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Questions tagged [higher-category-theory]

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

14
votes
2answers
502 views

What are the tangent $\infty$-categories to $\mathrm{Top}^\mathrm{op}$ and $E_\infty$-$\mathrm{Ring}^\mathrm{op}$?

Given an $\infty$-category $\mathcal{C}$ with finite limits and finite colimits, there are two ways to make it stable. One is to pass to the pointed objects and take the category of $\Omega$ objects $\...
13
votes
3answers
517 views

What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?

Alternate formulation of the question (I think): What's a precise version of the statement: "In a stable $\infty$-category, finite limits and finite colimits coincide"? Recall that a stable $\infty$-...
3
votes
0answers
62 views

Extending morphisms in an $A_{\infty}$ category to natural transformations

Suppose we are given a small (enriched) category $C$, and for $a,b \in C$ an isomorphism $m:a \to b$. It is always possible to find a functor $F: C \to C$, with $F(a)=b$ and a natural transformation $...
5
votes
0answers
118 views

“Characteristics” (thick subcategories) in $n$-groupoids

$ \newcommand{\Ab}{\mathbf{Ab}} \newcommand{\Sp}{\mathbf{Sp}} $In 0-groupoids (sets), the thick subcategories of the category of abelian groups $\Ab$ are given by the primes $p$ and $0$, which we can ...
23
votes
2answers
2k views

Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy programme

(Disclaimer: I'm no expert in homotopy theory nor in higher categories!) If I understand it correctly, Grothendieck's homotopy hypothesis states that there should be an equivalence (of $(n+1)$-...
2
votes
0answers
83 views

Classification of unitary pointed monoidal category

I wonder if the following classification results are true (and are there any references): Unitary pointed monoidal categories (the fusion rule of the objects is given by a finite group $G$) are ...
19
votes
0answers
449 views

Is the category Idem filtered?

I have recently been reading Lurie's "Higher Topos Theory", and come upon what I believe to be an erronous claim. However, the author goes to some pain as a result of that claim, and the error seems ...
33
votes
0answers
753 views

Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?

Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...
7
votes
1answer
323 views

Homotopy theory of non-test categories?

Let $\mathcal{C}$ be a category with pullbacks, and consider the functor $i_\mathcal{C}: \mathcal{C}^\mathrm{op} \to \mathsf{Cat}$, $X \mapsto \mathcal{C}/X$. We can equip $\mathcal{C}$ with a class $\...
7
votes
2answers
204 views

Algebras in a bicategory of spans

$ \newcommand{\Span}{\mathbf{Span}} \newcommand{\cE}{\mathcal E} $Given a category $\cE$ with plenty of limits, let $\Span(\cE)$ denote the bicategory of spans in $\cE$. It is known that monads in $\...
8
votes
0answers
208 views

Identifying and reconstructing the derived category from its auto-equivalences

Background: Given a smooth irreducible algebraic variety $Y$ with $\omega_Y$ or $\omega_Y^{-1}$ ample. Then Bondal-Orlov theorem states that if there exists any other smooth algebraic variety $Y'$ ...
5
votes
2answers
296 views

Higher and lower analogues of Yoneda's lemma

Here's a statement of Yoneda's lemma for n-category. Let C be a n-category and $C^{\wedge}=[C^o,n-1Cat]$ be the n-category of presheaves on C. $C^o$ is the opposite n-category of C and $n-1Cat$ is ...
7
votes
2answers
288 views

Explicit generating acyclic cofibrations and right properness of a model category

Let $\mathcal{C}$ be a cofibrantly-generated model category. My impression is that the following two conditions are highly correlated: $\mathcal{C}$ is right proper. There is an explicitly-...
2
votes
0answers
92 views

Does twisted arrows commute with the simplicial nerve construction?

Let $\mathcal{C}$ be a simplicial category, and let $N(\mathcal{C})$ be its simplicial nerve. We can form the category of twisted arrows as a simplicial category $TwArr(\mathcal{C})$ Now Lurie's ...
12
votes
0answers
246 views

How can you unitalize a higher category?

