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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G be a finite group, C^× the non-zero complex numbers with trivial G-module structure. then H^2(G, C×) is finite.

Let G be a finite group, C^× the non-zero complex numbers with trivial G-module structure. Show that H^2(G, C^×) is finite. H2(G/N, C*); C* is the G/N-trivial module of the group of nonzero complex ...
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Cocartesian edge gives contractible choice of filling in commutative diagram

This appears on p. 192 of Land's Introduction to Infinity-Categories which is the first page of his section on Straightening-Unstraightening. Let $p:\mathscr{E} \to \mathscr{C}$ be a cocartesian ...
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Is There a Notion of Diagram in Multicategories and/or Operads?

In ordinary category theory there is a notion of a diagram in a category $\mathsf{C}$ which is usually described as a functor $F: \mathsf{J \to C}$ where $\mathsf{J}$ is some small category. Based on ...
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Can you internalize the single sorted definition of a category without pullbacks?

nlab suggests falling back to pullbacks to internalize the single sorted definition of a category. But can we internalize single sorted categories here without pullbacks? I have an idea but I'm not ...
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What is problematic with formalizing higher category theory directly in HoTT

This is a soft-question. But please let me know if its not suitable. My question is essentially what the title asks but I want to elaborate on a few things. I know Emily Riehl uses a type theory to ...
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($\infty$ ,1) construction of Spectra and stabilization

I am trying to understand Spectra as an attempt to make the infinity category of pointed spaces stable. I don't know the details of some $\infty$,1 constructions so I have several questions. From now ...
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HA, 1.1.1.7, Lurie

In this remark, Lurie states that applying proposition HTT 4.3.2.15 twice, we deduce that $\theta$ is a kan fibration. How is this assertion deduced?
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Squares in additive infinity categories

I need to show that a commutative square in an additive infinity category, where the vertical maps both are split epimorphisms and both admit fibers/kernels, is cartesian if and only if it is ...
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Universal characterization of the arrow (interval, ordinal, 1-simplex) object? (Is $2$-category theory actually needed?)

In the "category of categories" $Cat$ (see e.g. here for a rigorous definition), there is a category $A$ with two objects and exactly one non-identity morphism. (Cf. "two-valued object&...
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Slice and Internal Hom; On the Definition of a Map $\underline{\operatorname{Hom}}(A,X/x)\to \underline{\operatorname{Hom}}(A,X)/x$

Recall that the join of two simplicial sets $X,Y$ are defined by $$(X\ast Y)_n=X_n\amalg X_{n-1}\times Y_0\amalg\cdots\amalg X_{0}\times Y_{n-1}\amalg Y_n,$$ with face and degeneracy maps defined ...
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How is the internal hom in $\mathsf{sSet}/\mathcal C$ defined?

Fix a simplicial set $\mathcal C$. We have a category of simplicial sets over $\mathcal C$, denoted by $\mathsf{sSet}/\mathcal C$. This is supposed to have an internal hom, denoted $\underline{\mathrm{...
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Motivation for Quasicategories

Recall that a simplicial set $X$ is called a quasicategory if for every $0<k<n$, every map $\Lambda^n_k\to X$ admits an extension to a map $\Delta^n\to X$. It is good to have a concise ...
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Does the left adjoint of the homotopy coherent nerve have a name?

The simplicial nerve $N: \operatorname{Sset-Cat} \to \operatorname{Sset}$ has the left adjoint $\mathfrak{C}: \operatorname{Sset} \to \operatorname{Sset-Cat}.$ Wherever I see it, it is always referred ...
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Categorical Intuition of Path Induction

I am trying to understand path induction from the trinitarian point of view. So far I understand the informal intuition of path induction from a homotopical and computational point of view. But I am ...
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Bimodules as algebra morphisms explicitly

My question is a more detailed version of the last paragraph in this previous one that I asked. Given a ring $k$, in the bicategory of $k$-algebras, bimodules (as the 1-morphisms) and intertwiners (2-...
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What's the difference between a higher sheaf on a regular ($1$-) site and a sheaf on a $2$- (or $\infty$-)site?

A higher site's coverage satisfies If $\left\{f_: U' \rightarrow U\right\}$ is a covering family and $g: V \rightarrow U$ is a morphism, then there exists a covering family $\left\{h: V' \rightarrow ...
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Morphisms set of sub-$\infty$-category closed under equivalences

In Markus Land's Introduction to Infinity-Categories, he defines sub-$\infty$-categories the following way (p. 56): Definition. A sub-$\infty$-category $\mathscr{C}'$ of an $\infty$-category $\mathscr{...
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Initial algebra in $\text{Cat}$ is a monad not a category?

I've been trying to figure out recursively constructed categories so I think initial algebras in $\text{Cat}$. However, some of the types don't seem to line up and I'm confused. An algebra/module in $\...
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Composition in an $\infty$-category

In the following, I'm following Markus Land, Introduction to Infinity-Categories (p. 81). Let $\mathscr{C}$ be an $\infty$-category and $x,y,z \in \mathscr{C}$ be objects, then I want to understand ...
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Mapping space of simplicial sets

In "Higher Topos Theory" by Lurie (Section 1.2.2, "Mapping spaces in Higher Category Theory") the notion of mapping space is defined as such: Given a simplicial set $S$ and two ...
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What is a coproduct in an enriched category?

