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March 9, 2014

Review of the Elements of 2-Categories

Posted by Emily Riehl

Guest post by Dimitri Zaganidis

First of all, I would like to thank Emily for organizing the Kan extension seminar. It is a pleasure to be part of it. I want also to thank my advisor Kathryn Hess and my office mate Martina Rovelli for their revisions.

In the fifth installment of the Kan Extension Seminar we read the paper “Review of the Elements of 2-categories” by G.M Kelly and Ross Street. This article was published in the Proceedings of the Sydney Category Theory Seminar, and its purpose is to “serve as a common introduction to the authors’ paper in this volume”.

The article has three main parts, the first of them being definitions in elementary terms of double categories and 2-categories, together with the notion of pasting. In a second chapter, they review adjunctions in 2-categories with a nice expression of the naturality of the bijection given by mates using double categories. The last part of the article introduces monads in 2-categories, and specializing to 2-monads towards the end.

Double categories and 2-categories

The article starts with the definition of a double category as a category object in the (not locally small) category of categories CAT\mathbf{CAT}. (I think that there might be some set theoretic issues with such a category, but you can add small everywhere if you want to stay safe.)

The authors then switch to a description of such an object in terms of objects, horizontal arrows, vertical arrows, and squares, with various compositions and units. I will explain a bit how to go from one description to the other.

A category object is constituted of a category of objects, a category of morphisms, target and source functors, identity functor and a composition.

The category of objects is the category whose morphisms are “the objects” and whose morphisms are the vertical arrows. The category of morphisms is the category whose objects are the horizontal morphisms and whose morphisms are the squares, with vertical composition.

Since the functors Obj,Mor:CATSET\mathrm{Obj}, \mathrm{Mor}: \mathbf{CAT} \longrightarrow \mathbf{SET} preserve pullbacks, by applying them to a double category seen as a category object, we get actual categories. Applying Obj\mathrm{Obj} to the double category, we get the category whose objects are “the objects” and whose morphisms are the horizontal arrows. Applying Mor\mathrm{Mor}, we get the category whose objects are the vertical morphisms and whose morphisms are the squares, but this time with horizontal composition.

An interesting thing to notice is that the symmetry of the explicit description of a double category is much more apparent than the symmetry of its description as a category object.

One can define a 22-category as a double category with a discrete category of objects, or as a CAT\mathbf{CAT}-enriched category, exactly as one can define a simplicially enriched small category as either a category enriched over sSet\mathbf{sSet} or as a category object in sSet\mathbf{sSet} with a discrete simplicial set of objects.

The second viewpoint on 2-categories leads to definitions of 2-functors and 2-natural transformations and also to modifications, once one makes clear what enrichment a category of 2-functors inherits.

It is also worthwhile mentioning that the pasting operation makes computations easier to make, because they are more visual. The proof of proposition 2.1 of this paper is a good illustration of this.

The basic example of a 2-category is CAT\mathbf{CAT} itself, with natural transformations as 2-cells (squares).

As category theory describes set-like constructions, 2-category theory describes category-like constructions. You can usually build up categories with as objects sets with extra structure. In the same way, small V-categories, V-functors, and V-natural transformations form a 2-category.

My first motivation to learn about 2-categories was the 2-category of quasi-categories defined by Joyal and which has been studied by Emily Riehl and Dominic Verity in the article The 2-category theory of quasi-categories in particular the category-like constructions one can make with quasi-categories, such as adjunctions and limits.

Adjunctions and mates in 2-categories

It is not a surprise that 2-categories are the right framework in which to define adjunctions. To build the general definition from the usual one, you just need to replace categories by objects in a 2-category, functors by 1-cells of the 2-category, and natural transformations by its 2-cells.

Adjunctions in a 2-category 𝒞\mathcal{C} compose (as in CAT\mathbf{CAT}), and one can form two, a priori distinct double categories of adjunctions. Both of them will have the objects of 𝒞\mathcal{C} as objects and the horizontal morphisms being the morphisms of 𝒞\mathcal{C}, while their vertical morphisms are the adjunctions (going in the same direction as the right adjoint, by convention). The two double categories differ on the squares. Given adjunctions fu f \dashv u and fuf' \dashv u' together with 1-cells a:AAa:A \longrightarrow A' (between the domains of uu and uu') and b:BBb:B \longrightarrow B' (between the codomains of uu and uu'), the squares of the first double category are 2-cells buuab u \Rightarrow u'a while the squares of the second are 2-cells fbaff'b \Rightarrow a f.

Now, the bijective correspondence between these kind of 2-cells given by mates induces an isomorphism of double categories. This means in particular that the horizontal (or vertical) composite of mates is equal to the mate of the corresponding composite.

This is a very beautiful way to express the naturality of the mate correspondence, and it provides a one-line proof of the fact that two 1-cells that are left adjoints to a same 1-cell are naturally isomorphic.

Monads in 2-categories

2-categories are also the right framework to define monads. A monad in a 2-category 𝒞\mathcal{C} and on an object BB is a 1-cell t:BBt:B \longrightarrow B together with 2-cells μ:t 2t\mu: t^2 \Rightarrow t and η:1 Bt\eta: 1_B \Rightarrow t, verifying the usual equations μ(tμ)=μ(μt)\mu \circ (t\mu)= \mu \circ (\mu t) and μ(tη)=1 B=μ(ηt)\mu \circ(t\eta) = 1_B = \mu \circ(\eta t). Since 2-functors preserve both horizontal and vertical compositions, for all objects XX of 𝒞\mathcal{C}, tt induces a monad on 𝒞(X,B)\mathcal{C}(X,B), given by post-composition (t *,μ *,η *)(t_{\ast},\mu_{\ast},\eta_{\ast}). The authors call * an action of tt on s:XBs:X \longrightarrow B* a t *t_\ast algebra structure on ss.

The definition of monad morphism in this article is quite surprising for someone who has read the previous article by Ross Street, The formal theory of monads, which we read for the second meeting of the Kan extension seminar.