Given an associative nonunital algebra $A$, there are (at least) two standard ways to produce a unital algebra $A'$ together with a map $A \to A'$. Following the discussion in the comments below, ...
5
votes
0answers
113 views

Dense (∞,1)-subsites

So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of ...
11
votes
2answers
466 views

Categorical mapping class group action

Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A_\infty$-) category which admits a faithful action $$ MCG(\Sigma) \to Auteq(\mathcal{C})$$ by ...
1
vote
1answer
161 views

Morphisms of cotensors

Let $\mathcal{V}$ be a closed monoidal category, and $\mathscr{C}$ be a category enriched over $\mathcal{V}$. One says that the power or cotensor of an objec $A \in \mathscr{C}$ by an object $U \in \...
3
votes
1answer
140 views

A smash product of an inner anodyne map with a cofibration is inner anodyne

I'm reading through Lurie's Higher Topos Theory and I'm not conviced by the proof of corollary 2.3.2.4. it asserts that it $i:A\rightarrow A'$ is inner anodyne and $j:B\rightarrow B'$ is a cofibration,...
3
votes
0answers
175 views

What is a proper n-etale morphism?

Let $Y$ be a complex algebraic variety, and let $n\in \mathbb Z_{\geq 1}\cup \{\infty\}$. How do I think about a proper $n$-etale morphism $X\to Y$? If $n=1$, I think this should be a finite etale ...
2
votes
0answers
192 views

Interesting examples of large, accessible, non-presentable $\infty$-categories?

What are some interesting examples of accessible $\infty$-categories which are not presentable and not small? By interesting I mean a category which comes up naturally in a certain context and in a ...
1
vote
0answers
62 views

Underdetermined Polynomial $(\infty ,1)$-Functors

Is there any sense in which the full subcategory of n-excisive functors (on spectra or on pointed spaces) on those functors $F$ which satisfy a set of distinct (modulo homotopy equivalence) equations $...
4
votes
2answers
389 views

Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories

Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions ...
4
votes
1answer
249 views

Do coproducts of infinity-groupoids commute with pullbacks?

As stated in this question, coproducts commute with pullbacks in the category of sets. Let $Grpd_{\infty}$ denote the $\infty$-category of $\infty$-groupoids. Do coproducts commute with pullbacks in $...
4
votes
0answers
141 views

Are there obstructions to refining partially coherent natural transformations up-to-homotopy between quasicategories?

EDIT: It looks like I didn't define the simplicial set $R^{\sim \Delta^1}$ clearly below. It's not simply $\mathrm{Ho}(R)^{\Delta^1}$. Rather, an $n$-simplex of $R^{\sim \Delta^1}$ is a 1-simplex of $\...
4
votes
0answers
43 views

Coherence laws when composing 2-monads

To have the composition of two monads be a monad itself, we need a distributive law natural transformation satisfying certain coherence laws. I'm interested in the strict 2-monad case, i.e. a strict ...
5
votes
1answer
300 views

Colimits of cofibrations and homotopy colimits

Say C is a left proper model structure. I have a diagram where all maps are cofibrations. Is its colimit a homotopy colimit? I know this is true for pushouts. Is it true for sequential colimits? ...
3
votes
1answer
249 views

The “Family” functor for infinity-categories

Background Given a category $C$, one can consider the category $Fam(C)$ of set-indexed familiies of objects in $C$. Formally, the objects are pairs $(X,F)$ in which $X$ is a set and $F:X\rightarrow C$...
4
votes
0answers
164 views

Terminology for filtered $\infty$-categories

Often to prove that a simplicial set $X_\bullet$ is a contractible $\infty$-groupoid, we instead prove that $X_\bullet$ is a contractible Kan complex (i.e. satisfies the extension property for ...
8
votes
2answers
346 views

Is there a model structure on (strict?) Monoidal Categories?

Basically, what the title says. Presumably, one could use the fact that monoidal categories (resp. strict monoidal categories) are one-object bicategories (resp. 2-categories) and use the Lack model ...
2
votes
0answers
162 views

Connected families of objects in $(\infty,1)$-categories?

Background Let $C$ be an (extensive) category. An object $X\in C$ is called connected if the functor $Hom(X,-):C\rightarrow Set$ preserves coproducts. Given a category $C$, one can consider the ...
1
vote
0answers
100 views

Localizations and their underlying groupoids

Fix $\mathcal C$ an $\infty$-category and $W \subset \mathcal C$ the inclusion of an $\infty$-category containing all weak equivalences in $\mathcal C$. The localization along $W$ is then (a Joyal-...
7
votes
0answers
169 views

$S^3$ partition function from the 1-category of $S^1$ in Chern Simons theory

I am trying to learn about some categorical aspects of topological quantum field theories. For concreteness, I am considering Chern Simons theory with gauge group $G$ in three dimensions. As I ...
2
votes
1answer
222 views

Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck construction?

