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2d
revised Characterize the category of rings
added 4 characters in body
2d
answered Characterize the category of rings
Oct
21
comment Higher vector spaces
Well, there is a trivial observation: an idempotent-complete category $\mathcal{C}$ is (equivalent to) the category of finitely presented $R$-modules if and only if $\mathbf{Ind}(\mathcal{C})$ is (equivalent to) the category of $R$-modules.
Oct
15
revised $(LLP(Epi), Epi)$ is a WFS on any variety of algebras
added 1 character in body
Oct
15
answered $(LLP(Epi), Epi)$ is a WFS on any variety of algebras
Oct
15
comment $(LLP(Epi), Epi)$ is a WFS on any variety of algebras
$U$ should be only faithful, not fully faithful.
Oct
15
comment Cocomplete but not complete abelian category
This looks like the direct limit of an $\mathbf{On}$-indexed sequence of cocomplete abelian categories along colimit-preserving exact functors, so it should be just abstract nonsense that it is a cocomplete abelian category.
Oct
13
comment Homotopy uniqueness of some coends
Essential uniqueness here is referring to the fact that some classifying space is contractible. I wouldn't read too much into it.
Oct
13
comment Homotopy uniqueness of some coends
But the diagrams in question are indeed cofibrant?
Oct
11
comment Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?
This seems to be a folklore result. It is alluded to in [Higher topos theory, §6.5.3].
Oct
9
comment The Karoubi model structure on Cat
Yes, it is a left Bousfield localisation of the canonical model structure, almost by definition. The generating cofibrations are therefore the same as for the canonical model structure.
Oct
8
comment The Karoubi model structure on Cat
I describe the model structure here. There is just one extra generating trivial cofibration, namely the inclusion of the free idempotent into the free split idempotent.
Oct
2
comment When is the product of an infinite family of simplicial sets also a homotopy product?
Here's a simple observation: let us say that a simplicial set $X$ has property $C_n$ if two vertices are in the same connected component of $X$ if and only if there is a path of length $\le n$ connecting them. So for example Kan complexes have property $C_1$. Then for every positive integer $n$, $\pi_0$ preserves products of families of simplicial sets having property $C_n$. This seems to suggest that it suffices to satisfy the Kan condition up to some finite subdivision...
Sep
19
awarded  Enlightened
Sep
18
comment Commutation of simplicial homotopy colimits and homotopy products in spaces
What exactly are you asking, though? If you're asking whether the functor $\prod : \mathcal{S}^{I} \to \mathcal{S}$ preserves homotopy colimits, then the answer is no even for $I = 2$: after all, $(1 + 1) \times (1 + 1) \ne (1 \times 1) + (1 \times 1)$. But if you're asking whether it preserves homotopy colimits in each variable, then the infinite case reduces to the finite case and there's no problem.
Sep
10
answered Categories of spans from categories of fibrant objects
Sep
3
comment Is there an analog of the Barratt-Eccles construction for group-like E_∞-spaces and E_∞-ring spaces?
@QiaochuYuan It seems to me that there is a lot of interest in enriched algebraic theories coming from the theoretical computer scientists.
Sep
2
accepted Aspheric functors and Grothendieck fibrations
Aug
31
asked Aspheric functors and Grothendieck fibrations
Aug
30
comment Is there a general notion of semigroup action?
@QiaochuYuan It's a nice trick for $\mathbf{Set}$ but it doesn't work properly in general: you would need every subobject to have a complement in the strong sense.