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Questions tagged [higher-category-theory]

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. (Def: http://en.m.wikipedia.org/wiki/Higher_category_theory)

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Category Theory after CatsNSheaves

I’m currently reading MacLane’s “Categories for the Working Mathematician” and am loving it. So I bought a copy of Kashiwara & Schapira’s “Categories and Sheaves” and plan to tackle that next. My ...
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Monoidal categories in which some tensor products of morphisms are equal

I have come across a certain monoidal category $(C,\otimes,I)$ (let us say it is strictly monoidal to simplify the notations, as usual it does not matter very much) which satisfies the following ...
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1answer
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Ind-completion of a 2-category

If $\mathcal{C}$ is a category, there is a well known construction called the Ind-completion of $\mathcal{C}$, indicated by $\text{Ind}(\mathcal{C})$. This can be equivalently defined in several ways: ...
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Higher inductive type: what for?

The typical example of higher inductive type (HIT) is the circle $S^1$ that is nicely described here. I understand HITs are convenient if you want to do homotopy theory within type theory. But what ...
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Base change of topological operad to any symmetric monoidal model category and $E_n$-algebras outside of $\textbf{Top}$

I would like to ask about how we can talk about Algebra over little $n$-disk operad $D_n$ in a greater generality outside of $\textbf{Top}$. I know that in the topological context, an $E_n$-operad is ...
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0-truncation of infinity-presheaves

Let $\mathcal{C}$ be a small 1-category endowed with a Grothendieck topology. Is it true that the 0-truncation $\tau_0(\mathrm{Shv}_\infty(N(\mathcal{C})))$ of the $\infty$-topos of $\infty$-sheaves ...
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Bousfield-Kan Formula for Homotopy Limits

Let $\Lambda$ and $C$ be categories, with $\Lambda$ small and $C$ complete. Let $F : \Lambda \rightarrow C$ be a diagram, where $C$ is cotensored over simplicial sets; there is then a functor $[\Delta^...
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Explicit definition of the čech nerve in this paragraph

In the picture below, how is the čech nerve $C(U)$ defined? In the first hom-set, it is taken to be a simplicial sheaf. It would actually make sense if it was a simplicial simplicial sheaf, however. ...
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Quasi-isomorphism of an $A_\infty$ module and its cohomology

Let $A$ be an $A_\infty$-algebra over a field $k$. It is a well-known fact that $H^\bullet (A)$ also has an $A_\infty$-structure, and further one can construct a quasi-isomorphism $H^\bullet (A) \to A$...
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$\infty$-category of chain complexes via Dold-Kan correspondence

Suppose we are given the category of abelian groups $\mathsf{Ab}$. Then I am aware of two procedures how to turn the category of chain complexes $\operatorname{Ch}_{\geq 0}(\mathsf{Ab})$ that are ...
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Is $\mathbf{Cat}/\mathcal{C}\simeq\mathbf{Cat}^{\mathcal C^\mathrm{op}}$ when $\mathcal C$ is a $1$-category?

Given a small set $S$, we can define the overcategory $\mathbf{Set}/S$ to be the category whose objects are pairs $(A:\mathbf{Set},a:A\to S)$ and whose morphisms $(A,a)\to(B,b)$ are functions $f:A\to ...
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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{...
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Initial objects on $\infty$-categories

Let $X \in \mathbf{Set}_{\Delta}$ an $\infty$-category and $\tau_1$ the left adjoint functor to the nerve $\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$. Show that if $x$ is an initial ...
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1answer
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What is the fundamental category?

Given a category $\mathcal{C}$, we have a nerve functor $$\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$$ that assigns to $\mathcal{C}$ its nerve $\mathrm{N}(\mathcal{C})$. This functor ...
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Understanding 2-category theory

There are a lot of examples of categories, functors and natural transformations — one can find them anywhere. On the contrary (weak) 2-categorical stuff seems to be more subtle. I have comprehended ...
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Join of simplicial sets induces a functor.