In Ross Street’s original paper, a monad morphism (B,t,μ,η)(B,t,μ,η)(B,t,\mu, \eta) \longrightarrow (B',t',\mu', \eta') is a 1-cell f:BBf: B \longrightarrow B' together with a 22-cell ϕ:tfft\phi: t'f \Rightarrow f t verifying certain conditions.

In this paper, morphisms of monads are defined only for monads on the same object, letting the 11-cell part of a monad transformation of the previous article be the identity. This leads the authors to reverse the direction of the morphism, since the 22-cell seems to go in the reverse direction of the 11-cell!

One might think that fixing f=1f=1 is needed by the result which explains that there is a bijection between monad morphisms ttt \Rightarrow t' and actions of tt on tt' making tt' a “(t,t)(t,t')-bimodule”. In fact, in the case where ff is not necessarily the identity, there is a bijection between 2-cells ϕ:tfft\phi:t f \Rightarrow f t' such that (f,ϕ)(f,\phi) is a monad functor and actions of tt on ftft' making ftft' a “(t,t)(t,t')-bimodule”. A statement of the same kind can be also made for monad functor transformations (in the sense of the formal theory of monads). A 2-cell σ:ff\sigma : f \Rightarrow f' is a monad functor transformation (f,ϕ)(f,ϕ)(f,\phi) \longrightarrow (f', \phi') if and only if σt:ftft\sigma t': f t' \Rightarrow f' t' is a morphism of “(t,t)(t,t')-bimodules”.

A 2-category admits the construction of algebras if for every monad (B,t,μ,η)(B,t,\mu, \eta), the 2-functor X𝒞(X,B) (t *,μ *,η *)X \mapsto \mathcal {C}(X,B)^{(t_\ast, \mu_\ast, \eta_\ast)} is representable. The representing object is called the object of tt-algebras. By Yoneda, the free-forgetful adjunction can be made internal in this case.

The terminology is justified, because in the 22-category CAT\mathbf{CAT}, it specializes to the usual notions of the category of tt-algebras and the corresponding free-forgetful adjunction.

A monad in 𝒞\mathcal{C} is the same as a 2-functor Mnd𝒞\mathbf{Mnd} \longrightarrow \mathcal{C}, where Mnd\mathbf{Mnd} is the 2-category with one object and Δ +\Delta_+, the algebraist’s simplicial category as monoidal hom-category (with ordinal sum). Since moreover, 𝒞(X,B) (t *,μ *,η *)[Mnd,CAT](Δ +,𝒞(X,)),\mathcal {C}(X,B)^{(t_\ast, \mu_\ast, \eta_\ast)} \cong [\mathbf{Mnd}, \CAT]( \Delta_{+\infty}, \mathcal{C}(X,-)), (where Δ +\Delta_{+\infty} is the subcategory of maps of Δ\Delta preserving maxima, which is acted on by Δ +\Delta_+ via ordinal sum) one can see that the object of t-algebras can be expressed as a weighted limit.

As a consequence, it is not surprising that a 2-category admits the construction of algebras under some completeness assumptions.


In the last part of the article, the authors review the notion of a doctrine, which is a 2-monad in 2-CAT\mathbf{CAT}, i.e., a 2-functor D:𝒞𝒞D: \mathcal {C} \longrightarrow \mathcal{C}, where 𝒞\mathcal{C} is a 2-category, and 2-natural transformations mm and jj, which are respectively the multiplication and the unit, verifying the usual identities. The fact that it is both a monad on a 2-category and in another one can be a bit disturbing at first.

If (D,m,j)(D,m,j) is a doctrine over a 2-category 𝒞\mathcal{C}, then its algebras will be objects XX of 𝒞\mathcal{C} together with an action DXXDX \longrightarrow X, exactly as in the case of algebras over a usual monad.

Already with morphisms, we can take advantage of the fact that a 2-category 𝒞\mathcal{C} has 2-cells, and define DD-morphisms to be lax in the sense that the diagram DX DY X Y \begin{matrix} DX & \longrightarrow & DY \\ \downarrow & & \downarrow \\ X & \longrightarrow & Y \end{matrix} is not supposed to be commutative, but is rather filled by a 2-cell with some coherence properties.

As one might expect, we can actually form a 2-category of such DD-algebras by adding 2-cells, using again the 22-cells existing in 𝒞\mathcal{C}.

If we keep only the DD-morphisms that are strict, we obtain the object of algebras (which should be a 22-category) that we discussed before.

One example of a doctrine is Δ +×:CATCAT\Delta_+ \times - : \mathbf{CAT} \longrightarrow \mathbf{CAT} together with the multiplication induced by the ordinal sum, and unit given on 𝒟\mathcal {D} by the functor 𝒟Δ +×𝒟\mathcal{D} \longrightarrow \Delta_+ \times \mathcal{D} that sends dd to (,d)(\emptyset,d).

The algebras for this doctrine will be categories equipped with a monad acting on them, while the DD-morphisms are transformations of monads, and the DD-2-cells are exactly the monad functor transformations of Street’s article.

Here, since we have two different 2-categories of algebras (with strict DD-morphisms or with all of them), one can wonder if monad morphisms DDD \longrightarrow D' will induce 22-functors DD'-AlgD\mathbf{Alg} \longrightarrow D-Alg\mathbf{Alg} on the level of these 22-categories.

This is indeed the case, and one can actually go even one step further and define monad modifications, using the fact that 2-CAT\mathbf{CAT} is in fact a 3-category! These modifications between two given monad morphisms are in fact in bijective correspondence with the 2-natural transformations between the 22-functors induced by these monad morphisms on the level of algebras (with lax D-morphisms). Note that they are not the same as monad morphisms transformations of Street’s article.

This bijection is nice because it implies that you can compare 2-categories of algebras by only looking at the doctrines: if they are equivalent, so are the 2-categories of algebras.

