bio  website  math.mit.edu/~hoyois 

location  Cambridge, MA, USA  
age  27  
visits  member for  3 years, 1 month 
seen  19 hours ago  
stats  profile views  1,018 
I'm a postdoc at MIT.
16h

awarded  Enlightened 
17h

awarded  Nice Answer 
19h

revised 
Category of motivic spectra
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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 quasiisomorphisms. 
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 followup 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 protruncated 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 Seifertvan 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 jumpmeasure 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 topostheoretic 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... 
Nov 23 
awarded  Custodian 
Nov 21 
answered  What does an endomorphism in a triangulated category give rise to? 
Nov 17 
comment 
What does an endomorphism in a triangulated category give rise to?
No matter what the degree of $\phi$ is, it gives rise to a filtered object. If you apply a homological functor $H$ to it (e.g. map another object into it), you get two spectral sequences, potentially converging to $H$ of the limit and colimit of the filtered object: ncatlab.org/nlab/show/… 