# Questions tagged [infinity-categories]

The infinity-categories tag has no usage guidance.

447
questions

4
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### Reference for the equivalence between chain complexes and sequential diagrams in a stable $\infty$-category

Lurie's $\infty$-categorical Dold-Kan Correspondence relates simplicial objects and sequential diagrams in a stable $\infty$-category. Is there any reference for an equivalence to a category of ...

5
votes

2
answers

196
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### Density Theorem for $\infty$-Categories (HTT, Lemma 5.1.5.3)

The density theorem in the ordinary category theory asserts that every presheaf on a small category is a colimit of representables in a canonical way. In Lemma 5.1.5.3 of Higher Topos Theory, Lurie ...

12
votes

2
answers

543
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### sSet-enriched categories, quasi-categories and the model-independent theory

sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual ...

2
votes

0
answers

55
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### Derived category of a exact categories with (unusual) weak equivalences

Every exact category $\mathcal{E}$ has an attached derived category (for simplicity I will just refer to the bounded one) $D^b(\mathcal{E})$.
The construction is for example explained in
A. Neeman, ...

11
votes

1
answer

323
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### A step in Lurie's treatment of $L$-theory

I am looking at Proposition 3 of Lecture 6 from Lurie's course Algebraic L-theory and Surgery (https://www.math.ias.edu/~lurie/287xnotes/Lecture6.pdf). This involves a stable $\infty$-category $\...

4
votes

2
answers

194
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### Quillen pairs / $\infty$-adjunctions / adjunctions of homotopy categories

Some of the examples of $\infty$-categories are those arising from model categories. I would like to ask: what is the relationship between Quillen adjunctions between model categories and adjoint ...

5
votes

2
answers

364
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### Explicit description of the right adjoint

Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram.
Given the ...

3
votes

0
answers

126
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### Characterizing compactly assembled localizations of presheaf $\infty$-categories

After SAG Def 21.1.2.1, say that an $\infty$-category $\mathscr{C}$ is compactly assembled if there exists a small $\infty$-category $\mathscr{C}_0$ such that $\mathscr{C}$ is a retract of $\mathrm{...

2
votes

0
answers

101
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### Does the category of stable infinity categories form a "subtractive Waldhausen" category?

In "The $K$-theory spectrum of varieties", Jonathan Campbell introduces the notion of a subtractive Waldhausen category, a slight generalization of the notion of Waldhausen category that ...

3
votes

1
answer

209
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### How to simplify this homotopy totalization coming from an arc-cover into a pullback?

My question concerns the proof of Proposition 4.2 in Bhatt-Mathew’s paper on the arc-topology, but my confusion is completely general and anyone familiar with limits in $\infty$-categories would know ...

2
votes

1
answer

421
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### What is the dual of the stable infinity category of perfect complex on smooth proper variety?

Fix a commutative ring $R$.
Lurie proved that smooth proper $R$-linear stable infinity categories are dualizable in $\text{Cat}^\text{perf}_{R,\infty}$.
For a smooth proper variety $X$ over $R$, what ...

3
votes

0
answers

68
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### Detecting product preserving functors in the Grothendieck construction

The straightening/unstraightening equivalence lets us describe functors from an $(\infty,1)$-category $C$ into either the $(\infty,1)$-category of anima $\mathrm{An}$ or of $(\infty,1)$-categories $\...

3
votes

1
answer

284
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### $(n,1)$-dagger categories

In category theory, a dagger category is a precategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \...

7
votes

2
answers

328
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### Derived functors out of an unbounded derived $\infty$-category

Let $\mathcal A$ be an abelian category. In this lecture, Thomas Nikolaus
Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\...

2
votes

1
answer

188
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### $\infty$-groupoid iff Kan condition

I'm going through Chapter 4 of "Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects" which begins with an overview of $\infty$-categories. Theorem 2 states that a ...

12
votes

2
answers

620
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### When did the Joyal model structure on simplicial sets originate?

Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006,
as well as Joyal's own account in The Theory of Quasi-...

3
votes

1
answer

169
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### Question about the proof of Kerodon tag 030V (Proposition 7.3.7.1)

$\require{AMScd}$
Related to this, I have a question about the proof given in Kerodon of the following result:
Proposition 7.3.7.1: Let $C$ be an $\infty$-category, let $\bar{F} : C^\rhd \to D$ be a ...

6
votes

1
answer

445
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### How to prove that topological Hochschild homology of a smooth proper stable k-linear infinity category is dualizable?

