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Stream: learning: questions

Topic: Colimits in reflective subcategories

Bernd Losert (Aug 08 2024 at 20:39):

Screenshot-2024-08-08-at-22.32.37.png
This is taken from Joy of Cats. I was trying to prove this more rigorously but I'm got stuck. It also seems fishy to me because it makes use of the reflector (unit of the adjunction).
Screenshot-2024-08-08-at-22.29.37.png
This is taken from Riehl's book. The proof there makes more sense to me because it makes use of the counit, which happens to be an isomorphism.

Is the Joy of Cats version just plain wrong?

Kevin Carlson (Aug 08 2024 at 20:42):

Riehl's version is a pretty clear corollary of the Joy of Cats one, so for the latter to be wrong you'd have to claim that Riehl needs to use the existence of reflectors for more objects than just the colimit. But I'm pretty sure she doesn't. (What other object do you think she needs?)

Kevin Carlson (Aug 08 2024 at 20:44):

Anyway, the existence of the reflector $r$ in Joy of Cats means precisely that composition with $r$ induces an isomorphism $\mathbf{A}(A,B)\cong \mathbf{B}(C,E(B))$ natural in $B.$ By definition the latter is naturally isomorphic to the set of cocones $E\circ D\to E(B)$ (in $\mathbf{B}$) which by full faithfulness of $E$ is naturally isomorphic to the set of cocones $D\to B$ (in $\mathbf{A}.$) Composing these three isomorphisms looks like the whole proof to me.

Bernd Losert (Aug 08 2024 at 20:47):

I'm not seeing it. In my mind, the Joy of Cats version is not correct because you have (r ∘ cᵢ : EDi → EFC). If this thing is a colimit, it is a colimit of E ∘ D, not of D.

Bernd Losert (Aug 08 2024 at 20:50):

So by fullness, (r ∘ cᵢ : EDi → EFC) translates to a cocone (gᵢ : Di → FC). I was trying to show that this cocone is a colimit, but I get stuck.

Bernd Losert (Aug 08 2024 at 20:54):

If both Joy of Cats and Riehl are correct, then that means there are two ways to go about finding colimits? One using the unit and one using the counit?

Kevin Carlson (Aug 08 2024 at 21:05):

What is $F$?

Kevin Carlson (Aug 08 2024 at 21:07):

And you haven't included the proofs so I can't comment on the question about unit versus counit, but I doubt there are two really distinct proofs here.

Bernd Losert (Aug 08 2024 at 21:09):

Here's my proof attempt at the Joy of Cats proposition: Let (ε, η) : F ⊣ E be the adjunction, where F is the reflector (left adjoint) of the embedding E. You have a colimit (cᵢ : EDi → C) of E ∘ D. Then compostion with η_C : C → EFC, you get (η_C ∘ cᵢ : EDi → EFC). Since E is full, there exists a family (gᵢ : Di → FC) such that (Egᵢ) = (η_C ∘ cᵢ). That (gᵢ : Di → FC) is a cocone of D can be shown using the faithfulness of E. Now, suppose there is another cocone (fᵢ : Di → A) of D. Then (Efᵢ : EDi → EA) is a cocone of E ∘ D. Since (cᵢ : EDi → C) is a colimit, there is a unique h : C → EA such that (Efᵢ) = (h ∘ cᵢ). At this point I get stuck, but I know I need to use the reflector F somehow.

Kevin Carlson (Aug 08 2024 at 21:18):

There's no $F$ assumed in this statement!

Bernd Losert (Aug 08 2024 at 21:19):

True. I'm assuming the subcategory is reflective to begin with.

Kevin Carlson (Aug 08 2024 at 21:19):

But if we pretend there is, then just use the universality of $\eta_C,$ which is by definition the initial map from $C$ to $E$ of something.

Kevin Carlson (Aug 08 2024 at 21:20):

Or in other words, apply the adjunction isomorphism to transpose $h$ into a map $FC\to A.$

Kevin Carlson (Aug 08 2024 at 21:20):

(You should try to let people know when you're introducing notation and assumptions not in the statement you asked about!)

Kevin Carlson (Aug 08 2024 at 21:21):

For what it's worth I find all this fiddling with explicit components of cocones confusing and think a proof like the one I sketched might be easier to understand. But maybe you're not very happy that kind of argument at this point.

Bernd Losert (Aug 08 2024 at 21:22):

Indeed. That was sloppy of me. It did not occur to me to take the transpose of h. Good tip.

Bernd Losert (Aug 08 2024 at 21:23):

But maybe you're not very happy that kind of argument at this point.

Yeah, my sophistication hasn't reached the point where I can just see it without looking at what's going on with the components.

Bernd Losert (Aug 08 2024 at 21:34):

Thanks a lot though. Appreciate it.