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January 10, 2017

Category Theory in Barcelona

Posted by Tom Leinster

I’m excited to be in Barcelona to help Joachim Kock teach an introductory course on category theory. (That’s a link to bgsmath.cat — categorical activities in Catalonia have the added charm of a .cat web address.) We have a wide audience of PhD and masters students, specializing in subjects from topology to operator algebras to number theory, and representing three Barcelona universities.

We’re taking it at a brisk pace. First of all we’re working through my textbook, at a rate of one chapter a day, for six days spread over two weeks. Then we’re going to spend a week on more advanced topics. Today Joachim did Chapter 1 (categories, functors and natural transformations), and tomorrow I’ll do Chapter 2 (adjunctions).

I’d like to use this post for two things: to invite questions and participation from the audience, and to collect slogans. Let me explain…

Joachim pointed out today that category theory is full of slogans. Here’s the first one:

It’s more important how things interact than what they “are”.

As he observed, the question of what things “are” is slippery. Let me quote a bit from my book:

In his excellent book Mathematics: A Very Short Introduction, Timothy Gowers considers the question: “What is the black king in chess?”. He swiftly points out that this question is rather peculiar. It is not important that the black king is a small piece of wood, painted a certain colour and carved into a certain shape. We could equally well use a scrap of paper with “BK” written on it. What matters is what the black king does: it can move in certain ways but not others, according to the rules of chess.

In a categorical context, what an object “does” means how it interacts with the world around it — the category in which it lives.

Tomorrow I’ll proclaim some more slogans — I have some in mind. But I’d like to hear from you too. What are the most important slogans in category theory? And what do they mean to you?

I’d also like to try an experiment. The classes move rather quickly, so there’s not a huge amount of time in them for discussion or questions. But I’d like to invite students in the class to ask questions here. You can post anonymously — no one will know it’s you — and with any luck, you’ll get interesting answers from multiple points of view. So please, don’t be inhibited: ask whatever’s on your mind. You can even include LaTeX, in more or less the usual way: just put stuff between dollar signs. No tinguis por!

Posted at January 10, 2017 6:48 PM UTC

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Re: Category Theory in Barcelona

Today’s first slogan:

Isomorphism is the right notion of “sameness” of objects of a category.

This requires a bit of refinement, as objects of a category can also be objects of a 2-category… but it’s a starting point for the definition of equivalence, which I’ll be doing in a couple of hours.

Posted by: Tom Leinster on January 11, 2017 8:13 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Today (Thursday) the topic of my lecture was the category of sets, following Chapter 3 of Tom’s book. As a response to the first slogan posted by Tom, here is the analogous slogan one categorical dimension down:

Equality is the right notion of “sameness” of elements in a set

It’s a silly slogan on its own, because there is no other notion of sameness there, so its purpose is rather to put the higher slogans in perspective by providing a safe basis for the inductive ascent up through the categorical dimensions. After

Isomorphism is the right notion of “sameness” of objects of a category

comes

Equivalence is the right notion of “sameness” between categories

(in turn just a special case of saying that Equivalence is the right notion of “sameness” of objects in a 2-category. But that’s only a teaser for the moment. We will come to 2-categories at the end of the course.)

Posted by: Joachim Kock on January 12, 2017 8:30 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Actually, I think that that slogan for sets is not entirely contentless. It depends a little on what you mean by “set”. In set-theoretic foundations like ZFC or ETCS, equality is indeed the weakest notion of sameness for elements of a set, and automatically respected by everything. However, in his classic work on constructive analysis, Errett Bishop “defined” a set as follows

A set is defined by describing exactly what must be done in order to construct an element of the set and what must be done in order to show that two elements are equal.

In other words, equality on a “Bishop set” is a specified notion, not one given by the foundation. Bishop didn’t work in any specific formal system (a fact which has caused much confusion among later constructive mathematicians), but one formalization of his idea starts with a foundational language of “types” or “pre-sets” that don’t even have a notion of “equality” (or if they do, we ignore it), and defines a “set” (sometimes called a “setoid”) to be a type together with a sort of “equivalence relation” on it. In this context, your slogan for sets could be read as saying that just as we shouldn’t do anything to objects of a category that’s not invariant under isomorphism, we shouldn’t do anything to elements of a Bishop set that’s not invariant under the specified equality.

We don’t “see” this in ordinary mathematics because when we have a set equipped with an equivalence relation (e.g. equivalence of Cauchy sequences of rational numbers representing real numbers), we generally pass to the quotient set, making the equivalence relation into actual equality, hence automatically respected by all constructions. Categorifying that, in homotopy type theory (you knew I was going to get there eventually…) we can do something analogous for categories, passing to a “quotient” or “saturation” that makes isomorphism into “actual equality” and hence automatically respected by all constructions.

Posted by: Mike Shulman on January 12, 2017 10:39 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Hi Mike, good to see you here (in this little corner of the big Café).

Thanks for the remark about setoids – I was not quite sure myself why I liked the equality slogan. It seems in fact that many sets we use in our everyday life are really quotient sets. Most strikingly, perhaps, if we define the set of natural numbers as the set of equivalence classes of finite sets, so that “3” is actually the equivalence class of all 3-element sets.

In yesterday’s lecture, Property 8 was the existence of quotients for equivalence relations, and we also noted the universal property of such quotients: to define a map out of it is the same as defining a map out of the original set that respects the equivalence relation, that is, respects the ‘stipulated’ equality. So that is actually a setoid morphism.

