@ArunDebray i decided to post them on my webpage: mit.edu/~sanathd/goodwillie.pdf, although a link will probably also be provided on the juvitop website
Does it hold in any abelian category that filtered colimits commute with finite limits? I think it should be true since it holds for modules and thus, one should be able to apply say Mitchell's embedding theorem. But maybe I am being silly and am making some mistake
This is not true, a (presentable) abelian category where filtered colimits commute with finite limits is called a Grothendieck abelian category
And it's not known not every abelian category has this property, the reason Mitchell's embedding argument fails is that the embedding constructed is exact but does not need to preserve filtered colimits
An explicit example would be $Ab^{op}$, the opposite category of abelian groups. It is again abelian, but filtered colimits in $Ab^{op}$ correspond to cofiltered limits in $Ab$ which are known not to be exact
@AdrianClough i see, sounds interesting. yeah, i haven't ever thought much about particular enriched ($\infty$-)categories but i feel like the "internal category theory" might be very different when the enrichment isn't cartesian. for instance, i'm imagining that replacing the product (a limit) of hom-objects with a more general monoidal product might make universal constructions behave much more subtly. i don't actually know, but i'd bet somebody else here does...
