Questions tagged [infinity-categories]
The infinity-categories tag has no usage guidance.
552
questions
4
votes
2
answers
449
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}\...
5
votes
1
answer
213
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 $\...
28
votes
5
answers
5k
views
What is the motivation for infinity category theory?
To my understanding, most mathematical theories can be simply understood in the view point of Category theory and its derivative theories. But what exactly is the motivation to study infinity category ...
15
votes
1
answer
473
views
Well pointed endofunctors on $\infty$-categories
In $1$-category theory, a well pointed endofunctor of a category $C$, is an endofunctor $F:C \rightarrow C$ endowed with a natural transformation $\sigma : Id \rightarrow F$ such that the two natural ...
1
vote
1
answer
362
views
Limits of infinity categories and mapping spaces
Let $p:I\to Cat_{\infty}$ be a diagram of infinity categories, where $I$ is a small Kan complex. Let $C:=\lim p$ be the limit of $p$. For any two objects $x,y\in C$ and $i\in I$, let $x_i,y_i\in C_i=p(...
2
votes
2
answers
297
views
Does the homotopy category of finite spectra act on stable homotopy categories?
Assume that C is a stable infinity category; $SH_{fin}$ is the homotopy category of finite spectra. Is there a canonical bi-functor (action? module structure?) $SH_{fin}\times hC \to hC$?
Is there any ...
5
votes
1
answer
220
views
Cofinal maps from posets (HTT, 4.2.3.16)
I do not understand the proof of Variant 4.2.3.16 of Higher Topos Theory by Jacob Lurie, and I need help.
Variant 4.2.3.16 asserts the following:
($\diamond$) Let $K$ be a finite simplicial set. ...
2
votes
1
answer
168
views
Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique?
In Higher Algebra 4.2.8.19, Lurie shows that the symmetric monoidal structure on spectra is uniquely defined (on the $\infty$-category level) by the following properties:
The sphere spectrum is the ...
5
votes
2
answers
356
views
Monomorphisms of diagrams in an $\infty$-category
Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions:
If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a ...
4
votes
0
answers
95
views
Localization and space of morphisms
I have a question regarding the proof of Proposition 2.19 of Factorization homology of topological manifolds by Ayala and Francis. In the final paragraph of the proof (more specifically, in the second ...
6
votes
1
answer
142
views
Is the functor $O$ from the simplex category to the category of orientals cofinal
Let $\Delta$ be the full subcategory of the category of small categories spanned by the non-empty totally ordered sets of the form $[n]$ for $n \geq 0$. Let $\mathfrak{O}$ be the full subcategory of ...
4
votes
1
answer
169
views
Homotopy totalization and chains - reference
Simple case. Take $X_{\bullet}$ a cosimplicial space. Is it true that the chain complex of its homotopy totalization is quasi-isomorphic to the homotopy totalization of its chain complex? Because of ...
4
votes
1
answer
226
views
Gluing isomorphism in derived categories along filtered colimit
Let $X$ be a locally finite type algebraic stack $X$ (but feel free to pretend it's a scheme) with a presentation as the filtered colimit of finite type open substacks $U_i$. By descent, at the level ...
5
votes
1
answer
171
views
Pushforward of cocartesian fibrations
Let $\pi : \mathcal{E} \to \mathcal{C}$ be a cocartesian fibration of $\infty$-categories which straightens to a functor $F : \mathcal{C} \to \mathrm{Cat}_\infty$. If $G : \mathcal{D} \to \mathcal{C}$ ...
2
votes
1
answer
374
views
Why do we need enriched model categories?
As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, ...
4
votes
1
answer
351
views
Quasicategorical Construction of a Cosimplicial Map of Rognes
In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...
2
votes
1
answer
87
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 $...
9
votes
1
answer
342
views
Are (complete) 2-Segal spaces the same as Span-enriched infinity categories?
