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Results tagged with infinity-categories
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user 20233
Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.
5
votes
Accepted
descent implies hyperdescent
It is certainly true that descent implies hyperdescent whenever $\mathcal C$ is a $n$-category for some $n<\infty$ (it wasn't clear from your question whether you knew this or not). This is because, f …
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5
votes
Accepted
What is the Isomorphism subspace of the mapping space in an infinity category
Question 1: The space of isomorphisms is always a full subspace of the mapping space. In other words, it is a union of connected components of the mapping space.
Question 2: Is $S$ is a classical sche …
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6
votes
Accepted
Localizations of model categories and $\infty$-categories
There is also an existence theorem for right Bousfield localizations of presentable $\infty$-categories.
In fact, it follows from the existence theorem for left Bousfield localizations.
Let $K$ be a …
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3
votes
Accepted
Etale sheaves on algebraic spaces vs. Etale sheaves on affines
Yes, the two $\infty$-topoi are equivalent. Let $u: \mathrm{Aff} \to \mathrm{AlgSp}$ be the inclusion. Then $u$ preserves étale covering families and pullbacks, hence commutes with the formation of th …
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13
votes
What is the dual of the stable infinity category of perfect complex on smooth proper variety?
It is self-dual. In general the dual of a smooth proper $R$-linear $\infty$-category $C$ is always $C^{\operatorname{op}}$, but for a scheme $X$ we have $\operatorname{Perf}(X) = \operatorname{Perf}(X …
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9
votes
Accepted
Effective epimorphisms and 0-truncations (HTT, 7.2.1.14)
Here's an easy way to resolve the circularity. Proposition 7.2.1.13 is only used in the proof of 7.2.1.14 to establish the following statement:
(1) If $f\colon V\to X$ is a monomorphism and is surject …
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7
votes
Accepted
Relation between hypercompleteness and the property that Cech cohomology calculates sheaf co...
There is no relation between hypercompleteness and the property that Čech cohomology agrees with genuine cohomology, i.e., there is no implication either way. For example, étale cohomology of nice sch …
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6
votes
Are $\infty$-topoi determined by their localic points ?
The functor is conservative if $T$ is hypercomplete. This follows from DAG VII, Cor. 4.14, which says that any $\infty$-topos admits a surjection from a hypercomplete locale (where $f$ is a surjection …
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2
votes
Accepted
(Local) Homotopy dimension of $\infty$-topoi on paracompact spaces
I think if $X$ is paracompact of covering dimension $\leq n$ then $\mathrm{Shv}(X)$ is also locally of homotopy dimension $\leq n$:
First, the $F_\sigma$ open subsets of $X$ form a basis of the topolo …
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4
votes
Accepted
Generalizations of tangent $\infty$-topos
This is rarely true. For example, the axioms for ∞-topoi imply that, if $T_S\mathbf H$ is an ∞-topos, then the fibers of $p_S$ must have van Kampen pushouts (more generally van Kampen weakly contracti …
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13
votes
Accepted
Tensor products of $\mathbb{E}_\infty$-spaces
The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful poi …
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