bio | website | math.northwestern.edu/~hoyois |
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location | Chicago, IL, USA | |
age | 27 | |
visits | member for | 2 years, 3 months |
seen | Feb 7 at 17:35 | |
stats | profile views | 740 |
I'm a graduate student at Northwestern University.
Jan 21 |
awarded | Enlightened |
Jan 21 |
awarded | Nice Answer |
Dec 28 |
awarded | Yearling |
Dec 4 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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Loops and suspensions of higher categories
Could you say a bit more about the "standard finality argument"? |
Sep 21 |
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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 |
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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). |