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.
Morgan Rogers (he/him) (Apr 05 2020 at 09:46):I've seen a few talks about abelian categories at conferences etc. How are they classified? Like what extra properties do abelian categories built in different ways have? Is this something that people study?
Sam Tenka (Apr 05 2020 at 16:39):You probably know this, but every abelian category embeds (surjectively in morphisms) into a category of Rmodules. Perhaps that's a start?
Morgan Rogers (he/him) (Apr 05 2020 at 18:33):The Freyd-Mitchell embedding theorem, yes! It's interesting to compare this with toposes, actually. The definition of a Grothendieck topos can roughly be stated as "a subtopos of a presheaf topos"; Giraud's theorem giving an axiomatic definition came later. For abelian categories, it's the other way around: there was an axiomatic definition and later Freyd and Mitchell showed that any such is a subcategory of a category of modules.
Morgan Rogers (he/him) (Apr 05 2020 at 18:35):But I could even restrict to that special case. It's possible to tell (by the presence of a kind of generating object) when an abelian category is a category of modules for a ring. So what else can I tell about a ring from the properties of its category of modules?
Reid Barton (Apr 05 2020 at 18:46):Well, by definition, those features which are Morita invariant
Sam Tenka (Apr 05 2020 at 19:22):Yeah, e.g. different fields are indistinguishable from this perspective. And there are probably subtler ambiguities, too, since e.g. (as shown in Mel Hochster's notes) there exist commutative rings R,S such that R[x], S[y] are isomorphic but R and S are not.
Joe Moeller (Apr 06 2020 at 01:16):Hmm, this conversation is making me think about like manifolds and sheaves. What sort of category do you get if you consider like stacks of abelian categories or stacks of topoi? :thinking:
Morgan Rogers (he/him) (Apr 06 2020 at 10:02):Reid Barton said:
Well, by definition, those features which are Morita invariant
:face_palm: Yes, the properties I'm asking about are the Morita invariant ones, but what Morita invariants have been studied and what properties of rings to they correspond to (or more generally what motivated their study)
Morgan Rogers (he/him) (Apr 06 2020 at 10:05):Sam Tenka (naive student) said:
(as shown in Mel Hochster's notes)
Could you be more specific? Their homepage has a lot of links on it and it's not clear where I should be looking.
Sam Tenka (Apr 06 2020 at 18:09):Morgan Rogers said:
Sam Tenka (naive student) said:
(as shown in Mel Hochster's notes)
Could you be more specific? Their homepage has a lot of links on it and it's not clear where I should be looking.
Yep! See http://www.math.lsa.umich.edu/~hochster/614F17/614.pdf :
Last paragraph of page 29 for introduction of the R[x] vs S[y] question,
Last paragraph of page 73 for one nice solution to the question using hairy ball theorem.
Enjoy!
Morgan Rogers (he/him) (Apr 06 2020 at 19:19):Sam Tenka (naive student) said:
Yeah, e.g. different fields are indistinguishable from this perspective. And there are probably subtler ambiguities, too, since e.g. (as shown in Mel Hochster's notes) there exist commutative rings R,S such that R[x], S[y] are isomorphic but R and S are not.
btw this is not true! I can recover a field from its category of vector spaces as the endomorphism ring of a 1-dim vector space!
Sam Tenka (Apr 06 2020 at 19:22):Morgan Rogers said:
Sam Tenka (naive student) said:
Yeah, e.g. different fields are indistinguishable from this perspective. And there are probably subtler ambiguities, too, since e.g. (as shown in Mel Hochster's notes) there exist commutative rings R,S such that R[x], S[y] are isomorphic but R and S are not.
btw this is not true! I can recover a field from its category of vector spaces as the endomorphism ring of a 1-dim vector space!
Oh! Good point! I should have noted that I am considering categories up to equivalence. For fields, each category of vector spaces is equivalent to some cardinality-indexed skeleton.
Arghh I was saying many wrong things. You're right that different fields are distinguished by the inequivalence of their module categories!