The fact that this bijection does not hold when we restrict only to strict morphism was really surprising to me, but I guess this is the price to pay to use the 3-category structure.

During the last days of April, the Kan extension seminar will be reading the article “Two dimensional monad theory”, by Blackwell, Kelly and Powell. We will then have more to say about these 2-monads!

Posted at March 9, 2014 7:50 AM UTC

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Re: Review of the Elements of 2-Categories

In our discussion during the week, it was noted that the “naturality” of the mates bijection does not seem to resemble the more usual expressions of naturality, but is rather a matter of the correspondence respecting composition and identities, as mentioned in the paper and in Dimitri’s post. Emily mentioned that Eugenia Cheng suggests we call it “double functoriality”, which I think better conveys the result.

However, there are subtleties in understanding what we mean by this. We cannot expect the mate of an identity to be an identity (or an isomorphism): the familiar hom-set bijection of an adjunction is a special case of mateship (I wish I could draw the squares here!), hence the unit and counit of the adjunction are mates of identities, but they are not themselves isomorphisms in general.

Tim made a good comment on how to interpret the result in question in light of this subtlety, but I’ll leave that free for him to mention here.

Posted by: Alexander Campbell on March 9, 2014 9:53 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

The way I ended up putting it is like this – apologies if I just end up introducing confusion into something simple! The mate correspondence for a 2-category CC respects horizontal and vertical pasting of 2-cells in the double category Adj(C)\mathbf{Adj}(C) that Emily describes below: horizontal arrows are functors and vertical arrows are adjoints. But it doesn’t preserve composition in the “diagonal” direction. In particular, “diagonal isomorphims” are not preserved.

Posted by: Tim Campion on March 10, 2014 6:49 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

What exactly did you mean by diagonal composition? Actually, let me guess: (forgive me for writing without pictures)

A square auuba u \Rightarrow u' b in the right-hand double category, with uu and uu' right adjoints, also appears as a square (au)11(ub)(a u) 1 \Rightarrow 1 (u' b) with 11 and 11 the right adjoints of identity adjunctions.

Importantly, if the original auuba u \Rightarrow u' b was an isomorphism, its inverse appears as (ub)11(au)(u' b) 1 \Rightarrow 1 (a u) and these squares can be composed vertically along either the horizontal morphisms aua u or ubu' b.

But this composite operation is not entirely internal to the double category so is not respected by a double functor, or even a double functor isomorphism. Note in particular, that the identity on aua u cannot be interpreted as a square whose vertical morphisms encode the adjunction uu.

Was this what you had in mind?

Posted by: Emily Riehl on March 11, 2014 9:03 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

That’s a great tidying up of what I meant! The situation is when we have two 2-cells inverse in our underlying 2-category CC:

A a A u α u B b B\array{A & \overset{a}{\to} & A' \\ {}^u \uparrow & ^\alpha \seArrow & \uparrow^{u'} \\ B & \overset{b}{\to} & B'} A a A u α 1 u B b B\qquad \array{A & \overset{a}{\to} & A' \\ {}^u \uparrow & ^{\alpha^{-1}} \nwArrow & \uparrow^{u'} \\ B & \overset{b}{\to} & B'}

If fu,fuf\dashv u,f'\dashv u', then the first one square, α\alpha, can be regarded as a 2-cell in the double category RAdj(C)\mathbf{RAdj}(C) of left-adjoints. But its inverse cannot! It points in the wrong direction. Actually, I think I was originally under the impression that that the α 1\alpha^{-1} square, when flipped about its diagonal axis, would wind up in RAdj(C)\mathbf{RAdj}(C), but now that I actually write out the diagrams, I see this is not the case: the vertical arrows would have to be a,aa, a', which are not right adjoints. Anyway, in this case, the mate of α\alpha is a cell of LAdj(C)\mathbf{LAdj}(C):

A a A f α¯ f B b B\array{A & \overset{a}{\to} & A' \\ {}^f \downarrow & ^\bar\alpha \swArrow & \downarrow^{f'} \\ B & \overset{b}{\to} & B'}

Of course, as you say, we can regard u,uu,u' as ordinary functors and obtain 2-cells of RAdj(C)\mathbf{RAdj}(C) underlain by α,α 1\alpha, \alpha^{-1} in CC, which can be composed in RAdj(C)\mathbf{RAdj}(C), namely

B u A a A 1 α 1 B b B u A\array{B & \overset{u}{\to} & A & \overset{a}{\to} & A'\\ {}^1 \uparrow & & ^\alpha \seArrow & & \uparrow^1 \\ B &\overset{b}{\to} & B' & \overset{u'}{\to} & A'} B b B u A 1 α 1 1 B u A a A\qquad \array{B &\overset{b}{\to} & B' & \overset{u'}{\to} & A'\\ {}^1 \uparrow & & ^{\alpha^{-1}} \seArrow & & \uparrow^1 \\ B & \overset{u}{\to} & A & \overset{a}{\to} & A'}

These are vertical isomorphisms, and therefore preserved by the mate correspondence. But the mate correspondence in this case is trivial, because we’re using the identities as our adjoints rather than fuf \dashv u and fuf'\dashv u': the mates are the same arrows when regarded as living in CC:

B u A a A 1 α 1 B b B u A\array{B & \overset{u}{\to} & A & \overset{a}{\to} & A'\\ {}^1 \downarrow & & ^\alpha \swArrow & & \downarrow^1 \\ B &\overset{b}{\to} & B' & \overset{u'}{\to} & A'} B b B u A 1 α 1 1 B u A a A\qquad \array{B &\overset{b}{\to} & B' & \overset{u'}{\to} & A'\\ {}^1 \downarrow & & ^{\alpha^{-1}} \swArrow & & \downarrow^1 \\ B & \overset{u}{\to} & A & \overset{a}{\to} & A'}

Part of what makes it weird is that a single 2-cell of CC can have different mates depending on how you regard it as a 2-cell of RAdj(C)\mathbf{RAdj}(C).