Given a simplicial set $X$, denote by $(\mathrm{Set}_{\Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X \star Y$ of ...
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1answer
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Simplicial set as a colimit

Let $K \in \mathrm{Set}_{\Delta}$ be a simplicial set. Then $K$ is the colimit of the diagram $$F \colon \Delta/K \to \mathrm{Set}_{\Delta} \, $$ that assigns to each $\Delta^n \to K$ the ...
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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....
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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 $...
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Presentation of $hS$ as in page 29 of HTT

At the page claims 29 of HTT, Lurie claims that the category $hS$ (we ignore all the enrichement) admits the following presentation by generators and relation : The objects of $hS$ are the vertices ...
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VAR, Algebra and local presentability

Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-\mathrm{lim}$, BUT $k-\textrm{colim}$ must be filtered. Where have we used that $k$ is ...
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1answer
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What is the $\infty$-category associated to a model category?

It is often said that model categories are but a shadow of an $\infty$-category. It is also often said that model categories should give rise to an $\infty$-category via their homotopies. In fact, ...
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1answer
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Lurie's reformulation of symmetric monoidal tensor categories in HA

In the introduction to Chapter 2 of Jacob Lurie's Higher Algebra (entitled "$\infty$-Operads"), a category $\mathcal{C}^\otimes$ is constructed from an arbitrary symmetric monoidal category $\mathcal{...
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1answer
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Is $Cat_{\Delta}$ enriched over itself?

Question is just as in the title. Is the category of simplicially enriched categories enriched over itself? If not, is it enriched over another relevant category, e.g., simplicial sets?
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Weak complicial set from strict $2$-category

I am reading this paper: https://arxiv.org/pdf/1610.06801.pdf It says: If $C$ is a strict $2$-category, there is a unique saturated $2$- trivial complicial structure on $NC$, in which the $2$-cell ...
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1answer
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Bergner homotopy category of simplicially enriched caterories is cartesian closed

Let $Cat_{\Delta}$ be the model category of simplicially enriched categories with the Bergner model structure. In a paper I am reading, they state without proof that $Ho(Cat_{\Delta})$ of this model ...
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The category of functors if the codomain category has zero object is non-empty. Why?

Here, page 6, Daniel Murfet said, the category of functors $(\mathcal{A}, \mathcal{B})$ can be empty, although it is nonempty if $\mathcal{B}$ has zero object. Why? [Zero object: An object which is ...
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Definition of $\mathfrak{C}[\Delta^n]$

Lure's book defines the $\mathfrak{C}[\Delta^J]$, but it seems he didn't define $\mathfrak{C}[\Delta^n]$. I wonder if $\mathfrak{C}[\Delta^n]$ means $\mathfrak{C}[\Delta^{[n]}]$? And for $I$ above, ...
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1answer
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The left adjoint functor $\mathfrak{C}$ to the simplicial nerve functor $N$.

I am referring to Lurie's book. In the following, he introduces a functor, $\mathfrak{C}$ which is left adjoint to the nerve functor $N$, which sends a simplicial set to a simplicial category, which ...
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2answers
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Why do the $-1$-categories form a categeory rather than merely a set?

The collection of $n$-categories naturally has the structure of a $(1+n)$-category. For example $\mathbf{Set}$ is a $1$-category and $\mathbf{Cat}$ is a $2$-category. Therefore we would expect the $-1$...
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A 2-group $\mathbb{G}$, so that always exists $0 \to BG_b \to \mathbb{G} \to G_a \to 0?$

If $\mathbb{G}$ is a 2-group, does there always exists a short exact sequence for this $\mathbb{G}$, such that $$ 0 \to BG_b \to \mathbb{G} \to G_a \to 0? $$ where both $G_a$ and $G_b$ are nontrivial ...
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Fiber sequence, a group and an $n$-group

Given a short exact sequence $$ 1 \to B\mathbb{Z}_2 \to \mathbb{G} \to O(n) \to 1 $$ and the fiber sequence: $$ B^2\mathbb{Z}_2 \to B\mathbb{G} \to BO(n), $$ classified by $\beta \in H^3(BO(n), \...
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1answer
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The Whitehead product and $\pi_{\leq 3} S^2$

Why does "the non-vanishing of the Whitehead bracket" imply that the fundamental 3-groupoid $\pi_{\leq 3} S^2$ of the two-shere cannot be strictified (as claimed here)?
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1answer
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$\mathbf{B}A$ as a 2-group in a long fiber sequence

I am trying to digest the following statement about 2-group: From nlab Observation 4.2: "Let $A \to \hat G$ be the inclusion of a subgroup, exhibiting a central extension $A \to \hat G \to G$ ...
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1answer
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Span Categories

Question: It seems to me, the bicategory of spans, with canonical choice of units, has strict units, so it is not a general bicategory. Am I right? Below is the definition: Definition: Consider a ...
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1answer
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Is the map $\Lambda^n_n \sqcup_{\Delta^{\{n-1,n\}}} J \to \Delta^n \sqcup_{\Delta^{\{n-1,n\}}} J$ left anodyne?