Let $k$ be a perfect field of characteristic $p$. I heard that the Topological Hochschild homology of a smooth proper stable infinity category (or dg-category) is dualizable as a THH(k)-module ...

2
votes

1
answer

128
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### Colimits of DG-categories and functors between them

Suppose I have two diagrams $\{\mathcal{C}_i\}_{i\in \mathcal{I}}$ and $\{\mathcal{D}_i\}_{i\in \mathcal{I}}$ in the $\infty$-category of DG-categories over a field $k$ with continuous functors (i.e. ...

4
votes

0
answers

144
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### Composition in $\infty$-Categories

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 $...

1
vote

1
answer

147
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### Compact generation of the infinity category of stable infinity categories

I am reading the paper "A universal characterization of higher algebraic K-theory" by Blumberg, Gepner, and Tabuada, and I am stuck on Corollary 4.25:
…the fact that we have accessible ...

2
votes

2
answers

524
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### TR2 for homotopy category of stable $\infty$-category

I’m trying to understand Lurie’s proof that the homotopy category of a stable $\infty$-category is triangulated. In showing TR2, he constructs a diagram
$$\require{AMScd}
\begin{CD}
X @>f>> Y ...

5
votes

0
answers

167
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### Another model for $\infty$-operads?

There are several well-developed notions of $\infty$-operad in the literature, which are nowadays known to be equivalent (see e.g. the introduction of Chu-Haugseng-Heuts. However, another model is ...

3
votes

0
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139
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### Do we really need degeneracies for spectral sequence of homotopy simplicial chain complex?

Let's consider an homotopy simplicial chain complex, that is a functor of $\infty$-categories $X_{\bullet} = \textrm{N}(\Delta^{op}) \to \mathcal{C}_{\ge 0}$, where $\mathcal{C}_{\ge 0}$ is the $\...

3
votes

0
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71
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### Derived prestacks regarded as functors into spectra

If $k$ is a field (probably of characteristic zero), the usual definition of a derived prestack is a functor $ X \colon {\operatorname{CDGA}}_{k}^{\le 0} \to \operatorname{Spaces} $ from (graded) ...

3
votes

1
answer

152
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### Constructing sections of a cocartesian fibration

Suppose $\mathcal{E} \to \mathcal{C}$ is a cocartesian fibration over (the nerve of) a classical category, and there is a section on zero simplices that sends $C$ to $s(C)$ such that, for every edge $...

2
votes

0
answers

62
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### Defining the bivariant mapping space functor via the twisted arrow $\infty$-category

I have what is surely a very silly question involving the construction of the bivariant mapping space functor $map_C(y,x):C^{op}\times C\to C$ starting from an $\infty$-category C. I’m using Land’s ...

2
votes

0
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191
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### Stable $\infty$-category of motives

In nLab motive, it defines the derived category of motives as the full sub-$\infty$-category of the $\infty$-category of functors $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...

5
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0
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182
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### Two models for the tensor product of modules

Let $\mathcal{C}$ be an $\infty$-operad. Then Lurie in Higher Algebra, section 3.3.3 constructs a family of $\infty$-operads
$$\operatorname{Mod}(\mathcal{C})^\otimes\to \operatorname{Fin}_\ast \times ...

4
votes

1
answer

161
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### How to define the $\infty$-category of left fibrations?

In his book Introduction to Infinity-Categories, Land in his Theorem 3.3.16 asserts an equivalence of $\infty$-categories where one of the categories $\mathrm{LFib}(\mathcal C)$ is the full ...

3
votes

0
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78
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### Singular complex and homotopy coherent nerve as simplicial sets

Let $X$ be a CW complex.
Is the simplicial set $\ \mathrm{Sing}\ X$ isomorphic to the homotopy coherent nerve of some Kan enriched category?
Is this true for $X$ = the real line?

5
votes

0
answers

87
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### Counit functor associated to a bicartesian fibration

I would like to understand $\infty$ categorical adjunctions better. I am far from an expert, and so I would greatly appreciate published references (with no unproven foundational assumptions) ...

4
votes

1
answer

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### $\text{Mod}(A)$ is an $E_n$ category $\Leftrightarrow$ $A$ is an ??? algebra

Say we're working in a symmetric monoidal $\infty$-category $\mathcal{S}$, and $A$ is an associative algebra in it. For instance,
$$\mathcal{S}\ =\ \text{dg vector spaces},\ \ \ A\ =\ \text{a dg ...