By the way, in the lecture I had some serious trouble with the numbering of the 11 properties of the category of sets. A numbering is a bijective map from some of the first natural numbers, in this case from {0,1,2,3,4,5,6,7,8,9,10}. When I established that bijection (which essentially followed the order in Tom’s book, in turn chosen for pedagogical considerations), I did NOT define it by first defining a map from all finite sets and then check that this map respected equivalence of finite sets. I am pretty sure this is NOT how I did :-) I rather used another universal property of the natural numbers, namely the convenient property that one comes after the other! In other words, I exploited Property 9 of the category of sets, the existence of a natural-numbers object, and the definition of natural numbers as initial object in the category of configurations 1XX1 \to X \to X. By being an initial object (a special case of colimit – we’ll come to colimits in Lecture 5), this is also characterised by how to define maps out of it (just like quotients), in this case by recursion: to define a map out of N (or some initial segment of it), i.e. to enumerate certain things, it is enough to know where to start, and to know which thing comes after a given thing. I am pretty sure THIS is how I actually did: I decided that the basic property should be “sets form a category”, so I called that Property 0. After that, I followed some intuition about what should be the next property. (But the difficulty with counting is to remember what was the previous number.)

The conclusion of all this introspection is just that the definition of the natural numbers as the setoid of finite sets is not always the most practical.

So now I am thinking about the definition of the real numbers as the setoid of Cauchy sequences. I have no doubt it is good, perhaps even the best for many purposes, in analysis, for example. But suppose my use of the real line is mostly as a tool for parametrisation, of paths in a topological space, say, or as a model of time, so that I can say ‘earlier’ and ‘later’ and such things. I feel it would be a bit too bureaucratic to use Cauchy sequences for such purposes. Is there some other definition of the real line that could suit me better? Or something different, more practical than the real line? (I do notice that I said ‘real numbers’ when talking about Cauchy sequences, whereas I said ‘real line’ when talking about my use of it. Maybe this is a clue.)

Well, that was a long ramble. I guess I am compensating for the strict time limits of a lecture.

Posted by: Joachim Kock on January 13, 2017 9:09 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

I agree that “the quotient of the finite sets by isomorphism” is not a very good definition of the natural numbers. For one thing, if you don’t know what a natural number is yet, then how do you define “finite”?

Personally, I also don’t really like the definition of real numbers as equivalence classes of Cauchy sequences, but not because of the equivalence classes; rather I think it just doesn’t give the right answer in constructive mathematics. I prefer Dedekind cuts. Do you find those equally bureaucratic? If so, perhaps you would prefer the locale of formal reals? Or Freyd’s universal characterization of the unit interval?

Posted by: Mike Shulman on January 13, 2017 6:02 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

For one thing, if you don’t know what a natural number is yet, then how do you define “finite”?

In classical mathematics, one could point to Dedekind’s definition, that a set is finite if every injective endofunction on it is a bijection.

Posted by: Todd Trimble on January 21, 2017 4:29 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

That’s true. It’s worth remembering that for Dedekind’s definition to agree with the usual notion of “finite” one needs not only excluded middle but also countable choice. However, it is also possible to define finiteness in a constructively correct way without the natural numbers: first define the “finite subsets” of a set SS to be the smallest subset of P(S)P(S) containing \emptyset and closed under disjoint unions with singletons, then define SS to be finite if it is a finite subset of itself.

But I still wouldn’t really consider “the quotient of the set of finite sets by isomorphism” a very good definition of the natural numbers. I suppose you could probably prove that it has the correct universal property, but it would take a substantial amount of work. One foundational problem with it is that it’s actually a proper class, not a set at all; only if you assume some kind of universe can you get a version of it that’s a set. Moreover, in HoTT this quotient would presumably have to be a set-quotient, so that the universal property would only work for mapping into other sets, not more general types.

Posted by: Mike Shulman on January 22, 2017 11:23 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Those are all good comments, Mike.

Just one side remark: in ZF, a typical maneuver for dealing with class-sized entities is Scott’s trick. So there, you could take the set whose elements are finite sets which are all of minimal rank (among sets having the same cardinality) and then put the equivalence relation on that. (I don’t suppose this will be to the taste of all readers in this corner of the internet!)

Posted by: Todd Trimble on January 22, 2017 7:42 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Today’s second slogan:

Adjoint functors arise everywhere.

This is Mac Lane’s. I wish I had his pithy way with words.

Posted by: Tom Leinster on January 11, 2017 8:14 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

There are slogans in Lambek and Scott’s Introduction to Higher-Order Categorical Logic. Six are numbered and there’s another on p. 148.

They attribute the idea of conveying category theoretic precepts through ‘slogans’ to Lawvere’s Adjointness in Foundations.

Where else has this been pursued?

Posted by: David Corfield on January 11, 2017 8:51 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Thanks, David. Here are Lambek and Scott’s six slogans:

I. Many objects of interest in mathematics congregate in concrete categories.

Hard to argue with that.

II. Many objects of interest to mathematicians are themselves concrete categories.

Less obvious. I might prefer the slightly less specific statement that many objects of interest are small categories. I guess what they have in mind is things like: every group can be viewed as a category of a special kind, as can every poset and — if you allow enriched categories — every metric space. (If I had their book with me, guessing would be unnecessary, but I don’t.)

III. Many objects of interest to mathematicians may be viewed as functors from small categories to Set.

Right: so an action of a group or a sheaf on a topological space can be viewed as a functor into Set.

IV. Many important concepts in mathematics arise as adjoints, right or left, to previously known functors.

This is a bit less cryptic and a bit less dramatic than Mac Lane’s “adjoint functors arise everywhere”.

V. Many equivalences and duality theorems in mathematics arise as an equivalence of fixed subcategories induced by a pair of adjoint functors.

I’d have loved to have had time to cover this in my class today. If anyone from the class is reading this, I encourage you to try Exercise 2.2.11, which explains what this slogan means. This is probably my favourite example.

VI. Many categories of interest are the Eilenberg-Moore categories [== categories of algebras] of triples [== monads] on familiar categories.

No arguments with that!

David wrote:

They attribute the idea of conveying category theoretic precepts through “slogans” to Lawvere’s Adjointness in Foundations.

