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Stream: learning: questions

Topic: 'Slice Yoneda'

Fawzi Hreiki (Nov 20 2020 at 01:32):

Fix a category $\mathscr{C}$. Every map $f: X \rightarrow Y$ in $\mathscr{C}$ induces a functor $f_!: \mathscr{C}/X \rightarrow \mathscr{C}/Y$ given by composing with $f$ (i.e. the dependent sum, although no pullbacks are required in $\mathscr{C}$). Furthermore, this functor commutes with the domain projections $\mathscr{C}/X \rightarrow \mathscr{C}$ and $\mathscr{C}/Y \rightarrow \mathscr{C}$. So we can define the 'slice Yoneda' functor $\mathscr{Y}: \mathscr{C} \rightarrow \text{CAT}/\mathscr{C}$ by sending $X$ to $\mathscr{C}/X \xrightarrow{\text{dom}} \mathscr{C}$ and $f: X \rightarrow Y$ to $f_!$. This bears a strong resemblance to the usual Yoneda embedding $\mathscr{C} \rightarrow [\mathscr{C}^\text{op}, \text{Set}]$ although it doesn't seem to be mentioned in any of the usual textbook treatments.

This functor $\mathscr{Y}$ is obviously faithful since if we had $f_! = g_!: \mathscr{C}/X \rightarrow \mathscr{C}/Y$, we could apply them to $1_X$ to get $f = g$. I'm trying to prove that $\mathscr{Y}$ is also full but I can't seem to figure it out (although I feel like I'm missing something obvious).

Also, is there an analogue in this situation to the usual Yoneda lemma $\text{Hom}(\mathscr{Y}(X), F) \cong F(X)$?
In this case, $F$ would represent a functor $F: \mathscr{A} \rightarrow \mathscr{C}$ instead of a presheaf on $\mathscr{C}$ and so $\text{Hom}(\mathscr{Y}(X), F)$ would represent the set (or category?) of functors $\mathscr{C}/X \rightarrow \mathscr{A}$ commuting with the functors down to $\mathscr{C}$ (the domain projection and $F$ respectively).

Nathanael Arkor (Nov 20 2020 at 01:38):

There's an isomorphism between the category of presheaves on C and the category of discrete fibrations over C (which are in particular functors into C), so you can always rephrase results about presheaves (e.g. Yoneda) in terms of discrete fibrations.

sarahzrf (Nov 20 2020 at 01:54):

hmmmm, now i have a question myself

sarahzrf (Nov 20 2020 at 01:56):

reading this made it finally click for me that X ↦ C/X factors as yoneda embedding followed by category of elements

sarahzrf (Nov 20 2020 at 01:56):

but now im wondering... if you have pullbacks, you have the self-indexing and that has the same object mapping!

sarahzrf (Nov 20 2020 at 01:57):

so does it correspond in any meaningful way to something w/ presheaves?

Fawzi Hreiki (Nov 20 2020 at 01:57):

Surely showing that $\mathscr{Y}$ is full doesn't need all the machinery of fibrations and whatnot?

sarahzrf (Nov 20 2020 at 01:58):

(of course the pullback functors dont commute with the domain projections so this certainly shouldnt give any kind of functor C^op → PSh(C) immediately, but still...)

sarahzrf (Nov 20 2020 at 02:02):

Fawzi Hreiki said:

Surely showing that $\mathscr{Y}$ is full doesn't need all the machinery of fibrations and whatnot?

well, it's yoneda, right?

