bio | website | math.mit.edu/~hoyois |
---|---|---|
location | Cambridge, MA, USA | |
age | 28 | |
visits | member for | 3 years, 4 months |
seen | Apr 2 at 22:40 | |
stats | profile views | 1,115 |
I'm a postdoc at MIT.
Mar 17 |
answered | When is the pullback in Chow groups defined? |
Mar 16 |
answered | Does the kan extension preserves contractible presheaves? |
Feb 16 |
comment |
What are iterated cobar constructions?
A modern treatment of this is also in section 5.2.3 of math.harvard.edu/~lurie/papers/higheralgebra.pdf |
Jan 25 |
awarded | Enlightened |
Jan 25 |
awarded | Nice Answer |
Jan 25 |
revised |
Category of motivic spectra
added 940 characters in body |
Dec 28 |
awarded | Yearling |
Dec 10 |
answered | Localizations of model categories and $\infty$-categories |
Dec 8 |
comment |
Topological retraction vs categorical retraction
This only shows that $A'$ is a topological retract of $X$, but $A$ need not be. See Mark Grant's counterexample! |
Dec 7 |
comment |
unbounded derived category of a $\infty$-topos
Hypercompleteness is defined in HTT 6.5.2. The point is that $Shv^{hyp}(X)$ is the localization of $Shv(X)$ at morphisms which induce isomorphisms on homotopy groups. From this it's easy to show that $Shv^{hyp}(X,\mathfrak{C})$ is the localization of $Shv(X,\mathfrak{C})$ at the quasi-isomorphisms. |
Dec 6 |
comment |
unbounded derived category of a $\infty$-topos
I agree with Dylan: for general $X$, $D_\infty(Ab(Sh(X)))$ and $Sh_\infty(X,D_\infty(Ab))$ are not equivalent. That's the whole point of Lurie's remark: proper base change holds for the latter but not the former. $D_\infty(Ab(Sh(X)))$ is equivalent to the $\infty$-category of hypercomplete $D_\infty(Ab)$-valued sheaves. |
Nov 24 |
comment |
When is a topological space the homotopy colimit of an open covering?
As a follow-up to my first comment, the proof of Lemma A.4.14 in HA shows the following. If $X$ is locally contractible, then $Map(Sh(X),K)\to Map(Sing(X),K)$ is an equivalence provided that the constant sheaf with fiber $K$ is hypercomplete. In particular, $Sing(X)$ and $Sh(X)$ have the same pro-truncated reflections. I suspect they're not the same in general because the usual definition of "locally contractible" for a space corresponds to the topos being "locally $\infty$-connective" rather than actually "locally contractible". |
Nov 23 |
comment |
When is a topological space the homotopy colimit of an open covering?
I think it would be unfair not to call Lurie's theorem itself a higher Seifert-van Kampen theorem. Note that the result I've quoted is only a special case of half of the theorem. The other half (in the same special case) says that the actual colimit of the diagram of simplicial sets $Sing(C_U)$ is weakly equivalent to $Sing(X)$. |
Nov 23 |
awarded | Nice Answer |
Nov 23 |
reviewed | Approve Given a Levy Exponent find the jump-measure and drift |
Nov 23 |
revised |
When is a topological space the homotopy colimit of an open covering?
added 252 characters in body |
Nov 23 |
answered | When is a topological space the homotopy colimit of an open covering? |
Nov 23 |
comment |
When is a topological space the homotopy colimit of an open covering?
Well, not only is the shape homotopy invariant (Higher Algebra, A.2.10), but your question is actually answered completely in that appendix, see Remark A.3.8: the functor $Shv(X)\to \infty Gpd$ induced by the "underlying homotopy type" functor preserves all colimits! |
Nov 23 |
comment |
What does an endomorphism in a triangulated category give rise to?
What I mean by $\phi$-completion here is the inverse limit of the $cofib(\phi^n)$. For example, if $\phi$ is multiplication by $p$ on $\mathbb{Z}$, you get $\mathbb{Z}_p$. |
Nov 23 |
comment |
When is a topological space the homotopy colimit of an open covering?
This is true in particular when the topos-theoretic shape functor $Sh$ agrees with the classical "underlying homotopy type" functor, since it's always true that $Sh(X)$ is the colimit of $Sh(C_U)$. Lurie shows that $Sh(X)$ is weakly equivalent to $X$ when $X$ is paracompact and homotopy equivalent to a CW complex (Higher Algebra, A.1.4). I'm not sure if you can generalize this to "locally contractible". It's not even clear to me that the shape of a contractible space is contractible... |