6
votes

0
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### Universal property of dg-algebras

Let $k$ be a field. Does the fully faithful inclusion from $k$-algebras to dg-$k$-agebras concentrated in cohomological degrees $\leq 0$
$$\operatorname{Alg}_k\hookrightarrow\operatorname{dgAlg}_k^{\...

1
vote

0
answers

102
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### Computing the cotangent complex of morphisms of perfect complexes

In Lurie's Spectral Algebraic Geometry the cotangent complex of $\textbf{Perf}$ is computed as $ \Sigma^{-1}( \mathscr{F} \otimes \mathscr{F}^\vee)$ for some universal $\mathscr{F} \in \text{Qcoh}(\...

3
votes

0
answers

131
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### Augmented algebras over $\infty$-operads via the envelope

Let $\mathcal{O}^\otimes$ be an $\infty$-operad and $\mathcal{C}^\otimes$ a symmetric monoidal $\infty$-category, both in the sense of Lurie's Higher Algebra.
By augmented $\mathcal{O}^\otimes$-...

5
votes

0
answers

310
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### Is there an analogue of Kan's $\operatorname{Ex}^\infty$ functor for $(\infty,n)$-categories?

$\DeclareMathOperator\Ex{Ex}$Kan's $\Ex^\infty$ functor (see Why is Kan's $Ex^\infty$ functor useful?) $\Ex^\infty\colon\mathsf{sSets}\to\mathsf{Kan}$ produces a Kan complex from a simplicial set ...

6
votes

1
answer

227
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### How do the various homotopy 2-categories compare?

There are various models of $\infty$-categories floating around, so there are as many models of the associated homotopy 1- and 2-categories. Because the relations between the former are worked out in ...

5
votes

1
answer

232
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### What is the correct statement of Theorem 4.2 in Street's "Parity Complexes"?

Ross Street's 1991 paper Parity Complexes (apologies; I don't know how to find DOI links for Cahiers papers) develops some very useful tools for working with free strict $\omega$-categories. There is ...

8
votes

1
answer

800
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### Compactly supported sections of coherent sheaves and the dualizing complex

Suppose $U$ is a (possibly singular) scheme and $X$ is a compactification (potentially unnecessary at least in characteristic $0$). Let $\pi:X\to *$ be the map to the point (though one can consider ...

5
votes

1
answer

187
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### Are $E_k$ monoids higher categories?

The May Recognition Theorem establishes an equivalence between the $\infty$-categories
The $\infty$-category of grouplike $E_n$ monoids
The $\infty$-category of pointed $(n-1)$-connected spaces
...

7
votes

1
answer

328
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### The contravariant mapping space represented by a homotopical classifying space (e.g. BG)

In classical homotopy theory, there are a number of spaces which are important because they represent an interesting functor on $\operatorname{Ho(Top)}$; for example, $K(G,n)$ represents singular ...

5
votes

1
answer

334
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### 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 ...

9
votes

1
answer

506
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### Any news about equivalences of periodic triangulated or $\infty$-categories?

There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted ...

11
votes

1
answer

537
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### What are the conjugacy classes of the category of ($\kappa$-small) sets?

$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such ...

5
votes

2
answers

432
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### 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, ...

4
votes

0
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### Why is this class of right-anodyne maps closed under pushouts?

Let $S$ be the class of all right-anodyne maps $r$ such that the pullback of $r$ along any left fibration is again right-anodyne. According to Land's book on $\infty$-categories (specifically the ...

2
votes

0
answers

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### Why is $\operatorname{Hom}^R_{\mathcal{C}}(X, Y )$ the fiber of $\mathcal{C}_{\backslash Y} \to \mathcal{C}$ (Lurie's HTT)

Reading Jacob Lurie's Higher Topos Theory I not understand the proof of the "only if" part in Proposition 1.2.12.4. It states
Proposition 1.2.12.4. Let $\mathcal{C}$ be an $\infty$-category ...

2
votes

1
answer

92
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### Fibrations of fibrant marked simplicial sets

Let $\mathrm{sSet}^+ = \mathrm{sSet}^+_{/ \Delta^0}$ be the model category of marked simplicial sets over the point. By Theorem 3.1.5.1 in Higher Topos Theory, this model category is Quillen ...

6
votes

0
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### $\mathbb{E}_\infty$-refinements of the graded tensor product of $\mathbb{Z}$-graded spectra

The category
$$\mathsf{Gr}_\mathbb{Z}\mathsf{Mod}_R\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes(\mathbb{Z}_\mathsf{disc},\mathsf{Mod}_R)$$
of $\mathbb{Z}$-graded $R$-modules has a natural monoidal ...