The question is basically in the title. More generally, I would like to know if this, or any reasonable variant of it, is true. Or perhaps, to understand better the gap between 2-Segal spaces and Span-...
5
votes
2
answers
290
views
Reedy fibrancy and composition in Segal spaces
I am going through V. Hinich's "Lectures on Infinity Categories" and I have a (possibly trivial) question on Segal spaces.
We define Segal space to be a bisimplicial set $X$ which is fibrant ...
5
votes
1
answer
448
views
Homotopy groups of categories of elements as higher colimits
Given a diagram of sets $D\colon\mathcal{C}\to\mathsf{Set}$, we have a bijection (Proof)
$$\operatorname{colim}(D) \cong \pi_0 (\textstyle\int_\mathcal{C}D).$$
Is there any known application or ...
4
votes
0
answers
393
views
Formal properties of limits of $\infty$-categories
I want to understand the usage of $\infty$-categories
in the proof of Proposition 10.5 in the Condensed Mathematics
lecture notes available here: https://www.math.uni-bonn.de/people/scholze/Condensed....
5
votes
1
answer
189
views
Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$
Homotopy coherent Invertibility.
Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we ...
4
votes
2
answers
973
views
unbounded derived category of a $\infty$-topos
In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, ...
9
votes
2
answers
476
views
One colored infinity operads via symmetric sequences?
The question
One standard approach to the theory of 1-colored (symmetric) operads in classical 1-categorical theory is via monoids in symmetric sequences with respect to the composition product. Has ...
17
votes
2
answers
679
views
Homotopy theories of operads
I know of three homotopy theories of colored operads.
The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak ...
3
votes
0
answers
136
views
Higher theory o $C^{\ast}$-algebras and the $C^{\ast}$-algebra of a $\infty$-groupoid
Has someone already worked out what would be the infinity categorical analogue of the category of $C^{\ast}$-algebras? Given a groupoid $G$ we can associate a $C^{\ast}$-algebra $C^* (G)$, can we do ...
4
votes
1
answer
366
views
The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)
Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm ...
5
votes
0
answers
65
views
Does the restriction functor $j^* $ to Zariski open preserve the limit of $j^*$-split cosimplicial diagram?
This might be a trivial question but I could not find a satisfatory answer easily.
Let $X = \mathbb{C}$ and $U = \mathbb{C}^*$, and let $j: U \to X$ denote the open embedding.
Consider $j^* : QCoh(X) \...
6
votes
0
answers
116
views
Homotopy fibers in the Joyal model structure and the Kan–Quillen model structure
While playing around with $\infty$-categories, I ran into the following problem:
Let $p:\mathcal{C}\to\mathcal{D}$ be a functor of $\infty$-categories. Does one of the following condition imply the ...
13
votes
2
answers
2k
views
Can we use Mann's six-functor formalism with D-modules?
In a recent course in Bonn, P. Scholze explains a formalization of a six-functor formalism due to L. Mann. In this axiomatization, three of the functors $f_!,f^*,\otimes$ are "constructed" (...
3
votes
0
answers
124
views
Gluing data for $\infty$-sheaves?
Let $\mathcal{F}$ and $\mathcal{G}$ be two $\infty$-sheaves on $X$ resp. $Y$. I want to understand exactly when we can "glue" $\mathcal{F}$ and $\mathcal{G}$ to give a $\infty$-sheaf on $X\...
3
votes
1
answer
250
views
Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\infty$-ring spectrum?
Recently, Ching and Salvatore have proven that the $E_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E_n$ operad, I realized there is an obvious first ...
21
votes
1
answer
700
views
The derived category does not satisfy descent - example
One motivation for studying infinity categories is that the derived category does not satisfy Zariski descent, although the infinity categorical version does.
I would like to see an example of Zariski ...
19
votes
0
answers
834
views
What is the status of the cobordism hypothesis?