Posted by: Tim Campion on March 12, 2014 2:30 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

The expression of the “naturality” of mates as “double functoriality” is an extremely valuable tool. As Tim and Emily pointed out, it explains exactly how we can have an operation that is a “functor” and yet doesn’t preserve “isomorphisms”, because in general the property of “being an isomorphism” is not “visible” to the double category. (It is visible if either both horizontal boundaries or both vertical boundaries of a square are equivalences, and indeed in those cases the mate of an isomorphism is an isomorphism.)

Another closely related “double functor”, which exhibits similar behavior, is the passage from model categories and Quillen functors to homotopy categories and derived functors. Its domain is a double category whose objects are model categories, whose vertical and horizontal arrows are left and right Quillen functors, respectively, and whose squares are arbitrary natural transformations. In particular, every natural transformation α:fghk\alpha:f \circ g \to h \circ k, where ff and kk are left Quillen and gg and hh are right Quillen, has a derived transformation Ho(α):LfRgRhLkHo(\alpha):\mathbf{L}f \circ \mathbf{R}g \to \mathbf{R}h \circ \mathbf{L}k. But as is the case with mates, α\alpha might be an isomorphism without this being visible to the double category, so that Ho(α)Ho(\alpha) might fail to be an isomorphism. The double functoriality of HoHo is important because it “preserves mates” in a double-categorical sense; see here.

Posted by: Mike Shulman on March 12, 2014 12:57 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

I just want to state for the record that I really like this paper.

Posted by: Emily Riehl on March 22, 2014 5:21 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Something I’ve been trying to understand about double categories is the nature of how they are internal categories in Cat\mathbf{Cat}. Do we in any way use the 22-categorical structure of Cat\mathbf{Cat} or are we really just using the 11-category structure? (If we start using the 22-category structure are we then looking at things like pseudo double categories? I.e., categories which are somehow `weakly internal’ to Cat\mathbf{Cat}.)

My motivation for thinking about this is to do with cartesian 22-monads. Whilst reading John Bourke’s thesis, I noticed a lemma (7.4) that says if we start with a cartesian monad T:T \colon \mathcal{E} \rightarrow \mathcal{E}, then we can produce a cartesian 22-monad Cat(T):Cat()Cat()\mathbf{Cat}(T) \colon \mathbf{Cat}(\mathcal{E}) \rightarrow \mathbf{Cat}(\mathcal{E}) on the 22-category of categories internal to \mathcal{E}.

If we then have a cartesian monad on Cat\mathbf{Cat} then this should give us a cartesian 22-monad on DblCat\mathbf{DblCat}, the 22-category of double categories, double functors, and transformations. Now there are many nice (read cartesian) 22-monads on Cat\mathbf{Cat}, so what I’m curious about is whether anyone has looked at what structures these internalised 22-monads would give? (For all I know it may be mentioned later in John’s thesis and I just haven’t read far enough yet…)

I’m now going to see if any of this makes sense and see if we can apply the `internalizer’ Cat()\mathbf{Cat}(-) to a 22-monad, so I might update this later.

Posted by: Alex Corner on March 9, 2014 1:59 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Internal categories are defined for any category with pullbacks – so there is no interaction with the 2-categorical structure of Cat\mathbf{Cat}. However, there is a more general notion of double category where one of the directions is not strict. (So we end up with a bicategory in one direction and a 2-category in the other.)

Posted by: Zhen Lin on March 9, 2014 9:00 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

People with little, or no experience at all about effectively proving coherent commutativities are somewhat scared by 2-category theory diagrams. Now that I studied something I think that not only the difficulty of getting through coherence diagrams grows quadratically with dimensions, but is also utterly underestimated in research papers which conclude “by standard computations with pasting diagrams”.

As an example of this, take Prop. 2.1 (page 86); a paper I had to go through some times ago began with that precise statement (or something which is ϵ\epsilon-near, maybe taken from Kelly-Street work?). The authors sketched a proof; I tried to go through that proof explicitly drawing every single diagram needed and I came up with a 6 pages long argument, which I’m trying to pass to LaTeX for many months.

This led me to wonder: is there a way to manage this explosive computational complexity in the same way there are dozens of other symbolic calculus- programs? I came out with this question on math.SE (feel free to mock my total ignorance of computer science and related topics -it is surely offensive to use a C-like pseudocode!).

The only thing which seems to be on this vein is this… Can anybody go further?

Another issue, again from the first lines.

Kelly and Street begin with a useful comparison between double categories (i.e., categories internal to Cat\mathbf{Cat}) and 2-categories (categories enriched over Cat\mathbf{Cat}). I think that the importance of drawing this connection will always be underestimated; in particular, I feel utterly disoriented by the richness of this zoology, which prevents all of us poor children from having a simple answer to a natural question like “what is the precise link between these two notions”?.

I had a pleasant chat with M. Przybylek about this precise topic, since he faced this precise problem in his thesis (which is unfortunately written in Polish!):

M. Przybylek wrote: When I was a student, I noticed there were two different, but not completely disjoint, definitions of a category (the definition of a category enriched in a monoidal structure, and the definition of a category internal to a finitely complete category), and no-one could tell me how these two concepts are related to each other. At the time I asked this question to Martin Hyland. He drew some diagrams, thought for a while and said, that he didn’t know the answer, but it may be a good starting point for my Master thesis - if I discover anything I should write to him.

During the next month I developed a theory of categories relative to monoidal fibrations, which generalises both internal and enriched categories. The funny thing is that, at exactly the same time, Mike Shulman worked on monoidal fibrations and discovered almost the same concept; and moreover, at my local university, Marek Zawadowski started working on monoidal fibrations (but his concept was broken at the time). A less funny thing is that, Thomas Streicher, informed us that we rediscovered the old forgotten concept of relative categories intrduced by Gouzou and Grunig in a technical report in 1976 (they work has been never published).

Another interesting point is the following.

