All Questions
Tagged with infinity-topos-theory infinity-categories
52
questions
6
votes
0
answers
114
views
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 ...
- 6,684
4
votes
1
answer
237
views
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 $f:\widetilde{X}\to ...
- 2,154
2
votes
1
answer
145
views
"$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 ...
- 2,154
8
votes
0
answers
229
views
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{...
- 6,684
5
votes
1
answer
266
views
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 ...
- 6,684
10
votes
2
answers
633
views
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 ...
- 2,154
4
votes
2
answers
469
views
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
votes
1
answer
219
views
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
2
votes
1
answer
93
views
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 $...
- 489
3
votes
1
answer
422
views
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 (...
- 6,684
5
votes
1
answer
405
views
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 ...
- 42.7k
6
votes
2
answers
850
views
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, ...
- 709
9
votes
0
answers
579
views
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 $\...
- 894
2
votes
1
answer
496
views
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 ...
1
vote
1
answer
181
views
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{\...