# Questions tagged [infinity-categories]

The infinity-categories tag has no usage guidance.

519
questions

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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 ...

6
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0
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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 ...

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 ...

4
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0
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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, ...

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2
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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 ...

3
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1
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148
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### 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 $f$ ...

11
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1
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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 ...

4
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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) \...

4
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2
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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}\...

5
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1
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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 $\...

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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 ...

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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 ...

1
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1
answer

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(...

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2
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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 ...

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votes

1
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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. ...

2
votes

1
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### 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
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2
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343
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### 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 ...

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### 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 ...

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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 ...

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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 ...

4
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1
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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 ...

5
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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 $G : \mathcal{D} \to \mathcal{C}$ ...

2
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1
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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, ...

2
votes

1
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### 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?
...

2
votes

1
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### 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 $...

5
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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 ...

4
votes

1
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359
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### 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 ...

5
votes

2
answers

254
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### 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
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1
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433
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### 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
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### 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
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### 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 ...

3
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### 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 ...

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1
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### 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 ...

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### 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) \...

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### 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 ...

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### 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\...

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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" (...

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1
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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 ...

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### 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 ...

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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 ...

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### 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 ...

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1
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### 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
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1
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### 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
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1
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### 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 ...

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### 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 ...

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### $\infty$-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 $\infty$-...

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### $(\infty,2)$-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 ...

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### 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
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### 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?
...

3
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1
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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 ...