Every honest exposition on the subject of 2-categories must give Gray’s monumental work

  • John W. Gray, Formal category theory: adjointness for 22-categories. Lecture Notes in Mathematics, Vol. 391. Springer-Verlag, Berlin-New York, 1974. xii+282 pp.

as a reference. And in fact Kelly and Street do.

I became acquainted with Gray’s particular taste for Mathematics and Category Theory reading his paper “Closed categories, lax limits and homotopy limits”. I feel Gray’s work has been somewhat neglected, and I would like to understand why. His book, albeit extremely “dry” and outdated in some parts, is extremely deep and pervasive.

Googling a bit one reads that

[Gray’s] book was supposed to be the first part of a four volume [!!!] work, but unfortunately later volumes/chapters never appeared.

What happened then? Why did he stop working on it? Is there any clue about the topics covered by the other three volumes?

Posted by: Fosco Loregian on March 9, 2014 7:12 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

I want to give a shout-out to Proposition 2.2, described above by Dimitri. The isomorphism of double categories whose horizontal arrows are functors and whose vertical arrows are fully-specified adjunctions expresses the “double functoriality” of the mates correspondence.

Its consequences are elementary but can be quite useful: this result describes which diagrams involving natural transformations between composites of functors which include left adjoints will have a “dual” form involving their right adjoints.

Here is one example that I used in my thesis: given any lax monad morphism whose functor part is a right adjoint, the mate is a colax monad morphism. The data of either natural transformation then specifies a lift of the right adjoint to the Eilenberg-Moore categories and an extension of the left adjoint to the Kleisli categories.

Another famous example, I believe due to Kelly, is that an adjunction between monoidal categories lifts to a monoidal adjunction (meaning both functors are lax monoidal) if and only if the left adjoint is strong.

What are some other corollaries of this result?

Posted by: Emily Riehl on March 10, 2014 2:11 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

One instance of the mate correspondence related to something we’ve talked about earlier in the seminar is that there are two dual descriptions of the Cauchy completion C˜\tilde C of a category CC. We can describe C˜\tilde C as the category of left adjoint profunctors 1C1 \to C. Or by taking mates in the bicategory of categories and profunctors, we can describe C˜\tilde C as the opposite of the category of right adjoint profunctors C1C \to 1. I think we only need the horizontal dimension of the double-functorality of mates in order to see that these descriptions are equivalent.

I suppose since the mates are taken in a bicategory, the naturality of the mate correspondence will need to be updated: For a bicategory BB, the double category Adj(B)\mathbf{Adj}(B) will be non-strict… in both directions, I suppose – making it a double bicategory.

As another twist, in order to upgrade to the Cauchy completion of enriched categories, we take mates in the bicategory of 𝒱\mathcal{V}-categories and 𝒱\mathcal{V}-profunctors. In order to see that the two descriptions of C˜\tilde{C} are equivalent as 𝒱\mathcal{V}-categories, I think we need the mate corrspondence not just for bicategories, but for 𝒱\mathcal{V}-bicategories, and the naturality of the mate correspondence becomes an equivalence of 𝒱\mathcal{V}-double-bicategories.

Posted by: Tim Campion on March 10, 2014 3:34 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Of course we should enrich the mate correspondence! That’s a lovely observation about Cauchy completions that I didn’t pick up on last time around.

How do we define an adjunction in a 𝒱\mathcal{V}-2-category?

How about fuf\dashv u if there exist morphisms η:Ihom(1,uf)\eta\colon I\to\hom(1,uf) and ϵ:Ihom(fu,1)\epsilon\colon I\to\hom(fu,1) such that yadda yadda, yeah I guess that will work out.

Can we also define an enriched adjunction a a 2-functor from a free enriched adjunction 2-category, whose endo-hom-categories are the free 𝒱\mathcal{V}-category with monoid object?

Posted by: Joe Hannon on March 11, 2014 9:45 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

yeah I guess that will work out.

Yes, and it’s equivalently just an ordinary adjunction in the underlying ordinary 2-category of your 𝒱\mathcal{V}-2-category (obtained by applying the usual underlying-ordinary-category functor homwise).

Can we also define an enriched adjunction a a 2-functor from a free enriched adjunction 2-category

I would guess probably, assuming 𝒱\mathcal{V} is well enough behaved.

Posted by: Mike Shulman on March 13, 2014 5:17 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

I was wondering about this remark that Kelly and Street make on p. 97:

On the other hand the category of symmetric monoidal closed categories, and of morphisms preserving all the structure on the nose […] is not doctrinal over CAT. Indeed it is only a category, not a 22-category: there seems to be no natural definition of a 22-cell.

Firstly, is it still true that there is no “natural” way to come up with 22-cells for the category SMCC of symmetric monoidal closed categories and functors preserving the structure on the nose? What prompted Street and Kelly to say this? Why is the obvious definition of a strict monoidal transformation between strict monoidal functors not appropriate? Perhaps I’m missing something, e.g. that with these strictness constraints things don’t compose right, or there is some issue with coherence, but at least there seems to be nothing problematic with the first obvious consequence of having strict transformations, namely that α Uα V=α UV\alpha_{U} \otimes \alpha_{V}=\alpha_{U \otimes V}.

The above also puzzles me because I (think I) know that the 22-category CCC of cartesian closed categories, functors preserving the cartesian structure on the nose and all natural transformations is pseudomonadic (“pseudodoctrinal”?), albeit not over Cat but over its underlying groupoid Catgr_{gr}, the 22-category of (small) categories and functors but with 22-cells only the natural isomorphisms. Is the same true of SMCC (with 22-cells the strict monoidal transformations)? Is it pseudomonadic but not 22-monadic?

(I should also perhaps clarify, I am not the Dimitri who wrote the above blog post - I have been commenting under ‘Dimitris’ without realizing that there was also a Kan seminar participant with whom I share my first name.)