Fawzi Hreiki (Nov 20 2020 at 02:03):

I'm trying to figure this out because in the appendix to Conceptual Mathematics, Lawvere states that a subcategory is $\mathscr{B}$ of $\mathscr{C}$ is adequate iff every functor $\mathscr{B}/X \rightarrow \mathscr{B}/Y$ (for $X$, $Y$ in $\mathscr{C}$) commuting with the domain projections is induced by a unique map $f: X \rightarrow Y$ in $\mathscr{C}$. That is, if the restricted slice Yoneda $\mathscr{Y}: \mathscr{C} \rightarrow \text{CAT}/\mathscr{B}$ is full and faithful.

sarahzrf (Nov 20 2020 at 02:03):

try applying the "natural transformation" to the "identity morphism"

sarahzrf (Nov 20 2020 at 02:03):

yeah that is probably the same as restricted ordinary yoneda being fully faithful, i.e., the subcategory being dense

sarahzrf (Nov 20 2020 at 02:04):

i think isbell originally called dense subcategories "left adequate" or sth

Fawzi Hreiki (Nov 20 2020 at 02:37):

Ok so my first thought is that the 'slice Yoneda lemma' should be as follows: Let $X \in \mathscr{C}$ and $F: \mathscr{A} \rightarrow \mathscr{C}$. Then $(\text{CAT}/\mathscr{C})(\mathscr{Y}(X), F) \cong F^*(X)$ where $F^*(X)$ is the pullback in $\text{CAT}$ over $\mathscr{A} \xrightarrow{F} \mathscr{C} \xleftarrow{X} 1$.

Fawzi Hreiki (Nov 20 2020 at 03:26):

In the case that $F = \mathscr{Y}(Y)$ for some $Y \in \mathscr{C}$, we get $F^*(X) = \mathscr{C}(X, Y)$ and so every functor $\mathscr{C}/X \rightarrow \mathscr{C}/Y$ preserving the domain projections is induced by a unique map $X \rightarrow Y$.

Fawzi Hreiki (Nov 20 2020 at 03:28):

This is all probably totally trivial and well known but its a shame its not to be found in any of the introductory textbooks since it sheds some light on the relationship between slice categories and representable functors.

Matteo Capucci (he/him) (Nov 20 2020 at 09:47):

Fawzi Hreiki said:

Ok so my first thought is that the 'slice Yoneda lemma' should be as follows: Let $X \in \mathscr{C}$ and $F: \mathscr{A} \rightarrow \mathscr{C}$. Then $(\text{CAT}/\mathscr{C})(\mathscr{Y}(X), F) \cong F^*(X)$ where $F^*(X)$ is the pullback in $\text{CAT}$ over $\mathscr{A} \xrightarrow{F} \mathscr{C} \xleftarrow{X} 1$.

I don't get this. The left hand of the proposed isomorphism is a set, while the right hand is...an object of $\mathscr C$?

Matteo Capucci (he/him) (Nov 20 2020 at 09:48):

This a neat idea btw, maybe not as deep as Yoneda but surely well-motivated, imo

Matteo Capucci (he/him) (Nov 20 2020 at 09:53):

If one were to follow @Nathanael Arkor 's suggestion, one would think the approach is doomed since $\mathscr Y$ is not sending $X$ to discrete fibrations afaik

Fawzi Hreiki (Nov 20 2020 at 10:11):

The right hand side is a pullback in $\text{CAT}$, not an object of $\mathscr{C}$.

Fawzi Hreiki (Nov 20 2020 at 10:13):

I don’t know anything about fibrations or discrete fibrations. Is $\mathscr{C}/X \xrightarrow{\text{dom}} \mathscr{C}$ a discrete fibration?

Matteo Capucci (he/him) (Nov 20 2020 at 10:30):

A discrete fibration is a fibration with discrete fibers, i.e. every fiber is a set, i.e. there are no non-trivial morphisms. The fiber of $\mathscr C/X \to \mathscr C$ over an object $Y$ is given by the category of morphisms $Y \to X$, which is not a set since endomorphisms $Y \to Y$ induce morphisms between morphisms $Y \to X$

Matteo Capucci (he/him) (Nov 20 2020 at 10:32):

Fawzi Hreiki said:

The right hand side is a pullback in $\text{CAT}$, not an object of $\mathscr{C}$.

