Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Complete wellpowered co-wellpowered implies coequalizers

Bert Lindenhovius (Aug 12 2020 at 14:39):

Dear all, I hope I may ask here a question where to find a reference for the following statement:

Any complete, wellpowered and co-wellpowered category has all coequalizers.

I found this statement as Proposition 154 in the book Subtlety in Relativity by Sanjay Moreshwar Wagh, unfortunately without a proof. I hope someone knows a different reference with a (hopefully accesible) proof. Or maybe if things are not too complicated an indication how to proof it. Thanks!

Edit: My coworker Andre Kornell found the statement in https://www.sciencedirect.com/science/article/pii/S0924650908701600
Nevertheless we are quite surprised that this result does not seem to appear in other sources. If anyone knows another source, that would still be welcome.

fosco (Aug 18 2020 at 07:49):

The slickest proof I can think of is that all the assumptions yield the solution set condition for a left adjoint to $\mathcal{A} \to \mathcal{A}^\rightrightarrows$

fosco (Aug 18 2020 at 07:52):

Call $\Delta_\rightrightarrows$ the functor above; then you can compute its left adjoint $\text{coeq}$ provided the solution set condition holds, as the right Kan extension of the identity along $\Delta_\rightrightarrows$ (it doesn't really matter if you don't know what a right Kan extension is: it's just a certain limit, that $\mathcal{A}$ now must possess because of the completeness assumption)

fosco (Aug 18 2020 at 07:54):

I'd go as follows:

{cowell,well}-powered => solution set condition holds
complete => the relevant limit exists

let me know if by any chance it works!

সায়ন্তন রায় (Sep 08 2020 at 12:41):

Bert Lindenhovius said:

Dear all, I hope I may ask here a question where to find a reference for the following statement:

Any complete, wellpowered and co-wellpowered category has all coequalizers.

I found this statement as Proposition 154 in the book Subtlety in Relativity by Sanjay Moreshwar Wagh, unfortunately without a proof. I hope someone knows a different reference with a (hopefully accesible) proof. Or maybe if things are not too complicated an indication how to proof it. Thanks!

Edit: My coworker Andre Kornell found the statement in https://www.sciencedirect.com/science/article/pii/S0924650908701600
Nevertheless we are quite surprised that this result does not seem to appear in other sources. If anyone knows another source, that would still be welcome.

This follows from Exercise 12J of Joy of Cats. I haven't yet gone that far, but I thought that it would be useful to point out a reference for the result.