transformations: which is to say that the adjunction lifts along VV from CatCat to MonCat lMonCat_{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:𝔸𝔹W:\mathbb{A} \to \mathbb{B} satisfies ll-doctrinal adjunction if given a tight morphism f:XY𝔸f:X \to Y \in \mathbb{A} and adjunction (ϵ,Wfg,η)(\epsilon, W f \dashv g,\eta) in B λB_{\lambda} (ie. gg is allowed to be loose) then that adjunction lifts uniquely along WW to an adjunction (ϵ ,fg ,η )(\epsilon^{\prime},f \dashv g^{\prime},\eta^{\prime}) in A λA_{\lambda}. Then the above observation about monoidal categories amounts to the fact that the forgetful \mathcal{F}-functor V:𝕄onCat p,lCatV:\mathbb{M}onCat_{p,l} \to Cat from strong and lax monoidal functors to CatCat satisfies ll-doctrinal adjunction. Since strict monoidal functors are strong it follows that V:𝕄onCat lCatV:\mathbb{M}onCat_{l} \to Cat also satisfies ll-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 ll-doctrinal adjunction captures laxness, as in the orientation and non-invertibility of the comparisons (Fa)(Fb)F(ab)(F a)(F b) \to F(a b) (of tensor products) and i BFi Ai^{B} \to F i^{A} (of units) defining a lax monoidal functor F:ABF:A \to B: for whilst V:𝕄onCat lCatV:\mathbb{M}onCat_{l} \to Cat satisfies ll-doctrinal adjunction neither of the forgetful \mathcal{F}-functors V:𝕄onCat pCatV:\mathbb{M}onCat_{p} \to Cat and V:𝕄onCat cCatV:\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 CatCat does satisfy ll-doctrinal adjunction. Intuitively then ll-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 ll-doctrinal adjunction to be what was really needed and called \mathcal{F}-functors satisfying this refinement ll-doctrinal. For our purposes it doesn’t matter what precisely an ll-doctrinal \mathcal{F}-functor is: it suffices to say that any \mathcal{F}-functor that satisfies ll-doctrinal adjunction, is faithful on 2-cells and reflects identity 2-cells is ll-doctrinal. For example V:𝕄onCat lCatV:\mathbb{M}onCat_{l} \to Cat satisfies these two additional conditions concerning 2-cells because monoidal transformations are just natural transformations with properties — thus VV is ll-doctrinal. I call the class of ll-doctrinal \mathcal{F}-functors ldoctl-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:ABF:A \to B between categories we can form the comma category B/FB/F — the one with objects like (α:bFa,a)(\alpha:b \to F a,a). This is a kind of 2-categorical limit, the so-called colax limit of FF, and has a colax cone which looks like B/F P λ Q A F B \array{ && B/F \\ & {}_P\swarrow & \Downarrow^{\lambda} & &\searrow_{Q} \\ A && \underset{F}{\to} &&& B } Here PP and QQ are projection functors — P(α:bFa,a)=aP(\alpha:b \to F a,a) = a and Q(α:bFa,a)=bQ(\alpha:b \to F a,a) = b — and the natural transformation λ:QFP\lambda:Q \to FP has component at (α:bFa,a)(\alpha:b \to F a,a) given by α\alpha itself. The limit property of B/FB/F is that this is the universal such cone.

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

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

In other words the inclusion j:A τ𝔸j:A_{\tau} \to \mathbb{A} is orthogonal to each ll-doctrinal \mathcal{F}-functor. It follows from the theorem that the decomposition MonCat s j 𝕄onCat l V Cat\array{MonCat_{s} &\stackrel{j}{\to}& \mathbb{M}onCat_{l} &\stackrel{V}{\to}& Cat} of the forgetful 2-functor V s:MonCat sCatV_{s}:MonCat_{s} \to Cat is an orthogonal decomposition. Likewise for any 2-monad TT on CatCat the decomposition TAlg s j 𝕋Alg l U Cat\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 τA_{\tau} sitting over BB then A τA_{\tau} can be extended to an \mathcal{F}-category 𝔸\mathbb{A} over BB in at most one way such that the loose morphisms of 𝔸\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 sCatV_{s}:MonCat_{s} \to Cat is strictly monadic — which is to say that we have an isomorphism of 2-categories EE over CatCat as below MonCat s E TAlg s Vs U s Cat \array{ MonCat_{s} &&\stackrel{E}{\to}&& TAlg_{s} \\ & {}_V_{s} \searrow && \swarrow_{U_{s}} \\ && Cat } So strict monoidal functors correspond to strict TT-algebra maps. What remains is to show that the lax monoidal functors correspond to the lax TT-algebra maps.