Posted by: Dimitris on March 10, 2014 7:18 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Oh – you’re not Dimitri? Well, nice to meet you, Dimitris. It’s been a pleasure talking to you in the past, and I’m sure it will be in the future!

Concerning the non-doctrinality of SMCC\mathbf{SMCC} – I’d bet it’s entirely analogous to the non-doctrinality of CCC\mathbf{CCC}, and that doctrinality can probably be recovered by by moving to Cat gr\mathbf{Cat}_{\mathrm{gr}} like you say. In both these cases, I think the reason we have to move to Cat gr\mathbf{Cat}_{\mathrm{gr}} has something to do with the fact that one of the variables of the hom-functor is contravariant. But I don’t have a grasp on how this works, even if it’s true.

Posted by: Tim Campion on March 11, 2014 4:03 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Likewise - always look forward to your comments.

Posted by: Dimitris on March 12, 2014 2:25 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

The 2-category of cartesian closed categories, functors preserving the cartesian and closed structure on the nose, and all natural isomorphisms is indeed strictly 2-monadic over Cat grCat_{gr} (and the related 2-category where the functors preserve the structure only up to isomorphism is, I believe, pseudo-monadic). You can’t expect to get non-invertible transformations in the category of algebras if the base 2-category has only invertible ones. Similarly for the 2-categories of closed symmetric monoidal categories, strict or pseudo closed symmetric monoidal functors, and monoidal natural isomorphisms (once you’ve decided whether your functors are strictly or pseudo-ly monoidal, there’s no additional notion of “strictness” for transformations).

The reason we have to use Cat grCat_{gr} instead of CatCat does indeed have to do with the contravariance of the internal hom. Suppose that a forgetful functor from some 2-category of closed monoidal categories to CatCat had a left adjoint FF, and consider the categories 1=()\mathbf{1} = (\star) and 2=(aϕb)\mathbf{2} = (a\xrightarrow{\phi} b) with the two functors a,b:12a,b:\mathbf{1} \to \mathbf{2} and natural transformation ϕ:ab\phi:a\to b. In F1F\mathbf{1} we have the object [,][\star,\star], and clearly we should have Fa([,])=[a,a]F a([\star,\star]) = [a,a] and Fb([,])=[b,b]F b([\star,\star]) = [b,b]. So the component of FϕF\phi at [,][\star,\star] should be a map [a,a][b,b][a,a] \to [b,b] in F2F\mathbf{2}. However, there is no candidate for such a map; we can apply ϕ:ab\phi:a\to b on the codomains, but there is no map bab\to a to apply on the domains.

Posted by: Mike Shulman on March 11, 2014 8:41 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Right, yes, sorry I got my terminology all mixed up - that’s what I meant to write: you get strict 22-monadicity if the cartesian structure is preserved on the nose and pseudomonadicity if it is preserved up to isomorphism.

Similarly for the 22-categories of closed symmetric monoidal categories, strict or pseudo closed symmetric monoidal functors, and monoidal natural isomorphisms

OK yes, thanks. So there is no hidden complication here. But that still does not explain why Kelly and Street thought that SMCC is not even a 22-category, according to the last sentence in my quote above. This remark of theirs still baffles me…

However, there is no candidate for such a map; we can apply ϕ:ab\phi \colon a \rightarrow b on the codomains, but there is no map aba \rightarrow b to apply on the domains.

I see, of course. I wonder what kind of sufficient conditions you would have to impose on the monoidal structure to overcome this problem. Projection maps abaa \otimes b \rightarrow a that have sections? That’s probably too strong…I guess what I’m wondering about is this: what is the largest (full) sub-22-category of SMCC (resp. weak SMCC) that is 22-monadic (resp. pseudomonadic) over CatCat?

Posted by: Dimitris on March 12, 2014 2:24 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

I don’t think even sections of projections would be enough; in the example I described, ϕ\phi is a completely arbitrary map. Offhand, the only sub-2-category of SMCC I can think of that’s monadic over Cat is the 2-category of 2-groups. (-:

As for SMCC being a 2-category, you can of course choose any 2-cells you like, but what Kelly and Street say is that there seems no natural choice of 2-cells. I don’t think they mean “natural” in the precise sense of “natural transformation”; what I guess they have in mind is a definition of 2-cell that “respects all the structure present”. And there’s no evident way for a natural transformation to “respect the closed structure” unless it is an isomorphism.

Posted by: Mike Shulman on March 12, 2014 4:16 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

(Gotta get into the comments eventually!)

Dimitri, je te remercie beaucoup pour ton exposé magnifique!

For now, just a question with some explanation. Behind the doors of our seminar, many people have remarked about the interaction of ‘enriched’ and ‘internal’. For/in SetSet, we have that enriched and internal categories both coincide, and give us CatCat. For CatCat itself, it is no longer true that two different notions coincide, however we can still remark that the distance, from some point of view, is not that great: 2-categories are just certain double categories (those for which the object-category is discrete).

For a general monoidal category, this relation breaks down, internal and enriched categories are drastically different. I wonder if there is something deep going on behind this: in fact, for an ‘additive’ monoidal category (vector spaces or their derived analogues, say), I cannot think of any example of an internal category which would actually be present in the ‘working’ mathematics (objects would form a vector ‘space’, and this one could probably organize, but I am not sure if that’d be done in a natural way). From the other point of view, we see that enriched categories belong to a broader family of structures: one can include enriched categories in coloured operads or multicategories. Is there a similar way to generalise the notion of an internal category?

