Zulip Chat Archive

Stream: condensed mathematics

Topic: exact functor

Johan Commelin (Apr 13 2022 at 14:01):

Do we have exact functors somewhere?
cc @Adam Topaz

Adam Topaz (Apr 13 2022 at 14:03):

additive + preserves finite colimits + prserves finite limits

But we don't have the fact that it will preserve exact_seqs (although @Riccardo Brasca and I did half of that some time ago for the 0th derived functor iso)

Matthew Ballard (Apr 13 2022 at 14:05):

I don’t think there is distinguished triangles to distinguished triangles either

Riccardo Brasca (Apr 13 2022 at 14:06):

We have docs#category_theory.abelian.functor.preserves_exact_of_preserves_finite_colimits_of_epi and I think also the dual statement, so it's just a matter of putting things together.

Matthew Ballard (Apr 13 2022 at 14:07):

Exact also means “triangulated”

Adam Topaz (Apr 13 2022 at 14:07):

Matthew Ballard said:

Exact also means “triangulated”

Ah ok, yeah, we definitely don't have that, although I do think we have triangulated functors?

Matthew Ballard (Apr 13 2022 at 14:10):

Sorry, exact functor is a synonym for additive functor between triangulated categories that commutes with shifts and sends distinguished triangles to distinguished triangles

Matthew Ballard (Apr 13 2022 at 14:11):

Last I checked I didn’t find this

Johan Commelin (Apr 13 2022 at 14:14):

But it also makes sense between non-triangulated cats, right

Matthew Ballard (Apr 13 2022 at 14:15):

So any additive functor between homotopy categories is “exact” but descends to an exact functor in derived categories if it takes acyclic complexes to acyclic complexes