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Stream: learning: questions

Topic: different version of Yoneda Lemma. Why does it work?

Davi Sales Barreira (May 29 2022 at 14:08):

So, I'm reading the book on Polynomial Functors, and the authors wrote the following definition for Yoneda Lemma:

Given a functor $F:\mathbf{Set} \to \mathbf{Set}$, and a set $S$, there is an isomorphism

$$F(S) \cong Nat(y^S, F)$$

where $Nat$ is the set of natural transformations. Moreover, the equation above is natural in both $S$ and $F$.

So, the only thing that is making me confused about such formulation is in regards to "locally small". From other books, the Yoneda Lemma is usally written with regards to locally small categories, which is not the case of $\mathbf{Set}$, right?
Hence, why is this formulation correct?

Kenji Maillard (May 29 2022 at 14:15):

$Set$ is locally small: the set of maps between two (small) sets is a small set (at least as long as you don't consider strongly predicative variants of set theory).

Davi Sales Barreira (May 29 2022 at 14:22):

I see. I had read in a book that Set was big, so I guess the book placed "locally small" inside of "big". Hence why I was thrown away. Thanks for clarifying.

Matteo Capucci (he/him) (Jun 03 2022 at 13:38):

Small and large refer to categories whose collection of objects form, respectively, a set or something possibly larger than a set. The category of sets is large because the collection of all sets is not a set. Locally small and large instead refer to the size of the collection of morphisms between two given objects. So set is locally small since there's only a set worth of functions between any two given sets.

Matteo Capucci (he/him) (Jun 03 2022 at 13:39):

An example of locally large category is Span(Set), there's a class worth of [[spans]] between any two given sets

Matteo Capucci (he/him) (Jun 03 2022 at 13:40):

An example of small category is the category of matrices, whose objects are the natural numbers and arrows $n \to m$ are given by $m \times n$ real-valued matrices. This is also locally small

Mike Shulman (Jun 03 2022 at 14:46):

Of course, Span(Set) isn't a category but a bicategory...

Matteo Capucci (he/him) (Jun 10 2022 at 09:51):

Mike Shulman said:

Of course, Span(Set) isn't a category but a bicategory...

I thought you could quotient out 1-cells using the invertible 2-cells (= say two spans are equivalent if there is an iso between them) and get a 1-category. Am I mistaken?

Chad Nester (Jun 10 2022 at 09:51):

No, you can definitely do that.

Matteo Capucci (he/him) (Jun 10 2022 at 09:53):

Ok then I was referring to that. But maybe it's non-standard to still call it Span(Set). I sweeped this problem under the carpet for expositional clarity.

Mike Shulman (Jun 11 2022 at 00:12):

In the absence of context, if I hear Span(C) I think of it as a bicategory, and I would write something like Ho(Span(C)) for the homotopy 1-category. In some particular context, if it's been explained that what's meant is the homotopy 1-category, I think it would be fine to use the notation Span(C) for that 1-category.

The point isn't totally irrelevant to this question -- locally large bicategories are much more common "in nature" than locally large 1-categories. (-: