Questions tagged [infinity-topos-theory]
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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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$(\infty,1)$-topoi generated by $(n,1)$-categories
A (1,1)-topos (i.e. an ordinary Grothendieck topos) is called localic if the following two equivalent conditions hold:
It is the category of sheaves on a (0,1)-site with finite limits$^*$ (i.e. a ...
- 65.8k
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Fibrations of sites for $\infty$-topoi
For any geometric morphism $f:\mathcal{F} \to \mathcal{E}$ of Grothendieck 1-topoi, there exists a functor of small categories $\ell :D\to C$ and left exact localizations $\mathcal{F} \hookrightarrow \...
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Using higher topos theory to study Cech cohomology
It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
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Does the concept of a $\infty$-category have a natural definition in the $\infty$-world?
I start with a thesis: the natural notion of equality is additional data (paths/morphisms), not a binary relation (the fact that they exist). So, in particular, with such a constructivization (...
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Is every sheaf $\infty$-topos equivalent to sheaves on itself with respect to the canonical topology?
Let $(\mathcal C, J)$ be a small subcanonical $\infty$-site, and let $Sh_J(\mathcal C)$ be the $\infty$-topos of sheaves thereon. Then $Sh_J(\mathcal C)$ is itself an $\infty$-site with respect to the ...
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The tangent bundle and dual tangent bundle in topos theory
Let $\mathcal B = B \mathbb T$ be an $\infty$-topos, thought of as the classifying $\infty$-topos of some "$\infty$-geometric theory" $\mathbb T$. The notion of "$\infty$-geometric ...
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The hyperdoctrine of topoi and a near KZ-comonad
Allow me to waffle about a bit of topos theory, leading up to a few questions about KZ comonads and about the comprehension schema in hyperdoctrines.
Let $Pr^L$ denote the $\infty$-category of ...
- 62.6k
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When do geometric morphisms lead to periodic adjoints?
This may be a naïve question but I've been unable to locate a reference that addresses it. Any thoughts are appreciated!
Let $f:\mathcal{E}\to\mathcal{S}$ be a cohesive morphism of toposes. That is, ...
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Do topoi have injective hulls?
Recall that a topos $\mathcal I$ is injective (with respect to embeddings) if and only if $\mathcal I$ is a retract of $Psh(C)$ for some finitely-complete $C$. Say that an embedding $f : \mathcal X \...
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Does $Pr^L_\kappa \in Pr^L$ behave like an "object classifier" or "universe"?
Let $Pr^L$ denote the $\infty$-category of presentable $\infty$-categories and left adjoint functors. Let $Pr^L_\kappa \subset Pr^L$ denote the locally-full subcategory of $\kappa$-compactly-generated ...
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When is a stable $\infty$-category the stabilization of an $\infty$-topos?
Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ ...
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Is there a (Grothendieck) $\infty$-topos for which Whitehead's theorem only holds for maps between truncated objects?
We know that non-hypercomplete $\infty$-toposes exist. Is there such a topos $\mathcal{E}$ with the following property?
For any $X, Y \in \mathcal{E}$, if all weak homotopy
equivalences (or $\infty$-...
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Understanding the double negation modality under the "propositions as types" paradigm
$\DeclareMathOperator\Hom{Hom}$I'm trying to understand the double negation modality under the "propositions as types" paradigm, but I'm running into an apparent contradiction: let $T$ be a ...
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Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?
$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and ...
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