Alright, I assumed the relevant part of the pullback was the arrow $1 \to \mathscr C$, hence an object of $\mathscr C$. Anyway, it's still not a set ! :/

Matteo Capucci (he/him) (Nov 20 2020 at 10:34):

It now occurs to me that probably we shall look for an isomorphism between $\mathscr C / F(X)$ and $\mathscr Y(X) / F$ or smth like that

Matteo Capucci (he/him) (Nov 20 2020 at 10:35):

In the spirit of replacing representable functors with slices

Fawzi Hreiki (Nov 20 2020 at 10:35):

Isn’t the pullback over $\mathscr{C}/Y \xrightarrow{dom} \mathscr{C} \xleftarrow{X} 1$ just the discrete category $\mathscr{C}(X, Y)$

Fawzi Hreiki (Nov 20 2020 at 10:35):

(deleted)

Matteo Capucci (he/him) (Nov 20 2020 at 10:36):

Yeah it looks like it but there's endomorphisms of $Y$

Matteo Capucci (he/him) (Nov 20 2020 at 10:38):

Matteo Capucci said:

Fawzi Hreiki said:

The right hand side is a pullback in $\text{CAT}$, not an object of $\mathscr{C}$.

Alright, I assumed the relevant part of the pullback was the arrow $1 \to \mathscr C$, hence an object of $\mathscr C$.

This is wrong haha it doesn't have to be an arrow from $1$...

Fawzi Hreiki (Nov 20 2020 at 11:09):

Right this setup seems somewhat flawed so I’ll have to go figure it out. But I do feel like there should be a slice version of the Yoneda lemma if just for the sake of filling out the analogy between $\mathscr{C}/X$ and $\mathscr{C}(-, X)$.

Fawzi Hreiki (Nov 20 2020 at 11:10):

Another motivation is the definition of adjoint functors via comma categories: https://ncatlab.org/nlab/show/adjoint+functor#in_terms_of_graphs2sided_discrete_fibrations

Fawzi Hreiki (Nov 20 2020 at 11:11):

Which again looks just like the hom-isomorphism version of adjointness (but doesn’t require any local smallness)

Fawzi Hreiki (Nov 20 2020 at 11:12):

What books/papers are good for learning more about fibrations/pseudofunctors/etc..?

Matteo Capucci (he/him) (Nov 20 2020 at 14:37):

I think I've got it! :party_ball:
The idea is to notice that the usual Yoneda lemma is sending $X$ to the functor assigning to every $A$ it set of generalized elements, while the fiber of $\mathscr C/X \to \mathscr C$ over $A$ has more information, as noted above, namely it is a category.
Therefore instead of looking at the functor $\mathscr C(-, X) : \mathscr C^{op} \to \mathbf{Set}$, we should look at the functor $-/X : \mathscr C^{op} \to \mathbf{Cat}$, where I'm abusing the slice notation: now $A/X$, where $A \in \mathscr C$, is the full subcategory of $\mathscr C/X$ determined by those arrows with source $A$. Equivalently, $A/X$ is the fiber over $A$ of the domain fibration.
Then the Yoneda embedding is into the category of pseudofunctors $[\mathscr C^{op}, \mathbf{Cat}]$ (which is a bicategory whose 1-morphisms are pseudonatural transformations and whose 2-cells are modifications), and Yoneda lemma says $\mathscr C^{op}, \mathbf{Cat} \simeq F(X)$ where notice we replace 'natural isomorphism' with 'natural equivalence', since this is the appropriate notion of 'equality' at this level.
The above works since it's a special case of the bicategorical Yoneda lemma (see Yau and Johnson's book referenced there for a detailed account)! Indeed, every (1-)category is a bicategory with trivial 2-morphisms. Nevertheless, as you noticed, in doing so we resume a bunch of 1-categorical structure that's lost in the 1-Yoneda lemma.
Finally, a note on the fibrational interpretation: as someone noted above, presheaves over a category $\mathscr C$ are the same thing as discrete fibrations, i.e. fibrations over $\mathscr C$ such that its fibers are sets. This is however an unnatural way to describe the usual Yoneda embedding, which is very elegantly defined as sending $X$ to its representable functor, which is a functor $\mathscr C^{op} \to \mathbf{Set}$.
In the new setting though, the fibrational interpretation of $[\mathscr C^{op}, \mathbf{Cat}]$ is actually more natural. The Grothendieck construction says this category is equivalent to $\mathbf{Fib}(\mathscr C)$, the category of fibrations over $C$. These fibrations do not have any restriction on the nature of the fiber.
I said this is more natural since $A/X$, as noted above, is desperate to be defined as the fiber of $\mathscr C/X \overset{dom}\to X$, so the image of $X$ under the augmented Yoneda embedding is more readily defined as the domain fibration of the slice over $X$.