This question is about objects studied in chapter 2 of Lurie's Higher Topos Theory. Let $p : X \to S$ be a left fibration, and let $e \in X_1$ be an edge of $X$ such that $p(e)$ is an equivalence in $...
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Why is it that homotopy is better described by weak equivalences than by homotopies?

I've been reading about (abstract or not) homotopy theory, and I seem to have understood (correct me if I'm wrong) that weak equivalences describe homotopy better than homotopies, in the following ...
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Is there a simplicial set classifying subobjects of groupoids?

A $1$-groupoid can be thought of as a Kan complex in the usual way. Is there a simplicial set $\Omega$ such that the contravariant functors $\text{Sub}_{\mathbf{Gpd}}(-)$ and $\text{Hom}_{\mathbf{sSet}...
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Defining Category of Adjunctions

$\newcommand{A}{\mathcal{A}} \newcommand{B}{\mathcal{B}} \newcommand{C}{\mathcal{C}} \newcommand{ADJ}{\mathsf{ADJ}} \newcommand{id}{\mathrm{Id}}$Morphism between a pair of adjunctions $(F : \A \...
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1answer
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Underlying quasicategory of a model category through framings?

Consider a model category $\mathcal M$. Because it is a category with weak equivalences, we can use the following construction to obtain the "underlying quasicategory" of $\mathcal M$: taking the ...
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What are the loops by delooping a group to a groupoid?

I'm trying to understand the definition of delooping a group to a groupoid, but I don't see the loops in a group that aren't present in the groupoid. Then why is it called delooping?
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What is the right notion of separator in a 2-category?

A separator, or separating family, in a category is a full subcategory $\mathcal{S} \hookrightarrow \mathcal{E}$ of a category $\mathcal{E}$ which satisfies the following: For any parallel pair of ...
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1answer
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Limits vs homotopy limits in derived algebraic geometry

In derived algebraic geometry (Lurie's thesis) he defines a notion of cohesiveness for functors $F: SCR \to S$, where $SCR$ is the $\infty$-category of simplicial commutative rings, and $S$ is the $\...
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1answer
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A topological space as a $2$-($\infty$ ?) category

I'm just using this example to gain a bit of intuition concerning higher categories (about which I have essentially no knowledge); because it's the first example that came to my mind, and because it ...
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the Verlinde formula

The Verlinde formula writes the fusion coefficient in terms of S matrix. My question is that for fusion category without braiding, is there a similar formula which gives the fusion coefficient in ...
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1answer
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Definition: What is the under-category of an $\infty$-category?

When I say "$\infty$-category", I mean a simplicial set such that all inner horns have not necessarily unique fillers. In Lurie's Higher Topos Theory there's the following Definition. Given an $\...
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2answers
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A terminological question regarding bicategories

My question is very simple. I have come accross some literature regarding bicategories (Benabou, etc) and I am a little confused on the terminology: Could you please tell me if bicategories and 2-...
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Fibrant objects of marked simplicial sets

Im reading the nlab article one the Cartesian model structure on the category of marked simplicial sets over a simplicial set $S$. Here it is stated just below Proposition 3.1 which is a ...
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1answer
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Pseudolimits equivalent to limits

In Proposition 4.1 of this article is proved that if $F: I\rightarrow\mathbf{Cat}$ is a pseudofunctor and $I$ is a filtered poset, then the pseudo colimit of $F$ is equivalent to its strict colimit. ...
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3answers
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Higher homotopy groups in terms of the fundamental groupoid

Let $X$ be a topological space. Then we can construct the following structure. Let an $n$-morphism be a map $I^n\to X$. We can view $n+1$ morphisms exactly as homotopies between $n$-morphisms. Let $f,...