Questions tagged [higher-category-theory]

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

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5
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0answers
60 views

Explaining the “free left fibration” functor for infinity categories

This is a cross-post from here I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \...
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0answers
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A couple of points in a proof about of $\infty$-toposes

I wanted to have a better understanding of the geometric interpretation of $\infty$-toposes, and in particular learn something about étale morphisms, but I got stuck trying to unravel two points in ...
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2answers
215 views

Path integral derivation of extended TQFT

I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of ...
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53 views

Applications of Day convolution for monoidal bicategories

Day convolution is a very powerful tool to build monoidal structures on categories of functors from a pro/monoidal $\mathcal{V}$-category. For instance, it is used in stable homotopy theory to ...
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1answer
115 views

Steenrod operations from the delooping viewpoint

Let $F$ be a finite field, $Sq^i$ be the $i$-th Steenrod operation $$ H^*(-;F) \to H^{*+i}(-;F).$$ By Yoneda lemma, such operation is a map $\phi_i: B^{*}F \to B^{*+i} F$, where $B$ denotes the ...
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0answers
120 views

Bar-cobar for topological spaces

Let $(A, \{m_k\}_{k \geq 1} )$ be an $A_{\infty}$ algebra over a field $k$. Recall that this is the data of a $\mathbb{Z}$-graded $k$-vector space, along with a collection of $k$-nary operations which ...
3
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1answer
175 views

What can be an appropriate notion of principal bundle over a category (with an appropriate notion of local trivialisation)?

Motivation for my question: It is a well-known fact that there exists a bijection between the set of isomorphism class of principal $G$ bundles over a nice topological space $X$ and the set $[X,B'G]$...
3
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Can chain homotopy induce space homotopy at $E_\infty$ level?

Space-level homotopy induces (co)chain homotopy, but I've never heard of the converse. I am not sure if it is simply not true? However, for good enough spaces (finite type nilpotent), Mandell proved ...
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3answers
379 views

What is the geometric realization of the the nerve of a fundamental groupoid of a space?

It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows: Obj: $X \mapsto \pi_{\leq 1}(X)$, ...
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152 views

The $\infty$-category of natural transformations as an end

Let $\mathcal{C}$ be an $\infty$-category viewed as a fibrant scaled simplicial set with all 2-simplices thin and let $\mathfrak{C}\!at_{\infty}$ be the $\infty$-bicategory of $\infty$-categories. A ...
4
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1answer
294 views

What is the notion of a group object and its action in a 2-category?

It is well known that a group object in a category $C$ (with terminal object $1$ and such that any two objects of $C$ have a product) is defined as an object $G$ in $C$ with the following morphisms: $...
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Does forgetting colimits preserve colimits?

For each regular cardinal $\kappa$ let $\operatorname{Cat}_{\kappa}$ be the $(2,1)$-category of small categories with $\kappa$-small colimits, and functors that preserve those colimits. For each pair ...
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116 views

Confusion in understanding the notion of $G$ Principal bundle where $G$ is a geometric group over a site

The first paragraph of the section Overview in the paper Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson https://arxiv.org/abs/1207.0248 precisely reads the following: ...
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1answer
133 views

Can a locally presentable category have a proper class of accessible localizations?

Question: What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D_i)_{i < ORD}$? In other ...
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Categorification of “Every domain embeds into a field”?

In the category of commutative rings, every domain embeds into a field. Is this true in the category of presentably symmetric monoidal stable $\infty$-categories? Here's what I mean by that. Let $...
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1answer
73 views

What is the definition of a prorelation?

In the context of quasi-uniform spaces, what is a prorelation? In the text I'm reading, they're defined as a down-directed upper set on relations X->Y. Now, I'm fine with a down-directed up-set, but ...
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100 views

A Whitehead theorem for 3-categories

Let $F:\mathscr{C}\rightarrow \mathscr{D}$ be a 3-functor between 3-categories. Are the following two properties known to be equivalent? $F$ is a 3-equivalence, meaning that there is a 3-functor $G:\...
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2answers
304 views

Can conservativity depend on the universe?

