Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
@SamGunningham can you say a bit more? By $Vect$ do you mean something like chain complexes on a field? Otherwise I'm not sure that's stable. And if you do mean something like that, can you say why you think it's the stability that's getting in the way here, and not something else? I'd rather like my question to have an affirmative answer in the case of spectra.
@MikeShulman ah yeah, right, I was actually slightly confused about the relationship between this tensor product and the cocartesian one (which is how I'm used to thinking about the tensor product of commutative algebras).
@MikeShulman ah sure, but then does this tell us explicitly what the monoidal structure on symmetric monoidal presentable categories is going to be, as described in Lemma 2.4 on the next page? It looks to me like they only agree on free algebras.
@MikeShulman I suppose I was just going off of the fact that "presentability" (by which I mean local presentability, since the term has been sadly mucked up) is one of the requirements on the categories addressed in the Berman paper cited in the question.
@NoahSnyder Ah I think maybe I misread your comment. Yeah, I agree that it's tensor with $Fin^{op}$, but what is that tensor? I don't think it's just the usual tensor of categories is it? It's some complicated tensor product of $Fin^{iso}$-modules?
@NoahSnyder maybe so! I guess I don't know what induction would be here. To me, I'd think it'd be something like $C\otimes_{Fin^{iso}}Fin^{op}$? But I'm not sure how this kind of thing works.