# Questions tagged [infinity-categories]

The infinity-categories tag has no usage guidance.

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### How should one approach reading Spectral Algebraic Geometry by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory. This MathOverflow question asked for a roadmap to Lurie's Higher Algebra. Still another question asked for a ...

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**1**answer

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### How should one approach reading Higher Algebra by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT ...

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**1**answer

224 views

### Is it possible to define a linear $A_\infty$-category as a special kind of an $\infty$-category?

A functor $N\colon\mathrm{Cat}_{A_\infty}\longrightarrow\mathrm{Cat}_\infty$ is constructed in a paper [1] by Faonte. This gives a way to get an $\infty$-category by starting with an $A_\infty$-...

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votes

**1**answer

191 views

### Link between homotopy equivalence of simplicial sets and categorical equivalences

In Higher Topos Theory, a map $f: S \rightarrow T$ of simplicial sets is a categorical equivalence if after applying the functor $\mathfrak{C}[-]$ we have an equivalence of simplicial categories.
In ...

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### Section 2.4.4 Higher Topos Theory

In section 2.4.4 of Lurie's Higher Topos Theory, it is said multiple times that using
Proposition: Let $p : \mathcal{C} \rightarrow \mathcal{D}$ be an inner fibration of $\infty$-categories. Let $...

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votes

**1**answer

159 views

### Precise reference for the equivalence of $E_n$ algebras and locally constant factorization algebra?

I've seen the following theorem attributed to Lurie:
Theorem. There is an equivalence of $(\infty,1)$-categories between $E_n$ algebras and locally constant factorization algebra on $\mathbf{R}^n$.
...

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**1**answer

234 views

### Functoriality of (co)limits in $\infty$-categories

I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) ...

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### Equivalences of categories of sheaves vs categories of $\infty$-Stack

Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e.
$$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$
And we want to ...

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### Reference Request: A “Chevalley-Eilenberg”-style formulation of the $L_\infty$ algebra minimal model theorem?

The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...

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### Free symmetric monoidal category of compactly generated category is compactly generated

Let $k$ be a field and let $\mathcal{C}=\mathbf{StLin}_k$ be the $\infty$-category of stable infinity categories enriched over the $\infty$-category $\mathbf{Vect}_k$, regarded as a symmetric monoidal ...

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### Testing for equivalences of $\infty$-categories on strictifications?

It is in general not too hard to show that maps between finite $CW$-complexes/finite simplicial sets are homotopy equivalences.
Question : Can we do something similar for:
quasi-categorical ...

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### Is there any survey of dg-categories from the $\infty$-category point of view?

I was reading this question on dg-categories and a comment by David Ben-Zvi says "An excellent pre-$\infty$-categorical overview is Keller's ICM address https://arxiv.org/abs/math/0601185".
I was ...

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### Is $Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))$ for an additive category $\mathcal{C}$?

Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category.
...

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### Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures

I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.
Suppose we are given a (strict) pullback square
...

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### Colimits of algebras for $\infty$-Monad

I would like to know in anyone has developed method for constructing colimits in the category of algebra for a monad in the $(\infty,1)$-categorical framework, using transfinite constructions.
I have ...

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### Localization of $\infty$-categories

In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...

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284 views

### Physical consequences of cobordism hypothesis?

Let $C$ be a symmetric monoidal $n$-category. An extended framed $C$-valued TQFT is a symmetric monoidal functor from the framed bordism category $\mathrm{Cob}^{fr}_n(n)$ to $C$.
The cobordism ...

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### Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack

I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.
...

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**1**answer

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### Coefficient (or target) category for factorization homology

In the article "Factorization homology of topological manifolds" by Ayala and Francis, a symmetric monoidal $\infty$-category $\mathcal{V}$ is fixed as the target or coefficient category. This ...

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**1**answer

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### Simplicial nerve functor commutes with opposites

There are two "opposite" functors:
$$ op_\Delta\colon sSet\to sSet$$
and
$$op_s\colon sCat\to sCat.$$
The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite ...

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**2**answers

310 views

### How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor
$$
d: \mathcal{C} \longrightarrow \mathcal{C}^{op}
$$
such that $d(x)$ is dual to $x$ for ...

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**1**answer

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### 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 ...

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395 views

### Floer cohomology from mapping spaces of $\infty$ categories

There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...

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422 views

### Uniqueness of quasi-inverses in infinity categories

I've been trying to learn some of the basic language of infinity-category theory (in the sense of Lurie), and in particular, to understand which basic statements in (1-)categories have analogues in ...

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votes

**1**answer

208 views

### Homotopy limit of model categories in the category of categories

Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or ...

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votes

**2**answers

263 views

### A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)

Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the ...

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**1**answer

232 views

### Compatibility of Grothendieck construction with pullback

Suppose $D$ is an $\infty$-category, then we have the equivalence
$$ \text{Fib} (D) \substack{ \text{St} \\ \longrightarrow \\ \cong \\ \longleftarrow \\ \text{Un}} [ D^\text{op}, \mathbf{Kan}]$$
...

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### Construction for algebras over little cubes operad

Recently I came across the following construction:
Fix a dimension $k$. Let $C$ denote the space whose points are disjoint rectilinear embeddings $c\colon I^k\to \mathbb R^k$ of the (closed) $k$-...