Now the isomorphism EE over CatCat just asserts the commutativity of the outside of the diagram

MonCat s j 𝕄onCat l V Cat E E l 1 TAlg s j 𝕋Alg l U Cat\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 (EE and 11) are isomorphisms we obtain a unique isomorphism of \mathcal{F}-categories E l:𝕄onCat l𝕋Alg lE_{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 TT-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 TT 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 TT-algebra map, 2-cell between, and their various compositions match those intended by a series of lengthy calculations backtracking through the construction of TT. 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 CC is a monoidal category then if the forgetful functor from monoids in CC has a left adjoint it is automatically monadic (by Beck’s theorem). Likewise if CC 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 TT-algebra maps for the induced 2-monad TT. Now our monadicity theorem allows us to show — assuming CC has some finite limits — that the pseudo, lax and colax monoid maps correspond to the pseudo, lax and colax TT-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.
Posted at January 11, 2013 7:29 PM UTC

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Re: Two dimensional monadicity

This is really exciting! One of the things I find most intriguing is what you touched on in your last bullet point: this may give us a way to define what is meant by a “lax morphism” even when there is no monad. For instance, suppose CC is a monoidal 2-category and the forgetful functor from monoids does not have a left adjoint. Then it still seems like the \mathcal{F}-category of monoids, strict morphisms, and lax ones (defined in the obvious way) satisfies the other conditions, so that it induces an ( ldoct,ldoct)(^\perp l doct, l doct) factorization. This could be viewed as an assertion that our naive notion of lax morphism is “correct” even though there is no 2-monad in sight.

You observe in the paper that ll-doctrinal \mathcal{F}-functors are the right class of a cofibrantly generated orthogonal factorization system. That implies that any 2-functor U:ABU:A\to B admits such a factorization A𝔸 l,BBA\to \mathbb{A}_{l,B} \to B, which we could regard as defining a notion of “lax AA-morphism relative to BB”. What conditions on AA, BB, and UU ensure that this notion is sensible? E.g. when is A𝔸 l,BA\to \mathbb{A}_{l,B} bijective on objects and fully faithful on tight morphisms? Always? Under what conditions does 𝔸 l,B\mathbb{A}_{l,B} have colax limits of loose morphisms, tight pullbacks, cotensors, or more generally the limits that Steve and I called “ll-rigged”?

Posted by: Mike Shulman on January 11, 2013 8:12 PM | Permalink | Reply to this

Re: Two dimensional monadicity

Hi Mike, Thanks. I’ve been meaning to think about some of your questions myself!

I agree with your first point - if I had a 2-category of “strict morphisms” AA over BB (via U:ABU:A \to B) and a candidate notion of lax morphism between the objects of AA, the first thing I would do to convince myself that this was the correct notion is to check that the corresponding \mathcal{F}-category 𝔸\mathbb{A} of strict and lax morphisms over BB was part of an orthogonal ( ldoct,ldoct)(^{\bot}l-doct,l-doct)-decomposition (regardless of any monadicity).

On your second point, note that any \mathcal{F}-functor which is 2-fully faithful on loose morphisms is ll-doctrinal. In particular each \mathcal{F}-fully faithful one is so. Therefore the enriched (bo/ff) factorisation system ensures that each member of ldoct^{\bot}l-doct is bijective on objects. Moreover any \mathcal{F}-functor FF with F τF_{\tau} an isomorphism is orthogonal to each GG with G λG_{\lambda} 2-fully faithful – if such orthogonal components formed a factorisation system on CAT\mathcal{F}-CAT it would follow that each FF of ldoct^{\bot}l-doct had tight part F τF_{\tau} an isomorphism. But thinking about it now I don’t see the factorisation system.

With regards understanding further properties of 𝔸 l,B\mathbb{A}_{l,B}: it would be great to know how to construct the factorisations 𝔸 l,B\mathbb{A}_{l,B} explicitly. Given F:ABF:A \to B my first thought was that (under certain assumptions) a loose map in 𝔸 l,B\mathbb{A}_{l,B} should be a span (p,q):XY(p,q):X \to Y in AA where the left leg pp is equipped with a right adjoint section (1,Fpg,η)(1,Fp \dashv g,\eta) in BB – modulo some equivalence relation on such spans. But I’m not sure – would be glad to hear any ideas!

Thanks, John.

