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Brendan Murphy (Mar 29 2024 at 02:55):In Characterizations of Abstract Stable Homotopy Theory by Moritz and Shulman they give the following equivalent conditions for a derivator (eg an infinity category) to be stable be7b0de7-bb91-4ef6-9955-c01872866648.png
The particular thing I'm interested in here is condition (iii), that stability is equivalent to finite limits and colimits commuting. Is there a similar characterization of abelian categories?
Eigil Rischel (Apr 12 2024 at 13:13):Abelian categories have finite limits and colimits. The condition of having biproducts - that the products and coproducts coincide, which is called either additivity or preadditivity, depending on the source - is equivalent to the condition that products commute with colimits and coproducts commute with limits.
The further condition of being abelian is that every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel. This amounts to a certain compatibility between pushouts and pullbacks - that in a square of this form:
if is a monomorphism and the square is a pushout, then it's a pullback, and dually if is an epimorphism, it's a pushout if it's a pullback. This is also some sort of compatibility condition between finite limits and colimits (if it was true for all squares that they were pushouts if and only if pullbacks, then finite limits and colimits would commute), but it's obviously a bit more complicated than the -categorical case.