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Stream: learning: questions

Topic: Two (equivalent?) ways of computing an end as an equalizer

Marco Vianello (Feb 27 2026 at 18:28):

Let $\mathcal C$, $\mathcal D$ be categories and let $P\colon \mathcal C^{\mathrm{op}}\times \mathcal C\to \mathcal D$ be a functor. Then

1. the end $\int_{c\in \mathcal C} P(c,c)$ of $P$ fits in an equalizer diagram

$$\int_{c\in \mathcal C} P(c,c) \hookrightarrow \prod_c P(c,c)\rightrightarrows \prod_{c\to c^\prime} P(c,c^\prime)$$

2. the same end can be characterized as the limit of the functor $\hat P\colon \mathrm{Tw}(\mathrm C)\to \mathcal D$ induced by $P$ on the twisted arrow category of $\mathcal C$, and as such it fits in an equalizer diagram

$$\int_{c\in \mathcal C} P(c,c) \hookrightarrow \prod_f \hat P(f)\rightrightarrows \prod_{f\rightsquigarrow g} \hat P(g)$$

where the first product in indexed by the objects of $\mathrm{Tw}(\mathcal C)$ and the second by the morphisms.

Two questions:

1. Are the two "$\prod \rightrightarrows \prod$" diagrams isomorphic?
2. Where does the first "$\prod \rightrightarrows \prod$" diagram come from? Is there really a better answer than the tautological fact that one find $\int_c P(c,c)$ by equalizing it?

Kevin Carlson (Feb 27 2026 at 18:51):

Well, the first diagram comes from the definition of an end as the universal extranatural transformation into $P.$ Is that more satisfying than the tautological explanation?

fosco (Feb 27 2026 at 19:43):

I had a conversation with Marco, about this problem, the other day, and I am still unsure what exactly is the explanation he is looking for, but the way I understand it is that 1. is the expression of an end as limit over the subdivision category of $\cal C$, in the way it splits as an equalizer of a map between products

fosco (Feb 27 2026 at 19:45):

This opens the way to other questions though :smile: why would an end be a limit over the subdivision category (and what is it exactly, and how does it relate exactly with wedges and with Tw(C).. see here), and ultimately why a limit is an equalizer of maps between products, but this is easier to justify.
Unfortunately, I realize, now that I'm a bit older, that I didn't understand at the level I'd like to some of these aspects, when I was writing!

fosco (Feb 27 2026 at 19:57):

At the heart of the "subdivision category" construction is the fact that one can define an endofunctor of $\Delta$ (the category of simplices) $\epsilon : \Delta \to \Delta$ sending $[n]$ to $[n] \star [n]^op$ (the join of [n] with its opposite). See here: https://arxiv.org/pdf/2008.03758

Marco Vianello (Feb 27 2026 at 20:32):

Well, I am asking how does one conconct that parallel-arrows diagram given only the definition of an end as a universal dinatural transformation. I can unwind the definitions and realize they are the same, but I was asking if something (?) was lurking behind.

I mean, when you write the limit of a functor $X\colon \mathcal I\to \mathcal X$ as the equalizer of $\prod_i X_i\rightrightarrows \prod_{i_1\to i_2} X_{i_2}$ you bring with you the intuition on $\mathrm{Sets}$-valued limits, right (* fear *)?

Kevin Carlson said:

Is that more satisfying than the tautological explanation?

Isn't this exactly the tautological explanation?

fosco said:

the way I understand it is that 1. is the expression of an end as limit over the subdivision category of $\cal C$, in the way it splits as an equalizer of a map between products

That's an answer I am more willing to accept. At least it explains the form of diagram 1.

Kevin Carlson (Feb 27 2026 at 23:25):

I would say one concocts that parallel-arrows diagram by simply writing down the definition of an dinatural transformation.

Kevin Carlson (Feb 27 2026 at 23:26):

Marco Vianello said:

I mean, when you write the limit of a functor $X\colon \mathcal I\to \mathcal X$ as the equalizer of $\prod_i X_i\rightrightarrows \prod_{i_1\to i_2} X_{i_2}$ you bring with you the intuition on $\mathrm{Sets}$-valued limits, right (* fear *)?

I don't think so? I think a map into this equalizer is just a cone, by definition: a map into each $X_i$ such that [various triangles commute].

fosco (Feb 28 2026 at 10:13):

@Marco Vianello Mac Lane puts it in a very simple way: "an end is manifestly a limit", meaning that whenever you have a family $\{Fc\mid c\in C\}$ indexed over the objects of a category, subject to certain commutativities $\{LHS_i=RHS_i\mid i\in I\}$ with respect to the arrows of said category, you are saying that you want to characterize a subobject of $\prod Fc$, such that the inclusion equalizes two maps, one defined in terms of LHS, one defined in terms of RHS. This is kinda obvious, and indeed, I suspect understanding how limits can be constructed out of products and equalizers only must be such an old realization that I cannot pinpoint who was the first to observe it.

fosco (Feb 28 2026 at 10:18):

So, the explanation of why an end is an equalizer feels tautological because it is tautological that subjecting things to equations means equalizing two things.

It is more or less the intuition that a limit is a locus of points making something true, and a colimit is a quotient obtained by forcing something to be true. They are dual concepts because all the way down in pre-mathematical intuition, finding what satisfies a property [inside a universe of discourse] is dual to imposing the same property on a system [and then count how many elements you are still able to tell apart].

I can see how you might feel with this mumbo-jumbo I am not dispelling the tautology, if anything, I am forcing you to embrace it without fear...

fosco (Feb 28 2026 at 10:35):

Again, Mac Lane observes that "a cone for the functor $P^§ : C^§ \to D$ induced by $P : C^o\times C \to D$ is a wedge for $P$ and vice versa", and I haven't been more wordy about this; at the time it seemed obvious, and sufficient to explain the relation

fosco (Feb 28 2026 at 10:38):

Your question is "yeah, I can see that, but what does this fact mean in terms of the relation between the limit over the subdivision category and the limit over the twisted arrow category?"

I have to admit I didn't do draw this connection at the time; I also believe, in the end, the fact that the categories of cones for $P^§$ and wedges for $P$, (or cones for $P^\tau : Tw(C) \to D$) are all isomorphic will mean that there is an isomorphism of diagrams of type 1, and 2.

Matteo Capucci (he/him) (Mar 02 2026 at 08:11):

fosco said:

the subdivision category of $\cal C$

what is this category? :thinking:

Matteo Capucci (he/him) (Mar 02 2026 at 08:17):

Marco Vianello said:

1. Are the two "$\prod \rightrightarrows \prod$ " diagrams isomorphic?

Well this one is easy to answer, they don't need to! You can have different diagrams inducing the same limit. In fact if $F:J \to D$ is a diagram, and you precompose it by an [[initial functor]] $G:I \to J$, then the limit of $FG$ will be the same as that of $F$, as it is the case here. Note this is a different kind of 'initial' than 'inital object'. So the above fact means the projection $tw(C) \to C^{op} \times C$ is initial.