Let $\mathscr{C}$ be a symmetric monoidal (weak) $n$-category. A framed extended TQFT of dimension $n$ with values in $\mathscr{C}$ is a symmetric monoidal functor from the framed bordism $n$-category ...
6
votes
0
answers
189
views
(Co)cartesian fibrations and left Kan extensions
Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite ...
1
vote
0
answers
114
views
What is the difference between stable ∞-categories $\text{perf}_ZX$ and $\text{perf}~Z$
Let $X$ be a quasi-compact quasi-separated scheme and $Z$ a closed subscheme.
One has a symmetric monoids stable infinity categories $\text{perf}_ZX$, which is generated by perfect complexes supported ...
2
votes
1
answer
117
views
Morphisms in category of left fibrations
I am trying to better understand the straightening-unstraightening equivalence of Lurie in the $\infty$-categorical setting. In the case that I am interested in, this equivalence states that
$$
\...
6
votes
1
answer
208
views
Homotopical properties of powersets of simplicial sets
Given a simplicial set $X_\bullet$, define its powerset simplicial set $\mathcal{P}_\bullet(X)$ as the composition
$$\Delta^\mathsf{op}\xrightarrow{X_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\...
4
votes
1
answer
217
views
The effect of straightening on morphisms
This is similar to another question on MO, but is different.
Let $p:\mathcal{E}\to\mathcal{C}$ be a right fibration between $\infty$-categories. As explained in Lurie's HTT, $\S$2.1, we can consider $...
4
votes
1
answer
268
views
HTT, Remark 4.2.4.5
In his book Higher Topos Theory, Lurie proves that in some favorable cases, functor categories of $\infty$-categories admits a rigid model. More precisely, he proves in Proposition 4.2.4.4 the ...
2
votes
0
answers
75
views
Hom-spaces of Segal spaces versus their realization in $\mathbf{Cat}$
Consider the $\infty$-category of simplicial spaces $s\mathcal{S} = \mathbf{P}(\Delta)$. The
inclusion $\Delta \to \mathbf{Cat}$ induces a left adjoint
$i_! : s\mathcal{S} \to \mathbf{Cat}$. It is ...
6
votes
0
answers
141
views
$\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$
In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-...
5
votes
0
answers
125
views
$(\infty,2)$-categories as colimits of orientals
Let $\mathcal{C}$ be an $\infty$-category represented by a fibrant simplicial set in the Joyal model structure. It is well known that $\mathcal{C}$ can be expressed as the (homotopy) colimit over its ...
7
votes
1
answer
426
views
Why does the tangent classifier classify the tangent (micro)bundle?
Let $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category ...
3
votes
1
answer
207
views
Can there be a cospan of symmetric monoidal $\infty$-categories whose maps are lax symmetric monoidal but the pullback is not symmetric monoidal?
Given symmetric monoidal $\infty$-categories $A, B, C$ and lax symmetric monoidal maps $F:A\to C$, $G:B\to C$, I am curious if the pullback (when I say pullback here I will really mean homotopy ...
3
votes
0
answers
53
views
Recognising absolute distributors in terms of simplicial model categories
Briefly, my question is the following:
Can we recognise when a simplicial model category $\def\cM{\mathcal M}\cM$ is an absolute distributor, using only the language of (simplicial) model categories?
...
14
votes
2
answers
929
views
Infinity local systems
I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems".
From what I've been told, given a good cover $\{U_i\}$ of $X$, ...
4
votes
1
answer
355
views
Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?
Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic ...
5
votes
1
answer
461
views
On the definition of infinity-category
On 8:38 of Session 9: Masterclass in Condensed Mathematics an $\infty$-category is defined as a simplicial set $\mathcal{S}$ (i.e a functor $\Delta^{op}\rightarrow Sets$) such that for every horn $\...
10
votes
2
answers
373
views
When is the homotopy category of an accessible $\infty$-category accessible?
Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible:
$ho(\mathcal C) = \...