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### Suspensions are H-cogroup objects

Do you know any reference where you have a formal justification for the following statement that appears in nLab?
https://ncatlab.org/nlab/show/suspensions+are+H-cogroup+objects
"Let $\mathcal{C}$ ...

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### Comonadicity of spaces over spectra?

As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma^...

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### Can an enriched functor be expressed as a colimit of representable functors?

Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors ...

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### Interpretations of Whitehead's $\Gamma_n$ functors

(This is related to my earlier question on Kan's simplicial formula as Curtis mentions the link with the Hopf map, which has a very pretty formula that links well with the Samelson / Whitehead ...

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### When is the derived category $D(A)$ locally cartesian closed?

Let $D(A)$ be the derived $(\infty,1)$-category of some abelian category $A$. For which $A$ is $D(A)$ locally cartesian closed?
Replace $D$ with $D^b$ or similar if appropriate.
I essentially want ...

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**1**answer

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### Is an ∞-topos of local homotopy dimension $\leq n$ of homotopy dimension $\leq n$?

[All references are wrt to Lurie's "Higher Topos Theory" in its latest online available version (March 10, 2012)]
Definition 7.2.1.8:
An ∞-topos $X$ is locally of homotopy dimension $\leq n$ if there ...

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228 views

### relative spectrum in derived algebraic geometry

I am trying to understand how much it is possible to extend the notion of spectrum of a qcoh sheaf of algebras to stacks.
More precisely, given a scheme $S$ and a stack $F$ of cohomological cdga's ...

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**1**answer

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### Example of a (presentable $k$-linear $\infty$-)category which is dualizable but not compactly generated?

Is there an example of a presentable, stable, $k$-linear $\infty$-category which is dualizable but not compactly generated, where $k$ has characteristic zero, and which is $\text{QCoh}(X)$ (by which I ...

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**1**answer

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### 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 ...

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votes

**1**answer

155 views

### When the global section functor is a Cartesian fibration?

Given a Cartesian fibration $p : \mathbf{E} \to \mathbf{B}$ over an $\infty$-topos the paper by Marc Hoyois mentioned in his answer to this question gives some sufficient conditions for $\mathbf{E}$ ...

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### Generalizations of tangent $\infty$-topos

If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\...

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### Group objects in $\infty$-categories

A groupoid object in an $(\infty,1)$ category $\mathcal{C}$ is a functor $G:N(FinSet)^{op} \to \mathcal{C}$ such that for any partition $[n]=S \cup S'$ intersecting in $s$, the object $G([n]$ is the ...

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### Cofiber of the inclusion of an $E_0$-algebra $M$ into the free $E_k$-algebra generated by it

Let $\mathcal{C}$ be the $E_k$-monoidal $\infty$-category of left modules over a fixed connective $E_{k+1}$-ring spectrum $A$. Suppose that $M$ is an object of $\mathcal{C}$ which is an $E_0$-algebra, ...

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### (Reference Request) Tensor product of chain complexes in terms of strict $\infty$-categories

(note: this question is essentially a reference request for the tensor product described at the end. the rest is context)
It is well known that the category of chain complexes (in positive degree, ...

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**1**answer

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### On HTT's Lemma 3.3.4.1

While studying the book Higher Topos Theory I have encountered some difficulty with Lemma 3.3.4.1, which says that the pullback along a cartesian fibration of a map q such that $q^{op}$ is cofinal is ...

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240 views

### Bar construction and the $\infty$-categorical Barr-Beck theorem

I am studying the proof of the $\infty$-categorical version of the Barr-Beck theorem in Lurie's Higher Algebra, but there is a step of the proof that is puzzling me. In Lemma 4.7.3.13, a simplicial ...

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### Functor of points definition of the Thom space

Let $X$ be a space (CW complex) and let $E \to X$ be a vector bundle.
Using the language of $\infty$-categories we can can define the Thom space $T(E)$ as the pointed space representing the ...

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### If the homotopy category is well-generated, must the $\infty$-category be presentable?

Suppose $\mathcal{C}$ is a stable $\infty$-category whose homotopy category is a well-generated triangulated category in the sense of Neeman's book. Must $\mathcal{C}$ be a presentable $\infty$-...

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### Complexes in stable categories

Generalizing from 1-category theory, there's a simple definition of a "naive complex" in a stable $\infty$-category. Considering bounded positive graded chain complexes, they are a sequence of maps
$$...

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### What's a (infinity-) semi-stack?

A stack is an object that mixes the notions of (algebraic) space and group. The key insight of stack theory is that most things you would want to do with spaces you can do with stacks: namely, you ...

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### Left adjoint of $I\colon \mathrm{Kan}\hookrightarrow\mathrm{WeakKan}$?

The inclusion $I\colon \mathbf{Grpd}\hookrightarrow\mathbf{Cat}$ of groupoids into categories has both a left and a right adjoint $L,R\colon \mathbf{Cat}\to \mathbf{Grpd}$, with $R(C)$ being largest ...

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### When does p-profinite completion commutes with maps from a $p$-finite space?

background
Let $\mathcal{S}$ be the ($\infty$-)category of spaces and $\mathcal{S}_{p-\text{finite}}$ the full subcategory spanned by the $p$-finite spaces (that is, the spaces with finitely many ...