# Questions tagged [higher-category-theory]

For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

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### Sheaves over a sheaf

Everything I write I mean in the in the sense of Lurie's HTT.
Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C}...

**16**

votes

**2**answers

821 views

### A multicategory is a … with one object?

We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly ...

**6**

votes

**1**answer

366 views

### Need help understanding comment in Higher Topos Theory

I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
Lemma 2.4.4.1. Let $p : \mathcal{C} \rightarrow \mathcal{...

**7**

votes

**0**answers

95 views

### Stability of accessible $\infty$-categories under some operations

I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.
In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...

**2**

votes

**1**answer

68 views

### Factorization of a map from a contractible Kan complex through a Kan complex

Suppose we are given a contractible Kan complex $S$ and a map of simplicial set $f : S \rightarrow T$. Under what conditions can we say that $f$ factors through the largest Kan complex $Z$ contained ...

**4**

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89 views

### Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?

Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...

**5**

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**1**answer

260 views

### Remark 2.4.1.4 Higher Topos Theory

In HTT, given a inner fibration $p : X \rightarrow S$ of simplicial, an edge $f : x \rightarrow y$ of the simplicial set $X$ is said to be a $p$-Cartesian if the induced map
$$ X_{/f} \rightarrow ...

**9**

votes

**1**answer

514 views

### Higher Topos Theory Theorem 2.2.5.3

The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.
We have a trivial Kan ...

**6**

votes

**1**answer

212 views

### Does the Dwyer-Kan model structure make dgCat a model $2$-category?

Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...

**5**

votes

**1**answer

271 views

### Universal property of sheaf category

Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...

**9**

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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) ...

**5**

votes

**1**answer

189 views

### Comonad for normalized pseudofunctors for strict higher categories

Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{...

**5**

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**0**answers

142 views

### Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?

In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$
Joyal's isofibration theorem says precisely
An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model ...

**17**

votes

**1**answer

313 views

### 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 ...

**5**

votes

**1**answer

285 views

### A question about HTT Lemma 5.5.2.1

I have a question about the statement of Lemma 5.5.2.1 in Lurie's `Higher Topos Theory'.
``Let $S$ be a small simplicial set, let $f: S\rightarrow \mathcal{S}$ be an object of $\mathcal{P}(S^{op})$, ...

**4**

votes

**1**answer

191 views

### A finite Whitehead Theorem for $\infty$-topos

Let's consider in an $\infty$-topos, we have an object $X$ of homotopy dimension $\leq n$ (in the sense of Lurie HTT), let $f: A\to B$ be an $n$-equivalence morphism. Can we conclude that $f$ induces ...

**6**

votes

**2**answers

610 views

### Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....

**2**

votes

**0**answers

93 views

### Compact generation of quasicoherent sheaves on mapping stack

Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...

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vote

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92 views

### Image of morphism of quasi-categories

I have two questions about images of morphisms of quasi-categories.
Suppose that $f\colon X \to Y$ is a morphism of quasi-categories.
Suppose that we calculate the image of $f$ in the category $\...

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**0**answers

136 views

### Higher homotopical information in racks and quandles

A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold.
Q1. a $\star$ a = a
Q2. (a $\star$ b) $\bar\star$ b = (a $...

**8**

votes

**2**answers

210 views

### “Closed bicategories”

I am interested in the following property that a bicategory may or may not have.
Let $\mathbf{B}$ be a bicategory. Every one-morphism $f\colon x\rightarrow y$ defines a functor $\mathbf{B}(y,z)\...

**4**

votes

**0**answers

90 views

### 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 ...

**13**

votes

**1**answer

335 views

### Why are finite cell complexes also finite as infinity-categories?

A quasicategory ($\infty$-category) $\mathcal{C}$ is finite if there is a finite simplicial set $K$ and a categorical equivalence $K\rightarrow\mathcal{C}$.
On the other hand, a Kan complex (space) $...

