Questions tagged [infinity-categories]
The infinity-categories tag has no usage guidance.
519
questions
5
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Are $\infty$-categories functorially colimits of their simplices?
Let $\mathcal C$ be an $\infty$-category. If $C$ is a quasicategory modeling $\mathcal C$, then we have a coend decomposition
$$\mathcal C = \int^{[n] \in \Delta} \Delta[n] \times C_n.$$
This allows ...
- 56.6k
6
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0
answers
176
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Higher categories using just simplicial sets
Is there a definition of $(\infty, n)$-category using just simplicial sets?
This is the case for $n \leq 2$.
Is the forgetful functor from saturated $n$-trivial complicial sets to simplicial sets an ...
- 763
7
votes
1
answer
220
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From the *usual* nerve of topological categories to $\infty$-categories
It is standard from work of Joyal and Lurie that there is a Quillen equivalence between the model category of simplicially enriched categories $Cat_\Delta$ and $\mathcal{S}\text{et}_\Delta$ with the ...
- 337
4
votes
0
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138
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Commuting homotopy colimits and arbitrary products in Spaces
Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, ...
- 1,823
10
votes
2
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461
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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 ...
- 1,440
3
votes
1
answer
148
views
A fiber-like method to show equivalence of infinity categories
Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...
- 1,772
11
votes
1
answer
372
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Recasting straightening/unstraightening equivalence as $(\infty, 2)$-adjunction
This is a vague set of questions that relies on (possibly non-existent) generalizations of low-dimensional results, mostly because I don't know many of the technical details underlying the ...
- 415
4
votes
0
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81
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Equalizer-product formula for $(\infty, 1)$-limits
If $F : K \to C$ is a functor of ordinary categories and $C$ has products and equalizers, then there is an isomorphism
\begin{equation*}
\lim F \cong \mathrm{eq} \left( \prod_{k_0 \in K_0} F(k_0) \...
- 415
4
votes
2
answers
398
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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}\...
- 1,826
5
votes
1
answer
190
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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 $\...
- 1,826
26
votes
5
answers
4k
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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 ...
- 1,179
3
votes
0
answers
63
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$n$-truncation of a Simplicial Model Category
I'm working in the category of rational $CDGAs$ and trying to find a reference/construction of a natural $2$-categorical structure via truncation of the mapping spaces.
In my head, the key point is ...
- 233
1
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1
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161
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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(...
- 483
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votes
2
answers
255
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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 ...
- 16.3k
5
votes
1
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197
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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. ...
- 1,440
2
votes
1
answer
125
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 ...
- 445
5
votes
2
answers
343
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
84
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 ...
- 1,440
9
votes
2
answers
345
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Simplicial sets with horn filling conditions up to some fixed degree
Let $X_\bullet$ be a simplicial set such that some horn filling condition (inner horns fill/inner horns fill uniquely/all horns fill) holds up to dimension $n$ (i.e. for $\Lambda_i[p]$ for all $p\leq ...
- 1,207
4
votes
1
answer
126
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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 ...
- 1,772
4
votes
1
answer
178
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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 ...
- 125
5
votes
1
answer
140
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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}$ ...
- 415
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votes
1
answer
317
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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, ...
- 5,627
2
votes
1
answer
212
views
Homotopy coherent nerve for algebraic model categories
Is a homotopy coherent nerve defined for algebraic model category that returns algebraic quasi-categories as Urs Schreiber wrote about? Or do we not know how to determine it / does it seem impossible?
...
- 5,627
2
votes
1
answer
77
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 $...
- 459
5
votes
1
answer
130
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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 ...
- 361
4
votes
1
answer
359
views
Proving Zariski descent
I want to understand why the functor $\mathscr{D}$
sending an affine scheme to its associated derived
$\infty$-category satisfies Zariski descent. My understanding
is that one has to show that given a ...
- 365
5
votes
2
answers
254
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 ...
- 1,679
5
votes
1
answer
433
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 ...
- 9,565
4
votes
0
answers
366
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....
- 365
5
votes
1
answer
167
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 ...
- 9,565
3
votes
0
answers
133
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 ...
- 1,057
4
votes
1
answer
299
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 ...
- 9,565
5
votes
0
answers
62
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) \...
- 51
6
votes
0
answers
113
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 ...
- 1,440
3
votes
0
answers
109
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\...
- 31
11
votes
2
answers
2k
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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" (...
- 1,024
20
votes
1
answer
607
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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 ...
- 303
17
votes
0
answers
643
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 ...
- 171
6
votes
0
answers
165
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(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 ...
- 1,826
1
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0
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110
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 ...
- 117
2
votes
1
answer
102
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
$$
\...
- 415
6
votes
1
answer
206
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}}\...
- 9,565
4
votes
1
answer
254
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 ...
- 1,440
2
votes
0
answers
71
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 ...
- 583
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votes
0
answers
130
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 $\infty$-...
- 837
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votes
0
answers
118
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 ...
- 295
7
votes
1
answer
385
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 ...
- 1,440
3
votes
0
answers
51
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?
...
- 587
3
votes
1
answer
175
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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 ...
- 352