Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
6
votes
1answer
192 views
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
2answers
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
1answer
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
0answers
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
1answer
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
votes
0answers
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
votes
1answer
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
$$ X_{/f} \rightarrow ...
9
votes
1answer
514 views
Higher Topos Theory Theorem 2.2.5.3
The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.
We have a trivial Kan ...
6
votes
1answer
212 views
Does the Dwyer-Kan model structure make dgCat a model $2$-category?
Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
5
votes
1answer
271 views
Universal property of sheaf category
Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...
9
votes
1answer
234 views
Functoriality of (co)limits in $\infty$-categories
I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) ...
5
votes
1answer
189 views
Comonad for normalized pseudofunctors for strict higher categories
Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{...
5
votes
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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}(...
1
vote
0answers
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 $\...
10
votes
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
votes
0answers
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
0answers
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
1answer
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
votes
1answer
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
2answers
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
votes
0answers
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.
...
10
votes
0answers
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
votes
1answer
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
votes
1answer
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 $...
9
votes
0answers
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
1answer
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, ...
36
votes
2answers
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
4answers
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
votes
0answers
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
1answer
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
votes
1answer
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. ...
3
votes
0answers
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 ...
8
votes
0answers
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
1answer
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
votes
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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 ...