That puzzles me, first, because I think of Lawvere as being rather anti-slogan, and second, because in that paper of his, the word “slogan” only seems to appear once, and not in an especially slogan-promoting way.

Posted by: Tom Leinster on January 11, 2017 1:31 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

That puzzles me…

Perhaps given the political dimension, it’s less surprising. Workers of the world, unite! is seen as a Marxist slogan.

Slogan has an interesting etymology

From Scottish Gaelic sluagh-ghairm ‎(“battle cry”)

Lambek and Scott also mention Lawvere 1967 which is an unpublished manuscript, Category valued higher order logic.

Posted by: David Corfield on January 11, 2017 6:48 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

I like the practice of summarizing important mathematical ideas as slogans. I think that some mathematicians get too fixated on only making statements that are completely precise and always true, and overlook the value of shorty, pithy statements that encapsulate big ideas, even if there are exceptions and addenda. I try to incorporate slogans in my teaching a lot, but I find that it often takes me several times through teaching a subject to figure out what I think the slogans should be.

Category theory may well be uniquely well suited to this kind of sloganeering, but it certainly has a role in other areas. It might be an interesting/useful project to collect such slogans for various areas somewhere.

Posted by: Mark Meckes on January 11, 2017 2:23 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

All diagrams commute!

David’s mention of Lambek and Scott’s led me (as internet searches do) to a book about the French philosopher Gilles Deleuze and education. It uses Lambek and Scott’s slogans as a case study of sloganeering in mathematics. Here’s a Google books link (p.150). Google books links don’t seem to be very shareable — often they work, but sometimes they don’t, so here’s an image of the relevant page.

Posted by: Tom Leinster on January 11, 2017 2:31 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

That looks interesting. I’ll try to take a closer look if I have a chance later (the link to that page works for me, but unfortunately I can’t see the next one), but one immediate reaction is that the author rather overestimates how specialized the terminology is. I’m anything but a “specialist in categorical logic” but the terminology in Lambek and Scott’s slogans is perfectly accessible to me.

(To be perfect honest, although I manage to hold on to the rough idea, I usually have to look up the definition of adjoint functors. Will I still be allowed to come here if I admit that?)

Posted by: Mark Meckes on January 11, 2017 3:17 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

one immediate reaction is that the author rather overestimates how specialized the terminology is.

Yes, and then immediately understates it too by saying “the thrust of them [the slogans] should be intuitive”. I can see how the plain-English meaning of category might lead someone without mathematical training to think they have an intuitive idea of slogans I and II, and their intuitive idea would probably be heading in the right direction. Just possibly you might say the same of III, but I have serious doubts about the others. In IV, they’ve cheated by mistranscribing “adjoints” as “adjuncts”!

(To be perfect honest, although I manage to hold on to the rough idea, I usually have to look up the definition of adjoint functors. Will I still be allowed to come here if I admit that?)

Thank you for saying that. I’d like it if people from our Barcelona class came here to ask questions, and in principle they only saw the definition of adjoint functor for the first time five hours ago. So, definitely yes!

Posted by: Tom Leinster on January 11, 2017 3:29 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Someone else can fill us in on whether this is why the notion is called “adjoint”, but some might find it a handy mnemonic.

If isomorphism is the right notion of sameness, then Hom(a,b)Hom(a,b) is a good notion of comparability; and in Inner Product Spaces, we like to compare vectors via their (I kind-of gave it away, didn’t I?) inner product a,b\langle a,b \rangle; meanwhile, functors act on categories and linear maps act on vector spaces (er… I shan’t try to make F arrows:Hom(a,b)Hom(Fa,Fb)F^{arrows} : Hom(a,b) \to Hom(F a, F b) correspond to anything linear…); and opposite maps F,GF,G between inner product spaces are said to be adjoint when Fa,b=a,Gb \langle F a , b\rangle = \langle a , G b \rangle and if given FF there is any such a GG, then there’s Only One GG — this boils down to the fact that the map 00 is adjoint only to the opposite 00. Similarly, functors F,GF,G are adjoint exactly when there’s a very natural isomorphism Hom(Fa,b)Hom(a,Gb) Hom(F a,b) \simeq Hom(a, G b) and furthermore, if given FF there’s any such GG and natural iso, then any two G,GG,G' with such natural iso’s are themselves naturally isomorphic: Hom(Gb,Gb)Hom(FGb,b)Hom(Gb,Gb) Hom(G' b, G' b ) \simeq Hom( F G' b , b ) \simeq Hom(G' b, G b) inducing the natural iso GbGbG'b \simeq G b pushing the identity Gb=GbG'b = G' b through the unnamed isomorphisms… and there’s one more triangle to check.

Posted by: Jesse C. McKeown on January 12, 2017 2:21 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

I discussed the idea of outlooks governed by broad principles around p. 206 of Towards a Philosophy of Real Mathematics. It would be interesting to make a collection. Along with Mac Lane, I was always drawn to the writings of people like Atiyah and Mackey because of their overarching visions.

Posted by: David Corfield on January 12, 2017 9:08 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Today’s third and final slogan:

Always consider the identity.

This was in the context of adjunctions. The first striking theorem about adjoint functors is that any adjunction is entirely determined by the unit and the counit. In other words, in the context of an adjunction FGF \dashv G, if you know the transpose

η A=id F(A)¯:AGF(A) \eta_A = \overline{id_{F(A)}}: A \to G F(A)

of the identity on F(A)F(A) for each AA, and also the transpose

ε B=id G(B)¯:FG(B)B \varepsilon_B = \overline{id_{G(B)}}: F G(B) \to B

of the identity on G(B)G(B) for each BB, then you know the tranpose of everything.

And better still, you only need to know one of the unit and the counit. The whole of the rest of the adjunction can be reconstructed from either η\eta or ε\varepsilon.