Given a locally presentable ($\infty$,1)-category $C$, can the fibrewise stabilization of it's codomain fibration, also called its tangent category $TC$, be given in terms of the Grothendieck ...
2
votes
1answer
273 views

Colimits in the category of simplicial categories

A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (...
7
votes
2answers
257 views

Equivalent definitions of Cartesian Fibrations between Quasi-Categories

In the paper by Verity and Riehl "Fibrations and Yoneda lemma in an ${\infty}$-cosmos" (https://arxiv.org/abs/1506.05500), they prove a Yoneda lemma that holds in any ${\infty}$-cosmos (see Corollary ...
3
votes
0answers
85 views

Is there a construction capturing indexed families of adjunctions?

I'm sorry in advance if this question does not belong on this site. I am curious as to what is "really" going on when you have a family of functors indexed by elements in a base category, all of which ...
2
votes
1answer
217 views

Is the property of being a dg generator open?

Suppose $\mathcal{C}$ is a dg category (over some base) with all colimits. We say that $X\in \mathcal{C}$ is a generator if $\mathcal{C}$ is equivalent to $\operatorname{End}_\mathcal{C}X$-modules (...
12
votes
0answers
294 views

Is there an analog of Kan's $Ex_\infty$ functor for quasicategories?

Is there a fibrant replacement functor in the Joyal model structure which can be described non-recursively, like $Ex_\infty$ for the Quillen model structure? I believe another way to put this is to ...
3
votes
1answer
121 views

Could $RHom(A,-)$ distinguish non-quasi-equivalent dg-categories?

One of the remarkable results in Toen's paper is the existence of internal homs of dg-categories. Actually for two dg-categories $A$ and $B$, there exists a dg-category $RHom(A,B)$ such that for any ...
2
votes
0answers
99 views

Why is the stabilization of augmented $\mathbb{E}_\infty$-algebras equivalent to $k$-module spectra?

(I have already asked this on Math.SE, but it didn't draw much attention there, so I am reposting it here.) Example 1.1.4 of Jacob Lurie's DAGX says that the stabilization $\operatorname{Stab}((\...
2
votes
0answers
77 views

Pullback along a Lax Functor

Consider a 2-category of bicategories, lax functors and transformations (I'm actually using double categories but the result should be the same). Given a cospan $\mathbb{C} \to \mathbb{E} \leftarrow \...
0
votes
2answers
203 views

Higher category theory (invertible morphism, Topological categories, monoidal n-category) [closed]

I'm Looking at a paper using higher category theory. Would you please answer these questions ? 1) A weak n-category (or $\infty$-category) has objects and k-morphisms. What is an invertible morphism ?...
2
votes
0answers
106 views

homotopy quotient categories [closed]

(Trying to rephrase an earlier question) In topology, a continuous map $f: X \to Y$ has a ``homotopy cofiber" $Cf$ included in a cofiber sequence $$ X \to Y \to Cf \to \Sigma X \to \Sigma Y \to \...
8
votes
0answers
204 views

Framed higher Hochschild cohomology

Given an $E_n$-algebra $A$, one can define its $E_n$-Hochschild complex $CH_{E_n}(A,A)$ by the formula $$Ch_{E_n}(A,A)=RHom_{Mod_A^{E_n}}(A,A)$$ where $Mod_A^{E_n}$ is the category of $A$-modules over ...
0
votes
0answers
94 views

Posets as (0,1)-categories

I am reading on the nLab that a poset can be seen as a (0,1)-category. I was assuming all along that an ($n$,$r$)-category were a category where all morphisms of order larger than $n$ are trivial. ...
0
votes
1answer
81 views

Equality of lax natural transformations in the constructive approach

In the constructive approach to category theory, a category comes equipped with an equality (an equivalence relation) between its morphisms but not between its objects. Let C and D be such categories,...
3
votes
0answers
131 views

Can representable presheaves be made injectively fibrant?

I suspect that the answer to my question is no, but let me give it a shot anyway. If $\mathcal{A}$ is a small simplicially enriched category, then the category of simplicial presheaves $\mathsf{sSet}^...
2
votes
0answers
134 views

Homotopy invariance of the moduli stack of smooth $G$-bundles?

This question ought to have a straightforward (perhaps even glaringly obvious) answer, but so far I've already wasted a few hours trying to untangle this web of inconsistent identifications. I'm sure ...
0
votes
0answers
178 views

The homotopy pullback of a point along itself is the loop space

I have seen on the nLab that we can view the loop space as a particular homotopy pullback, and that it is even the way a "loop space object" is defined in general (when it exists). Can someone give ...