bio | website | math.mit.edu/~hoyois |
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location | Cambridge, MA, USA | |
age | 28 | |
visits | member for | 3 years, 7 months |
seen | Jul 18 at 12:57 | |
stats | profile views | 1,267 |
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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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? |