Questions tagged [infinity-topos-theory]
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140
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If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?
I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.
We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos ...
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2
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Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$
$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...
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Subcategory of coherent objects in an $\infty$-topos forming a local $\infty$-pretopos
My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following:
Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the ...
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What is the connection between Lurie's definition of shape and Čech homotopy?
It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes).
For instance, Lurie [Higher topos theory] defines this one:
Definition 1.
The ...
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How to formulate the univalence axiom without universes?
The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot ...
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5
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1
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405
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Geometric realisation of smooth $\infty$-stacks
Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial ...
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A map that names itself
Call the walking arrow $\Delta_{1}$, containing exactly one nontrivial 1-cell $[0<1] : 0 \to 1$. I am interested in a map $\Phi : \Delta_{1} \to \mathrm{Type}$, such that $\Phi [0<1] = \Phi$ (...
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Learning roadmap to 'Differential cohomology in a cohesive $\infty$ topos'
I am very curious to study arXiv:1310.7930 (henceforth:DCCT) but am not sure if I have the pre-requisites. I am familiar with basic algebraic topology (singular cohomology, classifying spaces, ...
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Is every conservative, left exact left adjoint comonadic, $\infty$-categorically?
Consider a conservative left adjoint $G : C \to D$ between complete 1-categories. By Beck's theorem, the following are equivalent:
$G$ is comonadic.
$G$ preserves $G$-split equalizers.
(2) is ...
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Does every exponentiable ($\infty$-)topos have enough points?
The notion of a coherent topos is a somewhat "refined" finiteness condition to put on a topos. One reason it is considered a "fruitful" notion is the Deligne completeness theorem, ...
- 62.6k
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Coherent objects in a hypercomplete $\infty$-topos
In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\...
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2
votes
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496
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Equivalence relations, Segal groupoids and groupoid objects in an infinity category
There are three forms of "equivalence relations are effective" as part of Giraud's axioms in $1$-Grothendieck topoi, Model topoi and Infinity topoi. I am trying to understand how they relate ...
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1
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181
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n-truncation/n-connected factorization in an $\infty$-topoi
I want to prove that given an $\infty$-topos $\mathscr{C}$ and a morphism $f: X \to Y$ in $\mathscr{C}$, for each $k \geq -1$, there exists a factorization $X \xrightarrow{\eta_f} E_k^f \xrightarrow{\...
2
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1
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(Local) Homotopy dimension of $\infty$-topoi on paracompact spaces
I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a ...
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Join as a bifunctor
I have been reading these great notes by Charles Rezk, and one thing that has been bothering me is the join construction. To solve lifting problems in quasicategory theory we use the Leibniz ...