As many readers of this blog know, but not everyone in the class knows yet, this slogan will become super-important when we come to do the Yoneda lemma. We’ll get to that on Tuesday.

Posted by: Tom Leinster on January 11, 2017 2:02 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Today was the Yoneda lemma. As I mentioned in class, Always consider the identity has an important special case:

A natural transformation Hom(,A)XHom(-, A) \to X is determined by its value at id Aid_A.

Here AA is an object of a category 𝒜\mathcal{A} and XX is a functor 𝒜 opSet\mathcal{A}^{op} \to Set.

A curious thing: in the four classes so far, the number of students attending has been, respectively, 19, 17, 15, 13. Assuming that the arithmetic progression continues, our final class will have 1-1 student. Some of Joachim’s colleagues have expressed an interest in coming along to see what 1-1 student looks like. This presents problems of a philosophical type.

Posted by: Tom Leinster on January 17, 2017 9:25 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

15 today. For a few tantalizing minutes it was 11, but then more people turned up. Apparently category theory is just too much fun for the pattern to continue unbroken.

Today Joachim proclaimed the slogan:

The Yoneda lemma is a cornerstone in modern geometry.

And he justified it!

Posted by: Tom Leinster on January 18, 2017 7:59 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Hi Tom,

thanks for hosting this ‘forum’ at the café!

And thanks for helping with the course – it is a privilege to have you as co-teacher. (Although I was a bit intimidated by the situation of explaining Chapter 1 of a book with its author in the audience.)

I’d like to use this post for two things: to invite questions and participation from the audience, and to collect slogans.

Let me abuse of its purpose by also adding further remarks from one of the teachers. Here is one:

Yesterday, as an example of a category whose morphisms do not have underlying maps of sets of any kind, I briefly explained the category of 2-cobordisms (whose objects are closed oriented 1-manifolds and whose morphisms are diffeomorphism classes of oriented 2-cobordisms (that means 2-manifolds with boundary, interpreted as being a morphism from the ‘in-going’ part of the boundary to the ‘out-going’ part of the boundary). I chose this example out of inertia and lack of imagination.

Actually I should have taken a different example, which is more elementary, and somehow also more illustrative: the category Rel of relations: the objects are sets, and the morphisms are relations. A relation RR from AA to BB is a subset of A×BA \times B, writing aRba R b to say that (a,b)(a,b) belongs to RR, that is, aa is related to bb. The identity relation from AA to AA is the diagonal subset AA in A×AA\times A, and composition of relations is given like this: If RR is a relation from AA to BB and SS is a relation from BB to CC, then SRS\circ R is defined to be the relation from AA to CC that relates aa to cc iff the exists bb in BB with aRba R b and bScb S c.

This is a very hands-on examples, and it is fun to sit down and experiment with it. It is actually also an important example for applications, for example to database theory (namely so-called relational databases, such as SQL), and it is also used as a toy example in quantum mechanics, since the category of relations actually behaves a lot like the category of Hilbert spaces in many respects.

Here is a link to the nLab page on Rel: https://ncatlab.org/nlab/show/Rel

Posted by: Joachim Kock on January 11, 2017 2:06 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Hi Tom

It is clear that the naturality is very important in the definition of adjunction to make the theorems work. I tried to see it as really the naturality in the technical sens of natural transformations. For a FIXED object A in AA it works. The assignment sending B to HomBB(F(A),B) is a functor from BB to Set. Also the assignment sending B to HomAA(A,G(B)) is a functor from BB to Set. And the bijection for each B is a natural transformation between these two functors. But that is only for fixed A. If A varies it is not functorial in A. What am I missing something?

Posted by: Anonymous on January 11, 2017 6:45 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Hi, and thanks for the question!

For a FIXED object AA in 𝒜\mathcal{A} it works. The assignment sending BB to Hom (F(A),B)Hom_\mathcal{B}(F(A), B) is a functor from \mathcal{B} to SetSet.

Right! For any object XX of \mathcal{B}, there’s a functor

(X,):Set. \mathcal{B}(X, -): \mathcal{B} \to Set.

On objects, it sends BB \in \mathcal{B} to the set Hom (X,B)Hom_{\mathcal{B}}(X, B). On morphisms, it sends g:BBg: B \to B' to the map of sets

Hom (X,B)Hom (X,B) Hom_{\mathcal{B}}(X, B) \to Hom_{\mathcal{B}}(X, B')

defined by qgqq \mapsto g \circ q.

That’s a lot to swallow, and we’ll get to it on Tuesday. I didn’t want to attempt to explain all this today, which is why I didn’t refer to it when I explained the naturality part of adjointness.

In particular, all this is true when X=F(A)X = F(A), for some functor F:𝒜F: \mathcal{A} \to \mathcal{B}.

Also the assignment sending BB to Hom 𝒜(A,G(B))Hom_\mathcal{A}(A, G(B)) is a functor from \mathcal{B} to SetSet. And the bijection for each BB is a natural transformation between these two functors.

Yes indeed. Indeed, an adjunction between FF and GG (with FF on the left and GG on the right) defines a natural isomorphism

Hom (F(A),)Hom 𝒜(A,G()) Hom_\mathcal{B}(F(A), -) \to Hom_\mathcal{A}(A, G(-))

for each A𝒜A \in \mathcal{A}. This is a natural isomorphism between the two functors

Hom (F(A),),Hom 𝒜(A,G()):Set. Hom_\mathcal{B}(F(A), -), Hom_\mathcal{A}(A, G(-)): \mathcal{B} \to Set.

The naturality of this isomorphism corresponds, explicitly, to one of the two equations that I wrote on the board today and labelled as (#).

But that is only for fixed AA. If AA varies it is not functorial in AA. What am I missing something?

I think the thing you’re missing is contravariance. Exactly the same story can be told if we fix BB and let AA vary. We have two functors

Hom 𝒜(,G(B)),Hom (F(),B):𝒜 opSet. Hom_\mathcal{A}(-, G(B)), Hom_\mathcal{B}(F(-), B): \mathcal{A}^{op} \to Set.

