2,911 reputation
714
bio website math.mit.edu/~hoyois
location Cambridge, MA, USA
age 28
visits member for 3 years, 7 months
seen Jul 18 at 12:57

I'm a postdoc at MIT.


Jul
17
comment Are there connections between Homotopy type theory and Grothendieck's theory of motives?
Well, motivic homotopy theory is a presentable locally cartesian closed ∞-category, and therefore it admits a presentation by a type-theoretic model category. Of course, univalence does not hold...
Jul
16
comment What is the (co-)homology of $K(\mathbb{R}_\delta,n)$?
$K(\mathbb R_\delta,n)$ cannot be a retract of a manifolds since a manifold is a countable CW complex and hence has countable homotopy groups.
Jul
5
answered Does the forgetful functor from presentable $\infty$-categories to $\infty$-categories preserve filtered colimits?
Jul
4
comment What is the cokernel of a map of presentable stable $\infty$-categories?
Yes, exactly: you can factor the functor $D\to C$ as a colocalization $D\to C'$ followed by a conservative functor $C'\to C$, and you get the same pushout if you replace $C$ by $C'$.
Jul
4
answered What is the cokernel of a map of presentable stable $\infty$-categories?
Jun
25
answered Van Kampen colimits
Jun
22
comment Van Kampen colimits
Maybe your notion of 2-limit is different. According to the nLab definition ncatlab.org/nlab/show/2-limit, the inclusion Set → Cat does not preserve 2-colimits: the 2-colimit of $*\rightrightarrows *$ in Cat is the groupoid $B\mathbb{Z}$. Hence, the 2-limit of $Set\rightrightarrows Set$, where both arrows are the identity, is the category of $\mathbb{Z}$-sets.
Jun
22
comment Van Kampen colimits
In fact, the only locally presentable category in which all colimits are van Kampen is the terminal category, since such a category must be an $\infty$-topos.
Jun
22
comment Van Kampen colimits
The exponential functor $\mathbf{Set}^{(-)}$ does not send colimits of sets to 2-limits of categories. For example, consider the coequalizer of the diagram $*\rightrightarrows *$.
Jun
5
answered localization and $E_{\infty}$-spaces
Jun
4
comment localization and $E_{\infty}$-spaces
1 is true because the product of two homology equivalences is a homology equivalence, so $L$ preserves finite products. $X=S^1$ is a counterexample to 2.
May
15
answered higher Eilenberg-Moore-toposes of left exact derived comonads
May
15
comment higher Eilenberg-Moore-toposes of left exact derived comonads
Indeed, Theorem 6.3.8 does the trick! So coalgebras satisfy descent, but do they form an accessible ∞-category?
May
15
comment higher Eilenberg-Moore-toposes of left exact derived comonads
Yes, sorry, that's what I meant. The reflection part follows from conservativity of the forgetful functor, which is left adjoint, so I think the only thing to check is the preservation of pullbacks.
May
14
comment higher Eilenberg-Moore-toposes of left exact derived comonads
Do you know that the forgetful functor from coalgebras preserves and reflects colimits and pullbacks? If so, the characterization of ∞-topoi by descent (prop 3 here) immediately implies that coalgebras form an ∞-topos.
May
2
comment a (pseudo)adjunction for the functor sending a category C to PSh(C) the category of presheaves
The forgetful 2-functor from large categories with small colimits to large categories has a genuine left adjoint which sends a small category C to PSh(C). But this left adjoint is not a right adjoint: PSh(*) is not final, since it has many colimit-preserving endofunctors.
May
1
comment Why are unramified maps not required to be locally of finite presentation?
@jmc Well, you usually want closed immersions to be proper, that's what I had in mind. A specific situation where properness turns out to be appropriate is in the proper base change theorem (in $\ell$-adic cohomology, say): there's no need for finite presentability there.
May
1
comment Why are unramified maps not required to be locally of finite presentation?
You similarly don't want to require proper maps to be of finite presentation.
Mar
17
answered When is the pullback in Chow groups defined?
Mar
16
answered Does the kan extension preserves contractible presheaves?