Questions tagged [infinity-topos-theory]
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139
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Treatment of classes of mono/epi morphisms in $(\infty, 1)$-categories
In the classical theory of $(1, 1)$ categories, the chain of classes of mono/epi morphisms is well known: plain $\leftarrow$ strong $\leftarrow$ effective $\leftarrow$split ((I assume that the ...
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Why is the universal $n$-gerbe universal? (HTT, 7.2.2.26)
Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to ...
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"$X$ is $n$-truncated $\iff$ $\Omega X$ is $(n-1)$-truncated" for connected pointed $X$. (HTT, 7.2.2.11)
In the proof of Lemma 7.2.2.11 of Higher Topos Theory, Lurie makes the following claim:
($\ast$) Let $n\geq1$ be an integer, let $\mathcal{X}$ be an $\infty$-topos, and let $1\to X$ be a pointed ...
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Does every $\kappa$-compact topos embedd relatively $\kappa$-tidily into a presheaf topos?
Let $\kappa$ be a regular cardinal and say that a topos $\mathcal{E}$ is $\kappa$-compact if the global sections $\gamma_{\ast} : \mathcal{E} \to \mathsf{Set}$ preserves $\kappa$-filtered colimits.
My ...
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322
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$\infty$-topos as an internal $\infty$-category in itself
I'm interested (both autonomously and directly related to my work) in the natural internalization of $\infty$-topos sheaves in it (as usual, assuming Grothendieck universes). Is there any literature ...
- 6,684
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What is the exact definition of the $\infty$-topos of sheaves on a localic $\infty$-groupoid?
The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ .
The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{...
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Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely ...
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Coequalizers and pullbacks in $\infty$-topoi
In an $\infty$-topos, suppose we have two cartesian diagrams of the form
$$
\require{AMScd}
\begin{CD}
\overline{A} @>>> \overline{B} \\
@VVV @VVV \\
A @>>> B .
\end{CD}
$$
Let
$$
\...
- 66
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Sheaf of compact Hausdorff spaces but not a condensed anima
Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...
- 425
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630
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Effective epimorphisms and 0-truncations (HTT, 7.2.1.14)
In Proposition 7.2.1.14 of Higher Topos Theory, Lurie asserts the following:
Let $\mathcal{X}$ be an $\infty$-topos and let $\tau_{\leq0}:\mathcal{X}\to\tau_{\leq0}\mathcal{X}$ denote a left adjoint ...
- 1,979
4
votes
2
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466
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Categorical equivalences vs. categories of simplices
Let $j: K\to K′$ be a categorical equivalence of simplicial sets. By [HTT, Remark 2.1.4.11], we have a Quillen equivalence (with the covariant model structures)
$$
j_!:\mathsf{sSet}_{/K}\...
- 1,876
5
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219
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Connectedness of truncated version of cosimplicial indexing category
Let $F:\mathbf{\Delta}\to\mathcal{S}_{\leqslant n-1}$ be a cosimplicial object in the $\infty$-category of $(n-1)$-truncated spaces. Is it always a right Kan extension of its restriction along $\...
- 1,876
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How does nullification of $K(\mathbb{Z}, 2)$ compare to 1-truncation?
Let $B$ be a type (or space). A type (or space) is $B$-null if the canonical map $X \to X^B$ is an equivalence. The $B$-nullification of an arbitrary type $X$ is a $B$-null type $\bigcirc_B X$ ...
- 3,876
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Comparion theorem between symmetric monoidal $\infty$-functor
Let $T,T'$ be symmetric monoidal $\infty$-categories. And let $F_1,F_2:T\to T'$ be symmetric monoidal functors and let $t:F_1\Longrightarrow F_2$ be a symmetric monoidal natural transformation from $...
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Presenting geometric morphisms by geometric morphisms
It's known that any $\infty$-topos $\mathcal{E}$ can be presented by a Quillen model category $\mathbf{E}$ that is itself a 1-topos. For instance, if $\mathcal{E}$ is a left exact localization of a ...
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