Generally, for any object XX of any category 𝒜\mathcal{A}, the construction AHom(A,X)A \mapsto Hom(A, X) defines a contravariant functor from 𝒜\mathcal{A} to SetSet. In symbols, it’s a functor 𝒜 opSet\mathcal{A}^{op} \to Set. For instance, in the category of vector spaces over a field kk, the construction VHom(V,k)V \mapsto Hom(V, k) is contravariant in kk: a linear map VWV \to W gives rise to a map Hom(W,k)Hom(V,k)Hom(W, k) \to Hom(V, k) in the other direction. That’s why it’s contravariant.

But the contravariance doesn’t cause any problems! It’s just a fact of life that some things are contravariant, and it’s no big deal.

Does that answer it?

(Incidentally, if you want to do a curly/calligraphic A here, you can’t just type AA: as in Latex, you have to type \mathcal{A}, between dollar signs.)

Posted by: Tom Leinster on January 11, 2017 9:05 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

You might also enjoying looking at this alternate definition of “adjunction” which uses naturality in the sense of natural transformations.

An adjunction between F:𝒜F: \mathcal{A} \to\mathcal{B} and G:𝒜G: \mathcal{B} \to \mathcal{A} is a natural isomorphism between these functors: (𝒜 op×G)Hom 𝒜:𝒜 op×𝒜 op×𝒜Set(\mathcal{A}^{op} \times G) \circ Hom_\mathcal{A}: \mathcal{A}^{op} \times \mathcal{B} \to \mathcal{A}^{op} \times \mathcal{A} \to Set (F op×)Hom :𝒜 op× op×Set(F^{op} \times \mathcal{B}) \circ Hom_\mathcal{B}: \mathcal{A}^{op} \times \mathcal{B} \to \mathcal{B}^{op} \times \mathcal{B} \to Set

Posted by: jbs on January 12, 2017 12:28 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Oops, that was supposed to be two lines.

(𝒜 op×G)Hom 𝒜:𝒜 op×𝒜 op×𝒜Set(\mathcal{A}^{op} \times G) \circ Hom_\mathcal{A}: \mathcal{A}^{op} \times \mathcal{B} \to \mathcal{A}^{op} \times \mathcal{A} \to Set

(F op×)Hom :𝒜 op× op×Set(F^{op} \times \mathcal{B}) \circ Hom_\mathcal{B}: \mathcal{A}^{op} \times \mathcal{B} \to \mathcal{B}^{op} \times \mathcal{B} \to Set

I guess that’s what I get for previewing my post in a half-screen window :)

Posted by: jbs on January 12, 2017 12:33 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Here is a slogan to atone for my formatting mishap:

Category theory often lets us transform a complicated idea in simple setting into a simple idea in a more complicated setting.

The usual definition of an adjunction has a simple setting (just the hom sets your categories) but makes it seem like a complicated idea (a family of bijections, plus the naturality conditions).

In the definition I just gave, the setting is more complex (two composite functors in a functor category) but the idea is simple (just an isomorphism).

The benefit is not immediate because you have to come to grips with the more complicated setting, but once you are there the idea becomes easier to understand.

For example, it is pretty obvious that all terminal objects in a category must be isomorphic, but not so obvious that all limits of a diagram must be isomorphic. It becomes obvious once you are comfortable viewing limits as terminal objects (in a category more complicated than the original one).

Posted by: jbs on January 12, 2017 1:09 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

I like the slogan!

Joachim pointed out an example of this. At the end of yesterday’s lecture, I stated a result similar to the following:

A functor G:𝒜G: \mathcal{B} \to \mathcal{A} has a left adjoint if and only if for all A𝒜A \in \mathcal{A}, the category (AG)(A \Rightarrow G) has an initial object.

Here (AG)(A \Rightarrow G) is my preferred notation for what’s more commonly written as (AG)(A \downarrow G). It’s the comma category whose objects are pairs (B,f)(B, f) where BB \in \mathcal{B} and f:AG(B)f: A \to G(B).

The point is that “initial object” is a simple idea but (AG)(A \Rightarrow G) is a relatively complicated category. The result above formalizes a certain universal property of adjointness. In a familiar example: if you have the free group F(A)F(A) on a set AA, and another group BB, then any function ABA \to B extends uniquely to a homomorphism F(A)BF(A) \to B. This is a relatively complicated idea, made in a simple setting (groups and sets).

Posted by: Tom Leinster on January 12, 2017 9:50 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Category Theory in 10 Slogans on Slide 19 here, some reduced to a single word.

Posted by: David Corfield on January 11, 2017 7:09 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Well, that’s fun. In case that link expires, it’s a talk of Jan Rutten’s called “The Method of Coalgebra” (or to give it its full ominous and humorous title, “The Method of Coalgebra — In Some Detail”).

I can’t resist talking through Rutten’s “Category theory in 10 slogans”:

1. Always ask: what are the types?

I’m not 100% convinced that’s category theory rather than type theory, but yeah, in category theory you always have to think about where things live. Today I was introducing adjoint functors with the usual kind of free-forgetful examples: specifically, the free-forgetful functor between commutative rings and sets. And in order to do this, I had to be careful about the distinction between a ring and its underlying set, much more careful than one would normally be.

I spent years saying “where does this thing live?” rather than “what is this thing’s type?”, until a category theorist mocked me gently and I changed my way of speaking.

2. Think in terms of arrows rather than elements.

Agreed. But it’s also fine to define an “element” of an object AA as an arrow into AA, à la Lawvere.

3. Ask what mathematical structures do, not what they are.

Right. That’s the slogan mentioned in my post, which Joachim put in front of the class on day one.

4. Categories as mathematical contexts.

5. Categories as mathematical structures.

I guess these are the same as Lambek and Scott’s Slogans I and II.

