Questions tagged [infinity-categories]

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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) = \...

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