Posted by: John Bourke on January 11, 2013 10:35 PM | Permalink | Reply to this

Re: Two dimensional monadicity

So are you saying there isn’t a set of maps the things satisfying the orthogonal right lifting property with respect to this set are the ll-doctrinal maps? (This is what I’d presumed you meant by “cofibrantly generated”.)

Posted by: Emily Riehl on January 24, 2013 5:57 PM | Permalink | Reply to this

Re: Two dimensional monadicity

Hi Emily, Yes there is a set of small maps (maps between finitely presentable F-categories) such that the l-doctrinal F-functors are those which have the unique rlp with respect to them.

For example an F-functor satisfies l-doctrinal adjunction if it has the unique rlp with respect to the inclusion of the “free tight arrow” into the “free adjunction with tight left adjoint”.

The actual l-doctrinal F-functors are defined to satisfy a refined form of l-doctrinal adjunction but this can also be expressed as a unique rlp, and indeed the generating cofibrations are easier to describe explicitly than the above one: they are between finite F-categories rather than just finitely presentable ones.

Posted by: John Bourke on January 25, 2013 4:30 PM | Permalink | Reply to this

Re: Two dimensional monadicity

I’m a bit late to this party but very glad I took the time to read this. John, this all sounds very cool!

Would you mind explaining a bit more what you meant when you said that ll-doctrinal adjunction captures laxness but not coherence? I wasn’t sure how to relate the results you mentioned (about the lack of doctrinal adjunctions for strong or colax morphisms) with the case of incoherent lax maps.

Posted by: Emily Riehl on January 24, 2013 5:55 PM | Permalink | Reply to this

Re: Two dimensional monadicity

Hi Emily, Thanks for reading! Hope the following makes it clearer?!?

Just to be clear: by an incoherent lax monoidal functor I mean a functor F:ABF:A \to B equipped with natural comparisons FaFbF(ab)FaFb \to F(ab) and i BFi Ai^{B} \to Fi^{A} satisfying no further axioms.

These are “lax” in the same sense as lax monoidal functors are: the natural comparisons go “into F” and are non-invertible. However they don’t satisfy the additional “coherence” axioms asked of a lax monoidal functor.

Now let 𝕄onCat l\mathbb{M}onCat_{l} / 𝕄onCat il\mathbb{M}onCat_{il} be the F-categories of monoidal categories, strict and (lax/incoherent lax) monoidal functors respectively. Both sit over CatCat by forgetful F-functors and both of these F-functors satisfy ll-doctrinal adjunction – for the right adjoint of a strict monoidal functor is lax monoidal and so certainly incoherent lax monoidal! Therefore ll-doctrinal adjunction does not distinguish the strict and lax from the strict and incoherent lax: it does not capture coherence axioms.

On the other hand we have pseudo and colax monoidal functors. These are the loose morphisms of the F-categories 𝕄onCat p\mathbb{M}onCat_{p} and 𝕄onCat c\mathbb{M}onCat_{c}, with tight morphisms the strict monoidal functors in each case. Again both of these F-cats sit over CatCat by a forgetful F-functor but neither of these satisfies ll-doctrinal adjunction: the right adjoint of a strict monoidal functor cannot in general be given the structure of either a strong or a colax monoidal functor.

In this way the property of ll-doctrinal adjunction distinguishes 𝕄onCat l\mathbb{M}onCat_{l} from 𝕄onCat p\mathbb{M}onCat_{p} and 𝕄onCat c\mathbb{M}onCat_{c} and so captures the notion of laxness: distinguishes it from the pseudo and colax (and strict too). It doesn’t capture coherence though because it doesn’t distinguish the “coherent” 𝕄onCat l\mathbb{M}onCat_{l} from the “incoherent” 𝕄onCat il\mathbb{M}onCat_{il}.

The distinguishing feature of 𝕄onCat l\mathbb{M}onCat_{l} from 𝕄onCat il\mathbb{M}onCat_{il} is that the former admits an F-categorical limit that the latter does not: the colax limit of a loose morphism. So we can say that this limit recognises coherence axioms. Both F-categories admit “tight pullbacks” so these don’t have anything to say in the way of coherence: it turns out that there is an essentially unique F-category over CatCat having its tight morphisms the strict monoidal functors, admitting these limits and being ll-doctrinal (roughly satisfying ll-doctrinal adjunction). Therefore, roughly speaking:

ll-doctrinal adjunction captures the “laxness” and “colax limits of loose morphisms” capture the right coherence axioms.

Posted by: John Bourke on January 25, 2013 5:56 PM | Permalink | Reply to this

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