6. Make definitions and constructions as general as possible.

I disagree. I prefer Mac Lane’s remark at the end of Chapter IV of Categories for the Working Mathematician:

good general theory does not search for the maximum generality, but for the right generality.

Back to Jan Rutten’s list:

7. Functoriality!

8. Naturality!

9. Universality!

Good war-cries! Didn’t I hear Mel Gibson shouting these in some film, as he came running over a hill in blue face-paint?

10. Adjoints are everywhere.

Absolutely!

Posted by: Tom Leinster on January 11, 2017 7:29 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Here is a question about the category of matrices, from todays’ lecture (also exercise 1.3.33). What is an 0 x n matrix? It should be a matrix with zero rows but if it has zero rows then it also has zero columns. In linear algebra, when we prove that every linear transformation Rn to Rm is given by a matrix, we assume n and m not zero. May be this also is the case here?

Posted by: Anonymous on January 11, 2017 10:13 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Maybe it might help to ask: how many 0×n0\times n matrices are there?

Posted by: Jesse C. McKeown on January 12, 2017 2:25 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Let’s see. An m×nm \times n matrix over a ring kk is a function m×nk\mathbf{m} \times \mathbf{n} \to k, where m\mathbf{m} denotes an mm-element set. When m=0m = 0, the set m\mathbf{m} is empty, so m×n=\mathbf{m} \times \mathbf{n} = \emptyset. So, a 0×n0 \times n matrix must be a function k\emptyset \to k. There’s exactly one of these, so there’s exactly one 0×n0 \times n matrix. It doesn’t matter what we call it.

In linear algebra, the m×nm \times n matrices correspond in a natural one-to-one way with the linear maps k nk mk^n \to k^m. When m=0m = 0, there’s exactly one 0×n0 \times n matrix. There’s also exactly one linear map k nk 0k^n \to k^0, since k 0k^0 is a one-element set. So, the correspondence still works perfectly when you include m=0m = 0. The same is true for n=0n = 0.

So yes, I did mean to allow m=0m = 0 and n=0n = 0.

The way I just explained it makes it sound like it’s just luck that everything works for m=0m = 0 or n=0n = 0. But it’s not! In the usual proof that m×nm \times n matrices correspond to linear maps k nk mk^n \to k^m, there’s no reason to exclude 00 as a value of mm or nn. (I know this because I just taught it in my introductory linear algebra class last semester.)

Posted by: Tom Leinster on January 12, 2017 8:29 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

If it helps, vector spaces are like pointed sets (which in a sense are vector spaces over the mythical field with one element). An nn-dimensional space is then like a pointed set of n+1n+1 elements. Maps of such pointed sets must preserve the base point, so there is no choice when mapping out of or into the 00-dimensional space.

Posted by: David Corfield on January 12, 2017 9:20 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Anonymous wrote, anonymously:

What is an 0×n0 \times n matrix? It should be a matrix with zero rows but if it has zero rows then it also has zero columns.

No. It certainly has zero entries, but it still has nn columns. They’re just very skinny.

For example, suppose you have mm men and nn women. Then you can make an m×nm \times n matrix whose entries are all the ways a man and a woman can form a married pair. If there are 0 men and 3 women there are no possible married pairs, but don’t go berserk and claim there are no women!

Posted by: John Baez on January 14, 2017 12:28 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

C’est un joli exemple

Posted by: Fethi Kadhi on January 18, 2017 10:15 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

The computer scientist and semanticist Goguen even wrote a lovely “categorical manifesto”

Some slogans (he calls them dogmas) therein:

  1. To each species of mathematical structure, there corresponds a category whose objects have that structure, and whose morphisms preserve it.

1.1 Two objects have the same structure iff they are isomorphic, and an “abstract object” is an isomorphism class of objects.

  1. To any construction of structures of one species, say widgets, yielding structures of another species, say whatsits, there corresponds a functor from the category of widgets to the category of whatsits.

  2. To each natural relationship between two functors F,G : A -> B corresponds a natural transformation F=>G (or perhaps G=>F).

  3. Any canonical construction from widgets to whatsits is an adjoint of another functor, from whatsits to widgets.

  4. Given a category of widgets, the operation of putting a system of widgets together to form some super-widget corresponds to taking the colimit of the diagram of widgets that shows how to interconnect them.

A few more (not related to the paper):

Let me add one of the most famous slogans (probably not mentioned already because it is so famous): All Concepts are Kan Extensions!

In elements of the computer science community, the slogan is now notorious that “all told, a monad is just a monoid in the category of endofunctors.”

The informal slogan for n-category theory I thought, following Stone, was “Always categorify!” though googling I don’t seem to be able to find references for this.

Another, meta-slogan, which I can’t source (this one may be my own fault, in fact, which is probably why its so imprecise): “Naturality is about producing slogans of the form ‘The X of the Y is the Y of the X.’”

Posted by: Gershom B on January 12, 2017 9:02 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

The rhetorical form of this switching, usually not to equate but to contrast, is called ‘antimetabole’:

  • The weapon of criticism cannot replace the criticism of weapons.

  • It is not the consciousness of men that determines their being, but their being that determines their consciousness.

  • Philosophy can only be realized by the abolition of the proletariat, and the proleteriat can only be abolished by the realization of philosophy.

All due to Karl Marx.

Posted by: David Corfield on January 12, 2017 11:13 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

How about John Baez’s Every sufficiently precise analogy is yearning to become a functor?

Posted by: Mike Shulman on January 12, 2017 10:40 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Another one from John Baez: category theory is that branch of mathematics where the examples require examples.

Posted by: Simon Burton on January 13, 2017 2:51 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

That’s a nice one.

Here is another one, a bit related: in all areas of mathematics we ask:

What is an example of this?

In Category Theory, we also ask

What is this an example of?