**5**

votes

**1**answer

115 views

### Inductive folk model structure on strict ω-categories

There is a paper of Lafont, Metayer, and Worytkiewicz [1] that constructs a model structure on the category of strict $\omega$-categories that they call the folk model structure. This model structure ...

**7**

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**0**answers

166 views

### 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 ...

**3**

votes

**0**answers

161 views

### Tannaka duality for $DG$ indschemes

In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism
$$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$
where $X$ and ...

**3**

votes

**1**answer

129 views

### Cellularity of anodyne extensions?

Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)
If not, is there a known counterexample?
Similarly, does ...

**21**

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**1**answer

270 views

### Explicit Left Adjoint to Forgetful Functor from Cartesian to Symmetric Monoidal Categories

There is a forgetful functor from the category of (small) Cartesian monoidal categories (a symmetric monoidal category in which the tensor product is given by the categorical product) to the category ...

**8**

votes

**2**answers

313 views

### What is a spectrum object in $\infty$-topoi?

For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?
To ...

**3**

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**0**answers

148 views

### 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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228 views

### What are the monomorphisms of ($\infty$-)toposes?

There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...

**11**

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**1**answer

266 views

### Are there continua in $\infty$-topoi?

If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence....

**3**

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**1**answer

246 views

### Why must the essential image break the principle of equivalence?

I'm having trouble understanding why the "essential image" is defined the way it is.
The nlab article gives the following definition:
(A concrete realization of) the essential image of a functor $...

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212 views

### 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
...

**9**

votes

**1**answer

420 views

### Spectral and derived deformations of schemes

I'd like to understand how ordinary schemes deform or lift to spectral and derived schemes in two basic examples as well as what the structure of the space of deformations in general is.
Let $S = (X, ...

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votes

**2**answers

843 views

### What parts of the theory of quasicategories have been simplified since the publication of HTT?

It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression ...

**7**

votes

**4**answers

396 views

### 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 ...

**3**

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**0**answers

177 views

### Cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$

Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group.
(For example, given a short exact sequence
$$
1 \to BG_2 \to \mathbb{G} \to G_1 \to 1
$$
and the fiber sequence:
$$
B^2G_2 ...

**2**

votes

**1**answer

154 views

### Relaxing a natural isomorphism to a natural transformation to obtain a more general $2$-category

Many definitions of $2$-categories are given as categories equipped with some extra structure encoded by some functors and some natural (or extranatural) transformations between these functors (or ...

**7**

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**1**answer

156 views

### Direct comparison from the Rezk hom to the hom of a simplicial category along the coherent nerve?

Consider the following construction: Define $G_n$ to be the contractible groupoid on $n+1$ objects. Choosing a linear order on the objects of each $G_n$ turns $G_*$ into a cosimplicial object. ...

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votes

**0**answers

142 views

### A naive question about representations of group stacks

For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with ...

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**0**answers

212 views

### In what context can enriched category theory be done?

There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.
My question is what ...

**6**

votes

**1**answer

285 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 ...

**10**

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**0**answers

217 views

### 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.
...

**7**

votes

**1**answer

195 views

### Monoidal structures on modules over derived coalgebras

Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can ...

**4**

votes

**1**answer

138 views

### What is the idempotent completion of the (2,1)-category of spans of finite sets?

I don't believe the $(2,1)$-category $FinSpan$ has split idempotents.
Question: Is there a simple description of the idempotent completion of $FinSpan$?
Foundationally, we may think of $FinSpan$ as ...

**5**

votes

**0**answers

97 views

### Internal hom as 2-Kan lift of pseudofunctor

Consider a situation where there is a pseudofunctor from some category $C$.
Due to it's geometrical properties, it might induce some sort of an internal hom on the original category (making it a ...

**7**

votes

**2**answers

356 views

### What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

A paradox:
Goodwillie calculus considers only finitary functors.
$TC$ isn't finitary.
Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(...

**7**

votes

**1**answer

161 views

### 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 ...

**8**

votes

**0**answers

101 views

### Is every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?

Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For ...