Matteo Capucci (he/him) (Nov 20 2020 at 14:39):

@Fawzi Hreiki thanks for making me think about this!

Matteo Capucci (he/him) (Nov 20 2020 at 14:41):

Fawzi Hreiki said:

What books/papers are good for learning more about fibrations/pseudofunctors/etc..?

Mmmh the only centralized place that comes to mind is the nLab, but I don't know if it's actually good. Personally I picked them up (enough to being comfortable talking about them) by repeated exposure in many different contexts so now it's hard to pick a specific one

Fawzi Hreiki (Nov 20 2020 at 14:46):

Ok nice. I definitely like this version of it

Fawzi Hreiki (Nov 20 2020 at 14:47):

I’ll have to go and digest this a bit but it seems to work and quite elegantly too

Fawzi Hreiki (Nov 20 2020 at 14:48):

Then in the case of local smallness, everything ‘descends’ down to the category of sets giving the usual Yoneda lemma, which in practice is easier to work with

Fawzi Hreiki (Nov 20 2020 at 14:49):

Maybe this is blasphemous to say but somehow this version, even if a bit more involved, seems to me to be more ‘objective’ than the usual Yoneda lemma since it doesn’t presume any local smallness

Matteo Capucci (he/him) (Nov 20 2020 at 15:21):

Well, I'm not 100% sure there's no smallness requirement around, I've breen pretty handwavy

Fawzi Hreiki (Nov 20 2020 at 15:23):

Well it depends what you mean by ‘smallness’. The smallness requirement is that $\mathscr{C}$ lives in some category of categories $\text{CAT}$.

Matteo Capucci (he/him) (Nov 20 2020 at 15:23):

For example, if $\mathscr C$ were not locally small then $A/X$ wouldn't be a small category

Matteo Capucci (he/him) (Nov 20 2020 at 15:24):

Hence you should replaces are small cats with big cats in my long message above

Fawzi Hreiki (Nov 20 2020 at 15:25):

So it just depends on what foundation you take

Matteo Capucci (he/him) (Nov 20 2020 at 15:25):

Fawzi Hreiki said:

Well it depends what you mean by ‘smallness’. The smallness requirement is that $\mathscr{C}$ lives in some category of categories $\text{CAT}$.

Yes, I mean smallness somewhere in $\mathscr C$ (objects or morphisms)

Fawzi Hreiki (Nov 20 2020 at 15:26):

It’d maybe be interesting to replace $\text{CAT}$ with some sufficiently nice 2-category

Matteo Capucci (he/him) (Nov 20 2020 at 15:26):

Fawzi Hreiki said:

So it just depends on what foundation you take

In a sense, but so does local smallness in the usual Yoneda lemma.

Nathanael Arkor (Nov 20 2020 at 15:31):

Fawzi Hreiki said:

It’d maybe be interesting to replace $\text{CAT}$ with some sufficiently nice 2-category

Carrying out Yoneda arguments in general 2-categories is the motivation behind Yoneda structures, which you might find interesting.