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Stream: theory: category theory

Topic: yoneda lemma & yoneda embedding

Matteo Capucci (he/him) (Dec 02 2020 at 22:48):

A question that has been on my mind for quite a time is: can one recover Yoneda lemma from the existence of Yoneda embedding?
Clearly the question is really: how much should we know about the Yoneda embedding to be able to conclude Yoneda lemma holds?
A simple embedding is too weak. A natural thing that comes to mind is to see $よ$ as the unit of the free small cocompletion pseudomonad, i.e. the embedding of $\mathbf C$ into its own free small cocompletion. Since the latter is given by taking small presheaves over $\mathbf C$, then it is a category of presheaves and we can indeed embed $\mathbf C$ in it by sending $X$ to its representable $よX$.
Now it's tempting to say $\mathrm{Nat}(よY, P) = \mathrm{Nat}(よY, \mathrm{colim} よX_i) = \mathrm{colim}\, \mathrm{Nat}(よY, よX_i) = \mathrm{colim} \mathbf C(Y, X_i) = P(Y)$, but this is a circular argument: the interchange of $\mathrm{Nat}$ with $\mathrm{colim}$ is, afaik, a consequence of Yoneda lemma itself, so...
What do you think? Is that interchange true for some other reasons or is this approach just doomed? Shall I just be more coendy, i.e. ninja?

Dan Doel (Dec 02 2020 at 22:55):

What properties does "embedding" entail? Usually it is said that the important thing about the Yoneda lemma is that the embedding is fully-faithful.

Dan Doel (Dec 02 2020 at 23:00):

I'm not sure if even that is sufficient, though.

Matteo Capucci (he/him) (Dec 02 2020 at 23:35):

Embedding means fully faithful, yes. And it doesn't seem sufficient to say that is a an embedding since there are many embeddings for which Yoneda lemma doesn't work!

Dan Doel (Dec 03 2020 at 00:06):

I do think some sort of universality should be involved. For instance, there's Martin Escardo's writeup about how the Yoneda lemma is similar to the J rule for Martin-löf identity types.

Dan Doel (Dec 03 2020 at 00:07):

And you can instead present identity types with a Yoneda-like lemma.

Dan Doel (Dec 03 2020 at 00:12):

So it is not just saying that the embedding is an embedding, but it's somehow 'inductive', I suppose.

Dan Doel (Dec 03 2020 at 00:16):

This is the article.

Dan Doel (Dec 03 2020 at 00:22):

I think, running the exposition backwards, the Yoneda lemma is saying that the embedding is 'inductively generated' by the identity arrow.

Morgan Rogers (he/him) (Dec 03 2020 at 09:46):

Do you know about yoneda structures which axiomatise Yoneda in a way that generalises to a wider 2-categorical setting? I haven't read the paper in detail, but it seems to be providing sufficient conditions for the Yoneda lemma to be recoverable from a generalised Yoneda embedding?

Jens Hemelaer (Dec 03 2020 at 11:06):

I would guess that proving the interchange of $\mathrm{Nat}$ with colimits is more difficult than proving the Yoneda Lemma itself. So I agree that it is a bit a circular argument.

But I still think the interchange of $\mathrm{Nat}$ with colimits is a useful, geometric way to think about it. In a locally small category, an object $C$ is called tiny if $\mathrm{Hom}(C,-)$ preserves colimits. So Yoneda Lemma says that each $よX$ is tiny (or, using topos terminology, indecomposable projective).

Geometrically this means that they are really the most basic/rigid kind of "building blocks" (like bricks, maybe). As opposed to other, weaker kinds of building blocks, like the ones such that $\mathrm{Hom}(C,-)$ only preserves filtered colimits or direct sums.

Jens Hemelaer (Dec 03 2020 at 12:09):

I like the example of the category of directed graphs (digraphs), which is a category of presheaves.

Each directed graph is in a canonical way:

• a colimit of points and arrows; or
• a quotient of a "bunch of points and arrows" (seen as digraph); or
• a coproduct of connected digraphs; or
• a filtered colimit of finite digraphs.

This corresponds to the situations where you take as building blocks respectively:

• the indecomposable projective digraphs (aka the "tiny" ones, in this case only the point and the arrow);
• the projective digraphs (the ones such that each component is a point or an arrow);
• the indecomposable/connected digraphs;
• the finitely presented digraphs (which are the finite ones).

The Yoneda Lemma then says geometrically that you are in the first situation of the four.

Matteo Capucci (he/him) (Dec 03 2020 at 17:49):

@[Mod] Morgan Rogers I hoped I could avoid to resort to Yoneda structures, yet it's probably a lost fight

Matteo Capucci (he/him) (Dec 03 2020 at 17:49):

@Jens Hemelaer that's a neat perspective!

Fawzi Hreiki (Dec 03 2020 at 18:14):

It’d be interesting to see if you can assume what you’re looking for and try and derive a yoneda-structure-like thing

Morgan Rogers (he/him) (Dec 03 2020 at 19:10):

Matteo Capucci said:

[Mod] Morgan Rogers I hoped I could avoid to resort to Yoneda structures, yet it's probably a lost fight

It's just a matter of translation back into the specific case of Cat, which may even be included in the paper? I'm guessing here :wink: