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: theory: category theory

Topic: universal property of images


view this post on Zulip Morgan Rogers (he/him) (Nov 14 2023 at 15:48):

It depends on additional hypotheses on the factorization system (see for example "algebraic factorization system"). Often mm or ee will be universal in some way (so ii might be a kernel/cokernel, say).

view this post on Zulip Mike Shulman (Nov 14 2023 at 16:58):

It's always a reflection from the slice category C/c\mathcal{C}/c' into the subcategory M/c\mathcal{M}/c'...

view this post on Zulip Mike Shulman (Nov 14 2023 at 16:59):

(Assuming by "factorization system" you mean an "orthogonal/unique" factorization system.)

view this post on Zulip fosco (Nov 14 2023 at 20:02):

Sometimes ii is a pullback, and ee arises as the induced map into it; google "simple factorization system", and in particular a paper from Cassidy, Hebert and Kelly. Or read my PhD thesis, which is an account of torsion theories (particularly well-behaved FS, covered in non-abelian setting also in Borceux, handbook tome 1) in triangulated/stable-homotopic setting.

view this post on Zulip fosco (Nov 14 2023 at 20:04):

As Morgan says, AFS are another source of factorization obtained from universal properties; Garner's paper is the best reading.

In general, very little can be said, without knowing what are the specifics of the FS you have to handle. For example, is it right proper (=every mm is a monic)? Extremal-right proper (=every mm is an extremal monic)?

view this post on Zulip Matteo Capucci (he/him) (Nov 15 2023 at 06:53):

Mike Shulman said:

(Assuming by "factorization system" you mean an "orthogonal/unique" factorization system.)

Yeah I do!

view this post on Zulip Matteo Capucci (he/him) (Nov 15 2023 at 06:54):

Mike Shulman said:

It's always a reflection from the slice category $\mathcal{C}/c'$ into the subcategory $\mathcal{M}/c'$...

This sounds like what I need! Thanks Mike