# Questions tagged [higher-category-theory]

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

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### Strict 2-functoriality of lax-slices of 2-categories

$\DeclareMathOperator{\Hom}{Hom}$
I'm currently interested in the homotopy theory of categories "à la Grothendieck", as he developed it in "Pursuing Stacks". I'm trying to try and ...

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### Why is the universal $n$-gerbe universal? (HTT, 7.2.2.26)

Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to ...

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### "$X$ is $n$-truncated $\iff$ $\Omega X$ is $(n-1)$-truncated" for connected pointed $X$. (HTT, 7.2.2.11)

In the proof of Lemma 7.2.2.11 of Higher Topos Theory, Lurie makes the following claim:
($\ast$) Let $n\geq1$ be an integer, let $\mathcal{X}$ be an $\infty$-topos, and let $1\to X$ be a pointed ...

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### Lax Gray tensor product and opposite categories:

For any two strict (infinity, infinity)-categories $A,B$ let $A \otimes B $ be the lax Gray tensor product of $A$ and $B$. Let $A^{op}$ be the opposite (infinity,infinity)-category, where morphisms in ...

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### Homotopy 2-Category

I am reading Riehl's notes "Higher Category from scratch". I have come across the notion of a homotopy 2-category. I have not taken any course on Homotopy Theory before but I know that the ...

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### Compact objects in categories of categories

I am interested in the compact objects of various categories of categories.
For example, $Cat^{small}$ is presentable and has compact objects that are retracts of finite colimits of $\Delta^n$, the $n$...

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### Can Postnikov towers converge without Postnikov completeness?

In Higher Topos Theory, Lurie says that "Postnikov towers are convergent" in a presentable $\infty$-category $\mathcal{C}$ if $\mathcal{C}$ is equivalent to the $\infty$-category $\mathrm{...

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### New investigations on Homotopical Algebraic Contexts

Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II.
These are general abstract ...

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### Monoidal structure on presheaves

I am confused about the following monoidal structure, which gives a symmetric monoidal structure on R-modules (that I think is not Cartesian), even if R is not commutative.
Let $C$ be a small category....

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### A "lax Boardman-Vogt tensor product," or what object represents duoidal categories?

Let me preface this by saying I'm not sure what the fundamental examples should be, and perhaps that's part of my question.
The Boardman-Vogt tensor product of $\infty$-operads $\mathcal{O}$ and $\...

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### On Lemma 5.5.16 of Cisinski's "Higher Categories and Homotopical Algebra"

I have a question regarding Section 5 of Cisinski's
"Higher Categories and Homotopical Algebra".
Let us write $\mathbf{sSet}$ and $\mathbf{bisSet}$ for the categories of
simplicial sets and ...

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### Godement product of lax 2-natural transformations

Which of the two obvious choices for the Godement product of lax $2$-natural transformations is ‘correct’?
Specifically, recall that for natural categories, functors and natural transformations as ...

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### Bar construction in commutative algebras is calculated by pushout

$\DeclareMathOperator\colim{colim}$
Also asked in MathStackexchange here
This is a statement in Lurie's Higher Algebra 5.2.2.4.
Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...

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### What are the 2-categorical mono/epimorphisms in the 2-category of relations?

$\newcommand{\procirc}{\mathbin{\diamond}}\newcommand{\rightproarrow}{\mathrel{\rightarrow\mkern-17mu|\mkern7mu}}$The monomorphisms in the 1-category $\mathsf{Rel}$ of sets and relations are precisely ...

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### Predicting coherence diagrams one dimension up

Assume we have a good working knowledge of $n$-dimensional category theory for some fixed $n$. It seems like it should be possible to 'predict' what coherence diagrams we're going to encounter in the ...

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### Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions:
Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...

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### Presentability rank of tensor product of presentable categories

In this post category means $(\infty, 1)$-category.
Let $X, Y$ be two presentable categories. We can then form their tensor product $X \otimes Y \cong \operatorname{ContFun}(X^{\mathrm{op}}, Y)$. Can ...

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### Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\infty$-categories

Recently, in a conversation with Gabriel, the following question came up:
Question. Do co/limits of categories taken in the $\infty$-category of $\infty$-categories agree with the usual co/limits ...

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### (2,1)-limits vs 2-limits of categories

In the section 1 of these notes, Emily de Oliveira Santos gives an explicit construction of most usual (co)limits in the 1-category of categories. In the next section she affirms that the same ...

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### Computing nonabelian derived functors on fibrant-cofibrant objects

I am learning the process of "Animation" from Cesnavicius and Scholze's paper Purity for flat cohomology. In my understanding the animation of a category/functor is simply the nonabelian ...

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### Localizations that are endofunctors

Usually when $F: C \rightarrow D$ is a localization functor, the categories $C$ and $D$ are not equivalent. My question is when is it possible for $C, D$ to be abstractly equivalent but $F$ is not an ...

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### Self-enrichment for a closed monoidal bicategory

First, there are two possible generalization of the notion of closed category, vertical and horizontal.
I'm interested in the vertical one, something saying, I guess, that a monoidal bicategory $\...

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### Soft question: Deep learning and higher categories

Recently, I have stumbled upon certain articles and lecture videos that use category theory to explain certain aspects of machine learning or deep learning (e.g. Cats for AI and the paper An enriched ...

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### Why do we say IndCoh(X) is analogous to the set of distributions on X?

$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as ...

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### Completeness of comma $\infty$-categories

Let $\mathsf{A},\mathsf{B},$ and $\mathsf{C}$ be (ordinary) categories and $F : \mathsf{A}\to\mathsf{C}$ and $G : \mathsf{B}\to\mathsf{C}$ be functors such that
$\mathsf{A}$ and $\mathsf{B}$ are ...

