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Stream: learning: questions

Topic: colimits and connected components

Bruno Gavranović (Aug 15 2022 at 20:34):

I've learned something fascinating.

Let $F : \mathcal C \to \mathbf{Set}$ be a functor. Then the following holds

$$\textsf{colim} F \cong \pi_0(\textsf{El}(F)) \cong \textsf{colim}_{\textsf{oplax}} (F;\textsf{discr})$$

where $\textsf{El}$ is the category of elements of $F$, and $\pi_0 \dashv \textsf{discr}$ is the adjunction between the connected components functor and the discrete functor between $\mathbf{Cat}$ and $\mathbf{Set}$.

This tells us a few things:
1) A colimit of every $\mathsf{Set}$-valued functor -- which is a set -- arises in a canonical way from a category! This is surprising. I always thought of $\textsf{colim}$ as a functor $[\mathcal C, \mathbf{Set}] \to \mathbf{Set}$ and had no idea it can be factored through $\pi_0 : \mathbf{Cat} \to \mathbf{Set}$. In other words, the set given by the colimit is really a quotiented out version of something more intricate.
2) The category that it arises out of can be computed as the oplax colimit of our starting functor $F$ postcomposed with $\textsf{discr} : \mathbf{Set} \to \mathbf{Cat}$.

Does anyone know more about this? I really like this formulation of colimits as the connected componets of the category of elements, but I've never seen it before until now. I also never thought of the connected components functor as a mediator between (1-)colimits and oplax colimits. It feels surprising that the left adjoint of $\textsf{discr}$ appears here, as opposed to the right adjoint (this is what I'd have guessed, that the forgetful functor would play a role).

Paolo Perrone (Aug 15 2022 at 20:40):

This is related to some of the things I've written up in Section 3 of my Kan extensions paper. As far as I know it dates back to the work of Robert Paré, see the references.

Jonathan Weinberger (Aug 15 2022 at 21:12):

Bruno Gavranovic said:

$\pi_0 \vdash \textsf{discr}$ is the adjunction between the connected components functor and the discrete functor between $\mathbf{Cat}$ and $\mathbf{Set}$.

The adjunction should read $\pi_0 \dashv \textsf{discr}$.

Bruno Gavranović (Aug 15 2022 at 21:16):

Jonathan Weinberger said:

Bruno Gavranovic said:

$\pi_0 \vdash \textsf{discr}$ is the adjunction between the connected components functor and the discrete functor between $\mathbf{Cat}$ and $\mathbf{Set}$.

The adjunction should read $\pi_0 \dashv \textsf{discr}$.

Thanks, fixed.

Bruno Gavranović (Aug 16 2022 at 16:38):

Thanks @Paolo Perrone. During reading of your paper my question slowly crystalized.

Basically, I've noticed a pattern. I noticed people gave a name to the colimit of a particular functor $F : \mathcal C \to \mathbf{Set}$, and used it in various contexts. But then I noticed this colimit tends to clump a lot of things together - things I want to keep track of separately. I've learned that I can add more fidelity to this construction by not computing the colimit, but instead doing the (op)lax version thereof, after suitably postcomposing with $\textsf{discr} : \mathbf{Set} \to \mathbf{Cat}$.

My question is: this this a general recipe? Whenever I have a set, obtained in some universal way, but I want to not quotient things out, is the right way to compute the (op)lax version of that universal construction?

I suppose I'm learning a (in hindsight perhaps obvious) fact that given any universal construction in any context, somebody might be interested in a refinement of that construction by adding higher cells- and really, there's a straightforward way to refine it. It's by realizing that this construction is the result of applying the connected components functor to its refinement.

Bruno Gavranović (Aug 16 2022 at 16:43):

I'm actually not sure how clear this is. Maybe I'm just realising that all the higher-category theory is sort of always there, and that there's a canonical way to refine constructions using the $\pi_0 \dashv \textsf{discr}$ adjunction.

Bruno Gavranović (Aug 16 2022 at 16:44):

This works at least for colimits, but I'm expecting it holds in more generality.

Paolo Perrone (Aug 16 2022 at 19:29):

Yes, this idea seems correct. A lot of times traditional quotients "identify too much", and one loses track of some important information. I think the first people to realize this were algebraic topologists, who came up with the notion of homotopy quotient precisely for that purpose.
In general, one can have different formalizations of the same idea, which is "replacing identifications with arrows of some kind".
Other examples of the same idea, where the arrows are "invertible" (more like a pseudo colimit than lax) are the action groupoid, the idea of resolution, and cofibrant replacements.
By the way, this came up recently also in the work of Sean and me on Markov categories and dynamical systems (see the appendix here).

Bruno Gavranović (Aug 17 2022 at 10:42):

Ah, amazing. That's great to hear -- that it's not uncommon to try to add more fidelity to universal constructions in this way.