In this regard, let me explain one technical aspect of category objects which puzzles me. Recall that one way to describe a category object inside MM is to take a simplicial object X:Δ opMX:\Delta^{op} \to M which satisfies Segal conditions: X(n)X(1)× X(0)X(1)× X(0)...× X(0)X(1)X(n) \to X(1) \times_{X(0)} X(1) \times_{X(0)} ... \times_{X(0)} X(1) should be an isomorphism or an equivalence if we are working in a higher-categorical setting. It is well-known that working with simplicial objects in SSetSSet has allowed to introduce two models of higher categories, Segal categories and complete Segal spaces. A Segal category X:Δ opSSetX: \Delta^{op} \to SSet, in particular, has the property that X(0)X(0) is discrete and thus for the elements (objects) a 0,...,a na_0,...,a_n of X(0)X(0), one can extract maps X(a 0,...,a n)X(a 0,a 1)×X(a 1,a 2)×...×X(a n1,a n)X(a_0,...,a_n) \to X(a_0,a_1) \times X(a_1,a_2) \times ... \times X(a_{n-1},a_n) (this is done by taking fibers of the maps X(k)X(0) k+1X(k) \to X(0)^{k+1}) which are required to be weak equivalences.

However if one wonders how to define Segal category objects in a general monoidal (higher) category (so that there would be maps like X(a 0,...,a n)X(a 0,a 1)X(a 1,a 2)...X(a n1,a n)X(a_0,...,a_n) \to X(a_0,a_1) \otimes X(a_1,a_2) \otimes ... \otimes X(a_{n-1},a_n) for objects a ia_i), one runs into an immediate problem of incompatibility if the monoidal product is not Cartesian. While I am aware of some definitions in that setting (for instance this by Hugo Bacard, or the one I thought of in my Master thesis), I have not heard about a developed theory of such things (for instance, the existence of a model structure for enriched Segal categories). Overcoming the technical obstacles while dealing with such objects would potentially create another route in the field of Higher Algebra (in the sense of Lurie), but looking at the size of the book with the same name probably explains why this has not been done yet!

Posted by: Eduard Balzin on March 10, 2014 9:32 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

A category internal to Vect is called a Baez-Crans 2-vector space.

Posted by: Mike Shulman on March 11, 2014 2:48 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

… and was independently discovered by Forrester-Barker at the same time.

Posted by: Tim Porter on March 11, 2014 7:31 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

He also worked out the automorphism 2-groups of such 2-vector spaces.

Posted by: David Corfield on March 11, 2014 10:11 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

My thanks! I stand corrected and withdraw my babble about the non-naturality of internal category objects inside VectVect.

Posted by: Eduard Balzin on March 11, 2014 1:26 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

A category internal to Vect is called a Baez-Crans 2-vector space.

Or a “linear map”. :-)

Posted by: Tom Leinster on March 11, 2014 3:21 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Or, perhaps less misleadingly, a “2-term chain complex”. (-:

Posted by: Mike Shulman on March 11, 2014 6:45 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

If categories internal to Vect\mathbf{Vect} degenerate to linear maps, then what is the advantage of viewing them as internal categories? It’s hard to imagine that the internal functors are anything other than maps of chain complexes. Do internal natural transformations yield chain homotopies?

Posted by: Tim Campion on March 12, 2014 5:39 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Tim wrote:

If categories internal to Vect\mathbf{Vect} degenerate to linear maps, then what is the advantage of viewing them as internal categories?

Mainly it helps us think of interesting new things to do. If we think of 2-term chain complexes as ‘categorified’ vector spaces, this makes us want to categorify other chunks of math in a compatible way.

For example, it’s pretty easy, even without this viewpoint, to get interested in generalizing Lie algebras to L L_\infty-algebras, which are chain complexes obeying the Lie algebra axioms ‘up to coherent homotopy’.

But when we thought of 2-term L L_\infty-algebras as categorified Lie algebras, or Lie 2-algebras, it seemed very natural to look for the corresponding Lie 2-groups. The obvious guess is that they’d be categories internal to the category of Lie groups!

This is true in the simplest cases; in general they work out to be something subtler, but it was a good start. The correct concept of Lie 2-group, and the higher and ‘super’ versions of this concept, wind up being quite important in string theory. So, thinking of 2-term chain complexes as categories internal to Vect\mathbf{Vect} was a helpful line of thought.

I also think it’s useful to see general chain complexes of abelian groups as strict \infty-groupoids internal to AbGp\mathbf{AbGp}; this shows why they appear naturally as a way to simplify homotopy theory, which is about \infty-groupoids. When we work with chain complexes, we are strictifying and linearizing.

Posted by: John Baez on March 16, 2014 9:50 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

If categories internal to Vect\mathbf{Vect} degenerate to linear maps, then what is the advantage of viewing them as internal categories?

If you have a linear map A:VUA:V\to U it make sense to say that two vectors u 1,u 2Uu_1, u_2\in U are isomorphic if there is vVv\in V with

(1)u 2=u 1+A(v). u_2 = u_1+ A(v).

There are geometric situations where this is a very useful point of view. The one I have in mind is when UU is the space of invariant vector fields on some manifold MM with an action of a Lie group GG and VV is the space of infinitesimal gauge transformation (these are equivariant maps MLie(G)M\to Lie(G)). Then two vector fields isomorphic in the sense above induce the same flow on the orbit space. I learned this point of view from Richard Hepworth’s paper on vector fields on stacks. It’s useful for understanding relative equilibria, and without 2-vector spaces one wouldn’t think of invariant vector fields as objects in a category let alone of two different vector fields as being isomorphic.

Posted by: Eugene Lerman on March 16, 2014 7:50 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

This question might seem a bit off-topic in this thread since it is not about this particular paper, but it is a question about bibliography on 2-categories (and so I think that experts around here can help me).

I am a mathematician and I know somehow the basics of category theory, but up to know I have never read about 2-categories. Lately I have been working on computer science, and I have been reading some papers (for instance by Joseph Goguen) where they use something called 1.5-categories: which correspond to a simplification of 2-categories (where the arrows among “hom-sets” have the structure of a partial order, i.e., there is at most one arrow between two different elements of an hom-set).

Do your know any mathematical paper where they develop these 1.5-categories? For sure I can read about 2-categories having this particular case in mind, but I wonder whether any mathematical paper has focused on them (perhaps for pedagogical reasons before introducing 2-categories, etc.).

Thank you.

