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Jonathan Beardsley's user avatar
Jonathan Beardsley's user avatar
Jonathan Beardsley's user avatar
Jonathan Beardsley
  • Member for 14 years
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When is Fun(X,C) comonadic over C with respect to the colimit functor?
@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.
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When is Fun(X,C) comonadic over C with respect to the colimit functor?
@AlexanderCampbell I thought this was written somewhere, but it is perhaps just gossip... (oops...)
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Explicit left adjoint to forgetful functor from Cartesian to symmetric monoidal categories
@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).
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Explicit left adjoint to forgetful functor from Cartesian to symmetric monoidal categories
@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.
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Explicit left adjoint to forgetful functor from Cartesian to symmetric monoidal categories
@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.
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Explicit left adjoint to forgetful functor from Cartesian to symmetric monoidal categories
@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?
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Explicit left adjoint to forgetful functor from Cartesian to symmetric monoidal categories
@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.
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Non-commutative Formal Group Laws
@zzy unfortunately I have not.
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