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### Does the Gray tensor product exhibit Gray as a monoidal Gray-category?

Crans constructs a lax Gray tensor product for strict omega-categories that defines a monoidal structure on the category of strict omega-categories $\omega Cat$. We write Gray for this monoidal ...

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### Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?

Background
I am currently writing an MSc dissertation on TQFTs (and Khovanov homology, but that is unrelated to this question).
After having read most of Kock's book on the equivalence between 2D ...

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### Piecewise construction of a functor from an $(\infty,1)$-category with an orthogonal factorization system

For the simpler case of 1-categories, consider a 1-category $C$ and an orthogonal factorization system $(L,R)$ on $C$. Let $C_L$ and $C_R$ denote the wide subcategories of $C$ corresponding to the ...

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### Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories

By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).
An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...

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### How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?

I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful.
Let's say $C$ is a certain category, and ...

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### Colimits from van Kampen cocones

Let $\mathcal{C}$ be a category with pullbacks, $\mathcal{J}$ a small category, $F : \mathcal{J} \to \mathcal{C}$ a diagram and $\kappa : F \Rightarrow \Delta X$ a cocone in $\mathcal{C}$. Let $\...

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### Does the 2-category of double categories and vertical transformations have flexible limits?

Consider the 2-category of pseudo-double categories (with the weak composition in the horizontal direction and the strict composition in the vertical direction), strong double functors, and vertical ...

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### Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?

For ordinary category theory, we have the following fact.
A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor.
Specifically, the weighted colimit ...

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### Natural ways to make a functor adjoint

Let $F: C \to D$ be a functor between two categories without a right adjoint. What are some natural ways to create a right adjoint for $F$?
Of course, this does not make sense on the nose. One needs ...

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### Tangent $(\infty,1)$ topos

I am trying to understand the tangent $(\infty,1)$ category. It is the fiberwise stabilization of the codomain fibration (which is a functor from the arrow category to the category).
But, intuitively ...

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### Comparing notions related to $(\infty,2)$-categories

I am trying to understand two related notions:
$(\infty,2)$-category as in Definition 5.5.1.3, Kerodon
weak $\infty$-bicategory as in Definition 4.1.1 in "$(\infty,2)$-Categories and the ...

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### Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II

This is the second part to a previous question regarding left Kan extensions/lifts in the bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations, which has now been split into two ...

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### Higher lie groups and groupoids

I have come across the following notions while learning these things myself. Wanted to confirm these ideas.
Smooth ∞
stack or equivalently, lie (smooth) ∞
groupoid is an ∞
stack on the site SmoothMfd (...

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### Delooping higher groupoid

I am trying to learn the following ideas of groups and groupoids. Am a bit confused regarding the following ideas.
I have learned BA is a 2-group with a single object and the space of morphisms to be ...

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### Descent theory for higher sheaves

I am trying to understand the descent condition for sheaves from presheaves. Let Presh(S) be the $(\infty,1)$ category of presheaves on an $(\infty,1)$ site S, and Sh(S) be the corresponding category ...

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### Does every $\kappa$-compact topos embedd relatively $\kappa$-tidily into a presheaf topos?

Let $\kappa$ be a regular cardinal and say that a topos $\mathcal{E}$ is $\kappa$-compact if the global sections $\gamma_{\ast} : \mathcal{E} \to \mathsf{Set}$ preserves $\kappa$-filtered colimits.
My ...

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### (n,r)-Cat of (n,r) categories

The category of (n,r) categories is an (n+1, r+1) category (n,r)-Cat; where objects are (n,r) categories, morphisms are (n,r) functors, and k-morphisms for k>=2 are (k-1) transfors. I seem to know ...

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### Infinity stacks

I was going through some notions of stacks and higher stacks on nLab. $\infty$-stacks are usually $(\infty,1)$-sheaves which take values in $\infty$-groupoids. Now to recall, $(\infty,1)$-sheaf is a ...

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### Coevaluation for linear categories

For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...

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### Existence and characterisations of left Kan extensions and liftings in the bicategory of relations I

The bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations has right Kan extensions and right Kan lifts¹, however I believe it does not have all left Kan extensions/lifts.
Is it ...

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### Existence and explicit descriptions for left and right Kan extensions and lifts in bicategories of spans

Given a category $\mathcal{C}$, it follows from Proposition 4.1 of Day's
Brian Day, Limit spaces and closed span categories, Lecture Notes in Mathematics, 420, 1974 (doi:10.1007/BFb0063100).
That ...

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### $\infty$-topos as an internal $\infty$-category in itself

I'm interested (both autonomously and directly related to my work) in the natural internalization of $\infty$-topos sheaves in it (as usual, assuming Grothendieck universes). Is there any literature ...

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### Geometric realization of crossed square

Given a crossed square of groups, you can "totalize" it and get a 2 crossed module in the sense of Conduché "Modules croisés généralisés de longueur 2", then you can apply his ...

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### Does the simplex map to the cube?

For every $n \geq 0$ there is an inclusion of the ordered set $\{0<1<\dots<n\}$ into the product
$\{0<1\}^{\times n}$ sending $i$ to the increasing sequence $(0 < \dots<0<1<\...

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### Infinitesimal criteria for unramified morphism on stacks

In an Artin stack, the tangent complex is fundamentally a derived object, often viewed as a 2-term complex via some quotient presentation, as discussed in Raskin's note here, for example. In ...