Question 1: Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...
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0answers
28 views

Terminology for cubical boundary

In globular higher category theory, an $(n+1)$-cell is shaped like the $(n+1)$-disk $D^{n+1}$, and its boundary has the shape of the $n$-sphere $S^n$, given by two glued $n$-disks. In cubical higher ...
8
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1answer
218 views

DW, state sum models, and fully extended TQFTs

I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-...
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0answers
61 views

Oplax monoidal functors of $\infty$-categories

In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...
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Is there a n-category structure on algebras for $e_n$-like operads?

I'm fishing in troubled waters here and therefore the question is vague and meant to be as general as possible. In particular "$e_n$-like operad" can be an algebraic or topological $e_n$ operad, as ...
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$E_\infty$-maps of diagrams

I'm asking my question for a general symmetric monoidal $\infty$-category $ V$ and a general indexing simplicial set $I$, but my specific interest is for $ V = Sp$ with the usual smash product and $I =...
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Tensor algebras in the bicategory $\mathsf{2Vect}$

To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both ...
6
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1answer
192 views

Localization of symmetric monoidal categories and geometry

I have a series of vague questions, related to localization of symmetric monoidal categories. Here is the context. Say we are working over a field of characteristic zero. Then the "one category ...
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130 views

What kind of “skeleton” does a strict $n$-category admit?

The skeleton of a 1-category is a way to "normalize" it, because Any category is equivalent to its skeleton, and Any equivalence between skeletal categories is an isomorphism. For strict $n$-...
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1answer
127 views

Is the inclusion functor from gaunt strict $n$-categories to weak $(\infty,n)$-categories fully faithful?

I'm now second-guessing an assertion I made here so let me ask it as a question. Let $Cat_n$ be the 1-category of strict $n$-categories; Let $\widetilde{Cat_n}$ be the $(\infty,1)$-category obtained ...
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Classification of absolute 2-limits?

Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched ...
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0answers
168 views

$\mathbb Z \otimes_\mathbb S \mathbb Z$ is concentrated in degree $0$ : mistake in the argument

I'm not sure this is research level so if this is not appropriate, feel free to move the question to StackExchange. However, I post it here since my "fake proof" is based on a (recent) paper and I'm ...
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2answers
1k views

Understanding the definition of stacks

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
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1answer
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On equivalences of cartesian fibrations

Let $p:X^{\natural} \to S$, $q:Y^{\natural} \to S$ be two cartesian fibrations of simplicial sets (where the marking is given by cartesian edges) and assume that we are given an equivalence of ...
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Is there a model-independent characterization of the gaunt strict $n$-categories amongst the weak $(\infty,n)$-categories?

Recall that a strict $n$-category $C$ is called gaunt if every $k$-morphism in $C$ with a weak inverse is an identity, for all $k$; let $Gaunt_n$ denote the strict 1-category of gaunt $n$-categories. ...
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211 views

Simplicial model for $\mathcal{L}BG//S^1$ for a finite group $G$

$\require{AMScd}$For $X$ a (nice enough) topological space, the free loop space $\mathcal{L}X$ is the space of continuous maps from $S^1$ to $X$. This space has a natural $S^1$ action given by ...
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1answer
198 views

Understanding the adjunction $\mathfrak{C}:\mathbf{Set}^{\Delta^{op}}\rightleftharpoons \mathbf{Cat}_\Delta:\mathcal{N}$

Let $\mathbf{Cat}_\Delta$ denote the category of simplicially enriched categories (ie simplicial objects in $\mathbf{Cat}$ with all face and degeneracy maps bijective on objects). There is a canonical ...
5
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3answers
265 views

Why the third stage of Cech nerve a Lie 2-groupoid?

In the page https://ncatlab.org/nlab/show/Lie+2-groupoid the Lie 2-groupoid is defined as the 2 truncated $\infty$-Lie groupoid. I am not much comfortable with the language of higher category theory ...
5
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1answer
149 views

Which free strict $\omega$-categories are also free as weak $(\infty,\infty)$-categories?

