2,581 reputation
714
bio website math.mit.edu/~hoyois
location Cambridge, MA, USA
age 28
visits member for 3 years, 3 months
seen Mar 17 at 23:16

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...