I remember this question was actually asked by David Spivak at a very interesting workshop on Categorical Aspects of Network Theory in Torino a few years ago, to great surprise of some of the other participants, coming from fields such as Physics, Computer Science, Biology, Neuroscience. I think the speaker was actually John Baez – in any case, it was he who replied to the question (and expressed the shock on behalf of the audience).

(We could put the two questions together, and form an antimetabole!)

Posted by: Joachim Kock on January 13, 2017 4:00 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

David’s question was particularly shocking because he was asking what a particular theorem was an example of.

Posted by: Eugene on January 13, 2017 6:37 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Can anyone remember what the theorem was?

Despite the shock, this kind of question has a lot of potential to awaken people to larger patterns, and suggests imagination on the part of David to even think to ask. Sometimes it’s really hard to identify the subtle intimations and subconscious whisperings that prompt such questions.

Posted by: Todd Trimble on January 13, 2017 11:31 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

It’s been more than a year and a half. I think it was the theorem at the bottom of this post. I hope John will correct me if I am wrong.

By the way I have a strong impression that David always thinks this way :).

Posted by: Eugene on January 14, 2017 3:15 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

I don’t remember what theorem it was. I also don’t remember anyone being ‘shocked’ by David Spivak saying “What is this an example of?” But this probably says more about my memory, or lack thereof, than what actually happened.

Posted by: John Baez on January 14, 2017 4:26 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Joachim wrote:

We could put the two questions together, and form an antimetabole!

Right! To paraphrase Kennedy:

My fellow mathematicians: ask not what is an example of this, ask what this is an example of.

Posted by: John Baez on January 14, 2017 12:40 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

That should go on the category theorists’ purity test, right next to “what is the center of a set?”

Posted by: Mike Shulman on January 14, 2017 2:39 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

How about John Baez’s Every sufficiently precise analogy is yearning to become a functor?

For anyone in our class who doesn’t know the verb to yearn, it means something like to desire in a strong and dreamy way, maybe anhelar. It’s an unusually emotional word for a functor.

Anyway, I don’t get what John means. Here are the first two mathematical analogies that came into my head. Neither seems to yearn to become a functor, at least as far as I can see. But maybe someone can set me straight.

Suppose you’re an undergraduate learning ring theory. You already know some group theory, and the more you learn about rings, the more you notice similarities between the two subjects. A ring is a set equipped with some operations satisfying some equations; so is a group. You have homomorphisms between rings, just like you do for groups. You can take products of rings, which are rather like direct products of groups. Ring homomorphisms have kernels, just like group homomorphisms do. Ideals of rings seem to play the same role as normal subgroups of groups, and you can take quotients by ideals or normal subgroups in a similar way. There are first isomorphism theorems in both subjects, and they’re highly analogous too.

If I wanted to make this analogy precise, I’d use the notion of algebraic theory. Now it’s true that however you approach algebraic theories, there are functors kicking around: a model of a Lawvere theory is a functor out of it, and a monad is an endofunctor equipped with some stuff. And it’s also true that somewhat by coincidence, these two algebraic theories, rings and groups, are linked by forgetful functors

RingAbGrpGrp. Ring \to AbGrp \to Grp.

But I don’t think any of these can be the functors that John had in mind — the functor that the analogy between rings and groups is yearning to become.

The second analogy that came into my head was the one between metric spaces and categories. A metric space has a collection of points; a category has a collection of objects. For any two points a,ba, b of a metric space, there’s a real number d(a,b)d(a, b); for any two objects A,BA, B of a category, there’s a set Hom(A,B)Hom(A, B). For any three points a,b,ca, b, c of a metric space, there’s the triangle inequality

d(a,b)+d(b,c)d(a,c); d(a, b) + d(b, c) \geq d(a, c);

and for any three objects A,B,CA, B, C of a category, there’s a composition operation

Hom(A,B)×Hom(B,C)Hom(A,C). Hom(A, B) \times Hom(B, C) \to Hom(A, C).

Now as you and I know, this analogy is made precise by observing that both metric spaces and categories are special cases of enriched categories. But what’s the functor that the analogy between metric spaces and categories is yearning to become?

Posted by: Tom Leinster on January 13, 2017 3:13 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Hi Tom, maybe that’s how life is, also for analogies: you yearn and yearn without avail :-(

Posted by: Joachim Kock on January 13, 2017 4:04 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

This probably means that, to use John Baez’s language, those two analogies that you pointed are not “sufficiently precise”.

Posted by: Anonymous Coward on January 13, 2017 5:17 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Nah, they’re as precise as can be! Loads of work has been done on both things, and I’d say they’re completely understood and precisified. Maybe I’ll ask John to come and explain.

Posted by: Tom Leinster on January 13, 2017 5:57 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Well, I wouldn’t have thought John meant it too literally. There are other words which might fit the blank more gracefully than ‘functor’, depending on the circumstance. For the analogy between the categories of groups and rings, the slogan might be more like “every sufficiently precise analogy yearns to fit in a doctrine”, where a possible doctrine for groups and rings might be that of strongly protomodular algebraic categories, or something like that.
Posted by: Todd Trimble on January 14, 2017 6:41 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

What about the functor from algebraic theories to their category of models and the functor from monoidal categories to their category of enriched categories? I don’t think that an analogy between A and B has to yearn to become a functor from A to B or vice versa; it seems more likely that there is something of which both A and B are examples.

Posted by: Mike Shulman on January 13, 2017 6:08 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Perhaps more promising is the thought that an analogy is actually a span.

Gosh, was that really 10 years ago? I can’t say I looked further into 3/2 pushouts (next comment).

Posted by: David Corfield on January 13, 2017 9:26 PM | Permalink | Reply to this

Re: Category Theory in Barcelona

Spans are a nice way to think about generalization. As they say,

Every span yearns to be pushed out.

