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Stream: community: general

Topic: "Diagonal Yoneda"

Patrick Nicodemus (Nov 20 2021 at 07:32):

Mac Lane has a survey of category theory called "Categorical algebra" which is surprisingly broad. In only 60 pages he makes it up to the basic properties of the bar construction and PROP's!

One of the nice results here is the diagonal Yoneda lemma, which I have never heard of before:

Let $K : \mathcal{C}^{\rm op}\times\mathcal{C}\to \mathbf{Sets}$ be a functor. Then
$\operatorname{Nat}(\operatorname{Hom}(-,-), K) \cong \int_c K(c,c)$ .

How can we interpret this? $\operatorname{Hom}(-,-)$ is the unit object in the monoidal category of profunctors from $\mathcal{C}$ to itself. So if we regard maps from the unit to $K$ as "points" of $K$, then the end of $K$ counts its "points."

John Baez (Nov 20 2021 at 12:11):

Nice!

John Baez (Nov 20 2021 at 12:12):

I looked at that survey once, but I should do it again, because I'd understand it better now.

David Michael Roberts (Nov 21 2021 at 06:04):

In case people are looking for it:

Bull. Amer. Math. Soc. 71 (1965), 40-106, DOI: https://doi.org/10.1090/S0002-9904-1965-11234-4

fosco (Nov 21 2021 at 12:13):

Patrick Nicodemus said:

Mac Lane has a survey of category theory called "Categorical algebra" which is surprisingly broad. In only 60 pages he makes it up to the basic properties of the bar construction and PROP's!

One of the nice results here is the diagonal Yoneda lemma, which I have never heard of before:

Let $K : \mathcal{C}^{\rm op}\times\mathcal{C}\to \mathbf{Sets}$ be a functor. Then
$\operatorname{Nat}(\operatorname{Hom}(-,-), K) \cong \int_c K(c,c)$ .

How can we interpret this? $\operatorname{Hom}(-,-)$ is the unit object in the monoidal category of profunctors from $\mathcal{C}$ to itself. So if we regard maps from the unit to $K$ as "points" of $K$, then the end of $K$ counts its "points."

@Théo and I might want to expand on this :wink: https://arxiv.org/abs/2011.13881

Mike Shulman (Nov 21 2021 at 15:39):

I'm not sure if this is an "interpretation", but another way to say this is that the end of a functor defined on $C^{\rm op}\times C$ is a weighted limit where the weight is $\hom_C$.

David Michael Roberts (Nov 22 2021 at 04:06):

I find it curious that Mac Lane wrote that before "category theory", as a name for the field, was established; he never uses the phrase in the review. If you go back and look at the old seminars and proceedings from before the mid 1960s or so, none of them say "category theory". At best, they say "categorical algebra" or "theory of categories".

David Michael Roberts (Nov 22 2021 at 04:09):

But by 1971, Mac Lane uses it as the opening words of the preface of Categories for the Working Mathematician. It may have been used informally, but not in print, as far as I can see.

David Michael Roberts (Nov 22 2021 at 04:50):

I can find mentions in Math Reviews of the phrase "category theory" from the early 1960s, but not before. And there are relatively few, only 19 mentions before 1966, the year of the La Jolla conference on "Categorical Algebra". In the next two years, mentions more than doubled. In the next four years, mentions had more tripled that. And that brings us to 1971. In the next four years, it had more than doubled, again.

David Michael Roberts (Nov 22 2021 at 04:51):

Clearly the field itself was growing rapidly, but it wasn't entirely stagnant in the years 1945–1961. I just think it hadn't been named, yet.

Damiano Mazza (Nov 22 2021 at 09:21):

There's a set of typewritten notes from 1965 about a seminar Grothendieck gave on "functorial language" (Introduction au Langage Fonctoriel). When I took a look at it, I found it amusing to see that it's in fact nothing but an introduction to plain category theory! (With some emphasis on abelian categories). I guess "functorial language" may have been another temporary name for the subject at the time, at least in France.

John Baez (Nov 22 2021 at 12:32):

It might be interesting to compare the terms "group theory", "ring theory" and "measure theory", and how they got started. Maybe there needs to be a body of theory, well-organized with certain central definitions and theorems, that someone imagines teaching in a course before a subject of "X theory" gets announced.

