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bio website math.northwestern.edu/~hoyois
location Chicago, IL, USA
age 27
visits member for 2 years, 3 months
seen Feb 7 at 17:35
I'm a graduate student at Northwestern University.

Jan
21
awarded  Enlightened
Jan
21
awarded  Nice Answer
Dec
28
awarded  Yearling
Dec
4
comment Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology
The point of a category of fibrant objects is that you can compute a mapping space $Map(A,B)$ by resolving only the source $A$ and not the target $B$ (this is explained on the nLab page). Verdier's hypercovering is an instance of this for $B=K^{pre}(A,n)$. To compute $Map(A,B)$ using a model structure on $sPre(C)$ you need to replace $B$ by a fibrant object (for $B=K^{pre}(A,n)$ this means taking an injective resolution of $A$). So the hypercovering theorem is not a simple application of model category/∞-category techniques.
Dec
2
comment Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology
Actually if $X$ is a fibrant simplicial set, the map $X\to cosk_nX$ in $Ho(sSet)$ is the initial map to a simplicial set with no homotopy groups in degrees $>n$. So $cosk_n$ is left adjoint to the inclusion of that subcategory.
Dec
1
revised Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology
added 717 characters in body
Dec
1
comment Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology
Your $\pi_n F(X)$ is going to be $Hom(\pi_0(X),\mathbb{Z})$ in degree $0$ and trivial in positive degrees, so that's not the right way to define cohomology. To compute $H^n(C,A)$, where $C$ is a site with final object $X$ and $A$ is a sheaf of abelian group on $X$, you form the Eilenberg-Mac Lane simplicial presheaf $K^{pre}(A,n)$ and choose a fibrant replacement $K(A,n)$ (i.e. sheafify). Then $H^n(C,A)=\pi_0(K(A,n)(X))$.
Nov
30
answered Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology
Nov
15
comment On triangulated categories of pro-objects
Prop. 7.1.12 in Hovey's book says that the derived functor of a Quillen functor between stable model categories is always exact (for the induced triangulated structures).
Nov
15
comment Generalization of the Lefschetz fixed point theorem
@Qiaochu: Sorry for taking so long to reply; I don't log in often. The LHS is of the same nature as the RHS: the étale Euler characteristic is the trace of the identity in the derived category of étale sheaves over the base field. This is invariant under base change, so over $\mathbb{R}$ it equals the topological Euler characteristic of the $\mathbb{C}$-points. To recover the Euler characteristic of the $\mathbb{R}$-points you need something finer than étale cohomology, e.g. $C_2$-equivariant étale cohomology (traces then live in the Burnside ring of $C_2$).
Oct
19
answered Category of motivic spectra
Oct
16
comment Generalization of the Lefschetz fixed point theorem
@Qiaochu: Yes, étale cohomology with coefficients in $\mathbb{Q}_\ell$. Over an arbitrary field $k$, the RHS is replaced by a trace in the derived category of étale $\mathbb{Q}_\ell$-sheaves over $Spec(k)$. But both sides of the equation are integers and hence are invariant under change of base fields, so you can always make a change of base to an algebraic closure first.
Sep
28
comment Loops and suspensions of higher categories
Oh I see: putting $C'$ instead of $C$ in the lower right corner simply doesn't change the universal property of the pullback in the $(\infty,1)$-category of $(\infty,n)$-categories.
Sep
27
comment Loops and suspensions of higher categories
Could you say a bit more about the "standard finality argument"?
Sep
21
comment On triangulated categories of pro-objects
Actually, I guess it's possible to prove all this using Isaksen's model structure. I first thought of using $\infty$-categories because model categories don't make it easy to prove that a left Quillen functor preserves finite homotopy limits, but in fact when it's between stable model categories it's automatic (this is in Hovey).
Sep
20
answered On triangulated categories of pro-objects
Aug
23
awarded  Enlightened
Aug
23
awarded  Nice Answer
Aug
22
answered Which limits commute with filtered colimits in the category of sets?
Jul
27
comment Diffeology as a sheaf on the site of smooth manifolds
Actually Metzler references J. Giraud, Cohomologie non abélienne, p. 91, where the special case $n=2$ is proved (even for stacks of categories).