Thus, if A is a special case of B and also a special case of C, we can’t resist wanting to find some D of which both B and C and special cases. Ideally we get a commutative square, a pushout, so that we can say

A:B::C:D

or in words:

A is to B as C is to D.

This is a different kind of analogy than I originally had in mind, but it’s very nice! In this situation we say the relationship between A and B is analogous, or ‘parallel’, to the relationship between C and D. But also the relationship between A and C is analogous to that between B and D.

The idea of ‘parallel transport’ shows up in this vicinity, too.

Posted by: John Baez on January 14, 2017 12:23 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Tom wrote:

Anyway, I don’t get what John means. Here are the first two mathematical analogies that came into my head. Neither seems to yearn to become a functor, at least as far as I can see.

I’ve never really thought of groups as being ‘analogous’ to rings, they just seem ‘similar’ to rings in various way. To me an analogy is something like

A is to B as C is to D,

So, when I think about ‘analogies’, I think about situations where we have a bunch of objects (e.g. A and B) that are related in various ways, which are similar to a bunch of other objects (e.g. C and D) which are related in various analogous ways. So, I want there to be a category containing the objects A and B, and another category containing the objects C and D, and a functor from the first to the second, mapping A to C and B to D, and mapping the relation of A to B (for example a morphism in the first category) to the relation of C to D (for example a morphism in the second).

My favorite example is the analogy between classical mechanics and quantum mechanics, which is yearning to become a functor called ‘quantization’. Naively this functor might go from the category of symplectic manifolds and symplectomorphisms to the category of Hilbert spaces and unitary operators. Unfortunately this doesn’t work unless you add lots of structure to your symplectic manifolds—structure that gets quantization to work, but whose point is quite mysterious if you’re just doing classical mechanics.

This is the origin of Edward Nelson’s famous quote, roughly:

Quantization is a mystery but second quantization is a functor.

And that quote is the origin of mine:

Every analogy is yearning to become a functor.

I didn’t really mean ‘any’ analogy: in retrospect, that’s poetic exaggeration. I just meant analogies where a bunch of objects in some category are somehow ‘analogous’ to objects in another.

For example, the ring that’s ‘analogous’ to a given group GG is the group ring [G]\mathbb{Z}[G]: relationships between groups tend to give analogous relationships between their group rings. For example, a subgroup gives a subring. And taking the group ring of a group is indeed a functor.

Or, the ring that’s ‘analogous’ to a given Lie algebra 𝔤\mathfrak{g} is the universal enveloping algebra U𝔤U\mathfrak{g}, and this again becomes a functor.

But back to Nelson’s quote. What the heck is ‘second’ quantization???

‘Second quantization’ is something that at first seems more complicated than quantization but ultimately turns out to be much simpler. It amounts to taking a quantum system, thinking of it as a classical one, and then quantizing that. It’s actually a comonad!

In other words, if we put enough extra structure on our symplectic manifolds, we can actually succeed in making quantization into a functor, and then it has a right adjoint, and composing them gives the comonad called ‘second quantization’.

But why is it called ‘second’ quantization? The reason is that historically, people started by applying this functor to Hilbert spaces that were themselves obtained, nonrigorously, by quantization.

This may not make much sense to people who don’t know quantum physics, but most category theorists are quite familiar with the fact that if you have a left adjoint L:CDL: C \to D and a right adjoint R:DCR: D \to C, there’s an irresistible temptation to look at the functors L,LRL,LRLRL,...L, L R L, L R L R L, ... and also R,RLR,RLRLR...R, R L R, R L R L R...

Quantization, to the extent that you can actually turn it into a functor, is a left adjoint LL. Second quantization is LRL R. But historically, the first examples of second quantization involved the functor LRLL R L, where you’re quantizing twice.

And this leads to another of my mathematical mottos:

If it was fun doing something once, keep doing it again and again!

This presumably played a role in the origin of counting — especially counting to large numbers.

Posted by: John Baez on January 14, 2017 12:14 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Don’t forget the group ring functor. I can’t recall without calculation (and I haven’t time) what this does with the analogy. Perhaps it comes down to cohomology and vanishing Ext 1Ext^1s.

Posted by: David Roberts on January 14, 2017 12:19 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Thanks, everyone, for your thoughts on this. I think my favourite reading of

Every analogy is yearning to become a functor

is the one that Mike suggests. Let me expand on it very slightly (at the risk of changing Mike’s meaning). Whenever you discover an analogy between A and B, what you yearn to discover next is a common generalization, since this would explain why you’re seeing the analogy. So, you’re looking for some kind of general construction of which A and B are special cases. And many constructions are functors.

Posted by: Tom Leinster on January 16, 2017 9:00 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

All of which (well apart from the functor bit) was beautifully expressed by Polya in Mathematics and Plausible Reasoning. He spoke of the hope for a common ground underlying our analogical reasoning.

The strength of this hope features in the degree to which your confidence is boosted in a result when an analogue is proved, and this appears in his ‘Bayesian’ treatment of mathematical reasoning which I took up many years ago.

Posted by: David Corfield on January 16, 2017 9:50 AM | Permalink | Reply to this

Re: Category Theory in Barcelona

Consider finding a portion of a space which has no boundary of its own, and is not itself the boundary of another subspace. For instance, a spherical subspace, which does not bound a ball. This is analogous to finding a group element which gets sent to zero by a homomorphism, but which is not itself the image of another homomorphism. (The composition of the two homomorphisms is the zero map, just as the boundary of a boundary is empty.)

Of course the analogy I’m loosely describing turns out to be a (family of) nice functor(s): homology. Non-boundaries with no boundary are analogous to kernel elements which are not images, and we make it precise with free groups in a chain complex, and so on. At first glance this seems like a good reason for the slogan. Is it just as valid to think of a common generalization of the spaces and the groups?

Posted by: stefan on January 21, 2017 5:30 AM | Permalink | Reply to this

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