Patrick Nicodemus (Nov 22 2021 at 19:07):

Godement also uses this term in Topologie Algebrique et Theorie des Faisceaux.

dusko (Nov 23 2021 at 13:42):

Damiano Mazza said:

There's a set of typewritten notes from 1965 about a seminar Grothendieck gave on "functorial language" (Introduction au Langage Fonctoriel). When I took a look at it, I found it amusing to see that it's in fact nothing but an introduction to plain category theory! (With some emphasis on abelian categories). I guess "functorial language" may have been another temporary name for the subject at the time, at least in France.

category theory says that we should study mathematical structures in terms of their morphisms, without looking inside the objects. if category theory would practice what it preaches, it would be a theory of functors, without looking inside any categories.

Jules Hedges (Nov 23 2021 at 13:53):

I guess if you take that perspective seriously you end up inventing infinity-categories

dusko (Nov 23 2021 at 13:56):

Jules Hedges said:

I guess if you take that perspective seriously you end up inventing infinity-categories

but you only study them in terms of the infinity+1 morphisms.

Morgan Rogers (he/him) (Nov 23 2021 at 14:00):

The holistic principle we use to build categories will always force us to go to a higher ordinal..!

Ian Coley (Nov 23 2021 at 15:53):

I don't think category theory every tells us never to look inside the objects. It just gives us the tools so that we don't have to look inside the objects. For example, we can prove the Snake Lemma in any abelian category without every talking about elements. We can also prove the Snake Lemma in any abelian category by noting that every abelian category is locally isomorphic to R-Mod, and therefore we can use elements. Best of both worlds!

Fawzi Hreiki (Nov 23 2021 at 16:43):

I think this distinction between ‘inside’ and ‘outside’ is somewhat misfounded.

Almost all ‘inside’ descriptions of a thing are really just special kinds of ‘outside’ descriptions (eg how elements of a module are linear maps from the base ring).

John Baez (Nov 23 2021 at 17:59):

Lawvere argues that it's wrong to say that in category theory we don't look inside the objects.

He says we look inside an object $x$ using morphisms to it. So the object has elements $f: 1 \to x$, but also 'generalized elements' $f: y \to x$.

John Baez (Nov 23 2021 at 18:01):

And coYoneda says we can completely know an object from what's inside it.

Morgan Rogers (he/him) (Nov 23 2021 at 18:41):

Hmm but there are situations where the category can't see inside things in the sense you're describing, and as a result we can reduce the internal content of the objects to better understand the category. My intended example of this is Galois theory: we have a poset (or category, although one must be careful as I've found out) of field extensions, and that poset isn't rich enough to see all of the structure of the fields; as a result, there's the possibility of finding different objects forming an isomorphic poset, namely subgroups of a group, which makes computation a lot easier.

Patrick Nicodemus (Nov 24 2021 at 18:19):

dusko said:

Damiano Mazza said:

There's a set of typewritten notes from 1965 about a seminar Grothendieck gave on "functorial language" (Introduction au Langage Fonctoriel). When I took a look at it, I found it amusing to see that it's in fact nothing but an introduction to plain category theory! (With some emphasis on abelian categories). I guess "functorial language" may have been another temporary name for the subject at the time, at least in France.

category theory says that we should study mathematical structures in terms of their morphisms, without looking inside the objects. if category theory would practice what it preaches, it would be a theory of functors, without looking inside any categories.

mac laane addresses this here image.png

Jules Hedges (Nov 24 2021 at 18:27):

One I made earlier: Screenshot-2021-11-24-at-18.27.04.png

Nathanael Arkor (Nov 24 2021 at 18:29):

Nothing like the good old category $\mathbf{FunctionComposition}$.

Kenji Maillard (Nov 24 2021 at 18:42):

I'm afraid a lot of categories would end up being called $\textbf{Morphism}$...

dusko (Nov 24 2021 at 19:16):

Kenji Maillard said:

I'm afraid a lot of categories would end up being called $\textbf{Morphism}$...

... which would of course be abbreviated to $\textbf{Morph}$, which is nice because morph is also a verb, so we could write papers in pure syntax, from a one word lexicon, like https://isotropic.org/papers/chicken.pdf