Posted by: boumol on March 11, 2014 10:01 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

I remember looking at Goguen’s use of 3/2-pushouts to capture the notion of conceptual blending.

Goguen wrote

A 3/2-category is a category C\mathbf{C} such that each set C(A,B)\mathbf{C}(A, B) is partially ordered, composition preserves the orderings, and identities are maximal.

So this relates to what nLab has as 2-poset or locally posetal 2-category.

Posted by: David Corfield on March 11, 2014 10:45 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

It makes some sense to call a poset-enriched category (also known as a locally posetal 2-category) a ‘1.5-category’ because posets are in a certain sense halfway between sets and categories—this becomes clear in section 5.1 of ‘Lectures on n-categories and cohomology’, where various versions of the periodic table of n-categories are studied.

But we should be a bit careful, because ‘sesquicategory’ is used to mean something completely different, namely something like a 2-category but with the interchange law for horizontal and vertical composition dropped!

Posted by: John Baez on March 18, 2014 4:00 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Here are some thoughts that were prompted by Dimitri’s description of monads in a 2-category as 2-functors from another 2-category Mnd\mathbf{Mnd} (I haven’t checked the details of what I’ve said below very thoroughly, but I think it works). The 2-category Mnd\mathbf{Mnd} can be described as the “2-category generated by an monad”, that is, it contains exactly the structure necessary to have a monad in it and no more. There is also a 2-category generated by an adjunction: it has two objects, cc and dd, 1-cells F:cdF \colon c \to d and G:dcG \colon d \to c and all composites of these, and 2-cells generated by η:1GF\eta \colon 1 \to GF and ϵ:FG1\epsilon \colon FG \to 1, subject to the triangle identities. I don’t know if there’s a more concise description like the one for Mnd\mathbf{Mnd} in terms of the simplicial category. An adjunction in a 2-category 𝒞\mathcal{C} is just a 2-functor Adj𝒞\mathbf{Adj} \to \mathcal{C}.

There is an inclusion 2-functor I:MndAdjI \colon \mathbf{Mnd} \to \mathbf{Adj}, and restriction along II gives I :[Adj,𝒞][Mnd,𝒞]I^{\star} \colon [\mathbf{Adj}, \mathcal{C}] \to [\mathbf{Mnd},\mathcal{C}], a 2-functor sending an adjunction to the monad it generates (we take lax natural transformations in both functor categories). If this restriction has a right adjoint I :[Mnd,𝒞][Adj,𝒞]I_{\star} \colon [\mathbf{Mnd}, \mathcal{C}] \to [\mathbf{Adj}, \mathcal{C}] it is given by right Kan extension along II, and sends a monad to its Eilenberg–Moore adjunction. In the Formal Theory of Monads, Street defined construction of algebras by saying that [Mnd,𝒞]𝒞[\mathbf{Mnd}, \mathcal{C}] \to \mathcal{C} has a right adjoint. This is similar to the above but with the terminal 2-category replacing Adj\mathbf{Adj}. Using Adj\mathbf{Adj} means that the right adjoint gives the whole Eilenberg–Moore adjunction in one go, rather than just the object of algebras.

Posted by: Tom Avery on March 11, 2014 10:47 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Exactly, and maybe one should emphasize that the right Kan extension is enriched.

This has been written down by Auderset in 1974 in french:

Ajonctions et monades au niveau des 2-catégories, Cahiers de Topologie et Géométrie Différentielle Catégoriques.

Posted by: Dimitri Zaganidis on March 11, 2014 11:36 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

When I first learned the definition of an adjunction in an arbitrary 2-category (probably from nLab), I was told that the hom-set definition of adjoint functors was not suitable for a generic 2-category (1-cells do not act on objects of 0-cells), so the unit/counit/triangle definition is used instead.

Now I come to learn that there is a very nice hom-set statement. Ok, it’s a little more complicated than the classical equation hom(La,x)=hom(a,Rx)\hom(La,x)=\hom(a,Rx), for which the construction was named in analogy with adjoint operators on a Hilbert space.

Explicitly, Kelly’s and Street’s prop 2.1 states hom(bu,ua)hom(fb,af)\hom(bu,u'a)\cong\hom(f'b,af). If our 2-category has a terminal object, then taking A=1=BA=1=B, we get hom(fb,a)=hom(b,ua)\hom(f'b,a)=\hom(b,u'a), where now we can think of aa and bb as global elements 1A1\to A' and 1B1\to B', respectively. Perhaps it be more natural to consider arbitrary “elements”, i.e. morphisms into AA' and BB'? If the 2-category doesn’t have a terminal object then that’s your only choice.

So can we take this as the definition of an adjunction in a 2-category, is it equivalent to unit/counit, like we know it is for Cat?

Posted by: Joe Hannon on March 11, 2014 9:22 PM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

Using generalized elements, we can let a=fa = f', b=1 Ab=1_{A'} to get hom(f,f)hom(1 A,uf)\mathrm{hom}(f', f') \cong \mathrm{hom}(1_{A'}, u' f') and extract a unit as the mate of 1 f1_{f'}. Similarly we find a counit as the mate of 1 u1_{u'}. By using naturality assumptions in the right way, we should be able to get triangle equations…

Posted by: Tim Campion on March 13, 2014 1:10 AM | Permalink | Reply to this

Re: Review of the Elements of 2-Categories

This came up in the last discussion.

Posted by: Mike Shulman on March 13, 2014 5:02 AM | Permalink | Reply to this
Read the post An Exegesis of Yoneda Structures
Weblog: The n-Category Café
Excerpt: Motivates the notion of Yoneda structure as an expression of basic notions of category theory in a natural 2-categorical language.
Tracked: March 24, 2014 5:36 AM
Read the post Elementary Observations on 2-Categorical Limits
Weblog: The n-Category Café
Excerpt: Describes Kelly's "Elementary observations on 2-categorical limits" and the general theory of weighted limits and colimits, which are described here in a special case.
Tracked: April 18, 2014 8:42 PM

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