# Questions tagged [higher-category-theory]

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

**14**

votes

**2**answers

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

**3**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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 ...