There are a number of formalisms available for presenting free strict $\omega$-categories -- Street's parity complexes, Steiner's directed complexes, computads, polygraphs,... Typically one has a ...
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2answers
217 views

Do disjoint unions of stacks commute with finite fibre products?

Choose a big $\mathit{fppf}$-site $(\mathbf{Sch})_{\mathit{fppf}}$ and let $S$ be a scheme in that site. Let $\{\mathcal{X}_i\mid i\in I\}$ be a family of stacks in groupoids over $S$ and let $\...
13
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3answers
451 views

Cofinality for coends?

Recall that a functor $I \xrightarrow u J$ is cofinal if it has the property that for any functor $J \xrightarrow F C$, we have that $\varinjlim F \cong \varinjlim Fu$ via the canonical map, either ...
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699 views

Application of higher categories in algebra

Higher topos and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher categories ...
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1answer
144 views

Is the inclusion of its 2-skeleton into the walking idempotent homotopy cofinal?

Let $Idem = Idem^{(\infty)}$ be the walking idempotent [1], and let $Idem^{(n)}$ be its n-skeleton. Note that $Idem$ has one nondegenerate simplex in each dimension. Let $\iota_n^m: Idem^{(n)} \to ...
8
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1answer
237 views

Generalized “Homology Whitehead” — How much does stabilization remember?

Classically, the (non-local-coefficients) homology Whitehead theorem says that if $X \xrightarrow f Y$ is a map of simple spaces, and if the induced map $H_\ast(X;\mathbb Z) \to H_\ast(Y;\mathbb Z)$ ...
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0answers
148 views

Weak 2-groups and non-abelian gerbe over a manifold

In literature, a strict 2-group is defined as a group object in the category of categories or groupoids. It has many well known equivalent descriptions viz: 1. A strict monoidal category in which all ...
9
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1answer
360 views

HNN-extension as a 2-colimit

In the spirit of this question, it would be interesting to give a characterization of HNN extensions as a 2-colimit. If $G$ is a group and $\alpha:H \xrightarrow{\cong} K$ is an isomorphism between ...
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155 views

What do you call a map of spaces which is weakly left orthogonal to all $n$-connected maps?

$\let\op=\operatorname$In $\op{Set}$, we have an $(\op{Epi},\op{Mono})$ orthogonal factorization system. Strikingly, if we reverse the roles, we get the no-less-important $(\op{Mono},\op{Epi})$ weak ...
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1answer
228 views

When is the model structure on functors correct, i.e. when does localization commute with taking functor categories?

Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with ...
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0answers
187 views

Is every limit-closed, accessibly-embedded full subcategory of a presentable $\infty$-category reflective?

Let $C$ be a presentable $\infty$-category and let $D\subseteq C$ be a full subcategory closed under limits and sufficiently-filtered colimits. If $D$ is known to be accessible, then by the adjoint ...
11
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1answer
352 views

Does the homotopy category of spaces admit a weak generating set?

As a follow-up to this question, let $\mathcal C$ be a category and $\mathcal S \subseteq \mathcal C$ a class of objects. Say that $\mathcal S$ is weakly generating if the functors $Hom_{\mathcal C}(S,...
7
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1answer
204 views

Vanishing of Hochschild homology of a category

Let $A$ be a dg- or $A_{\infty}$-category (with $\mathbb{Z}$-graded Hom sets, over a field of characteristic $0$). Let $HH_*(A)$ be the Hochschild homology of $A$. Suppose that $HH_n(A)=0$ for all $n ...
11
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1answer
303 views

On the relation between categorification and chromatic redshift

In the introduction to the paper Higher traces, noncommutative motives, and the categorified Chern character, Hoyois, Scherotzke and Sibilla write the following. An important insight emerging from ...
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Characterization of geometric morphisms without referring explicitly to the left adjoint?

Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...

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