Does anybody know how to define coproducts in enriched categories ? For example I was wondering if they could be thought of as some kind of weighted colimits. I am particularly interested in if/how ...
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The class $l(F)$ of morphisms which have the left lifting property with respect to $F$ is stable under transfinite compositions.

I am reading Cisinski's Higher Categories and Homotopical Algebra and I am having trouble trying to verify some claims there. My background in category theory is not very solid. I would like some help ...
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Composition in quasi-categories

Let $C$ be a quasi-category. Then $C^{\Delta^2}\to C^{\Lambda_1^2}$ is a trivial fibration and we may choose a section $s$. The map $C^{\Lambda_1^2}\xrightarrow s C^{\Delta^2} \to C^{\{0,2\}}$ defines ...
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Singular simplicial sets that are (nerves of) categories

The simplicial sets that are both Kan complexes and nerves of a category are exactly the groupoids. Simplicial sets of the form $\mathrm{Sing}_\bullet(X)$ are a proper subclass of Kan complexes. ...
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The functor $\Omega^\infty : \operatorname{Sp}(\mathcal{C})\rightarrow \mathcal{C}$ is accessible

Here $\mathcal{C}$ is a presentable infinity category. In the proof of proposition 1.4.3.4 in higher algebra, the fact that $\Omega^\infty : \operatorname{Sp}(\mathcal{C})\rightarrow \mathcal{C}$ is ...
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Is every infinity-category equivalent to a category? To a weak 2-category?

As far as I understand it, the composition of morphisms in an $\infty$-category $\cal C$ does not have to be unique, so $\cal C$ does not have to be a category. Is $\cal C$ equivalent to a category (...
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A problem on algebraic topology.

In classical homotopy theory, there are a number of spaces which are important because they represent an interesting functor on $\operatorname{Ho(Top)}$; for example, $K(G,n)$ represents singular ...
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Complex of $n$-balls in analogy of simplicial complexes

I would like to know if there is a version of complexes (as in simplicial complexes/sets) with the role of simplices played by $n$-balls. I thought of the following: the analogue of a (-1)-simplex(?) ...
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Can we control the "degree" of accessibility of right adjoints between presentable $\infty$-Categories? (HTT 5.4.7.7)

Suppose $g: D \to C$ is a right adjoint between presentable $\infty$-categories. Then by the adjoint functor theorem, $g$ is accessible, i.e. there is a regular cardinal $\kappa$ such that both $C, D$ ...
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What are the important things to study in infinity categories?

I am reading about infinity categories. My source is http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf My aim is to think categorically so that all the constructions I deal with are natural (...
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What is the difference for a functor to have a left adjoint or a right adjoint?

I am wondering the meaning for a functor to have a left adjoint or a right adjoint in practice. Stardard results including Freyd's adjoint functor theorem (preserving colimits or limits) are not ...
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Connections between $0$-toposes and $1$-toposes (Grothendieck and elementary)

A Grothendieck $0$-topos is the same as a frame (see here). Another relationship between Grothendieck toposes and frames is the following: whenever $\mathcal O$ is a frame, then $\mathrm{Sh}(\mathcal ...
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Verification of proof of HTT Prop 3.1.1.6

My question is on the last 7th row of the proof of Proposition 3.1.1.6 of Lurie's book Higher Topos Theory where he states that 'for (3), we are free to replace $S$ by $(\Delta)^{\#}$'. I cannot see ...
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$\infty$-groupoids = Kan complexes

One direction is clear, a Kan complex has all its morphisms invertible hence it is an $\infty$-groupoid. Is it possible to show the opposite as a Corollary of the following theorem? Theorem. Let $p:X\...
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Bijection induced by Hom and inner Hom

For an $\infty$-category $X$ and a simplicial set $A\in sSet$. And for a category $\mathcal{C}$ we define $\mathcal{C}^\simeq$ whose objects are those of $\mathcal{C}$ and morphisms contain only ...
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Definition of strict 2-monad on Cat

The category $\text{Cat}$ can be thought of as a $2$-category. I was hoping somebody could help by telling me the explicit definition of a strict 2-monad $(T, \eta, \mu)$ on $\text{Cat}.$ In ...
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Category of Monads on $PreOrd$

In a recent paper by Adamek ("Finitary Monads on the Category of Posets", see it on Arxiv) the author introduces the category of finitary monads on $Pos$ and analyzes it in great detail. ...
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How to prove the fully-faithfulness of nerve functor?

I know it might be a bit elementary but I‘ve just read about the defintion of nerve functor $N: Cat \rightarrow sSet$ in nLab : https://ncatlab.org/nlab/show/nerve And Proposition 3.12 states that ...
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"Functors" which map objects to morphisms

I'm just beginning to learn category theory, and this question popped up in my mind, which Googling has not been able to resolve (probably because I am not searching the right terms). A functor ...
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internal hom between nerve of categories is nerve of functor category

The following statements are true:\ if $A,B$ are small categories, then: $\underline{Hom}(N(A),N(B))= N(Fun(A,B))$ if $A\in sSet$ and $B$ a small category, then $\underline{Hom}(A,N(B))\cong N(Fun(\...
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Understand the internal hom simplicial set

I am currently reading the paper "a short course on $\infty$-categories" by M.Groth. The Theorem 1.18 in this paper states: A simplicial set $X$ is an $\infty$-category if and only if the ...
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The boundary of an open set as the homotopy limit of the open set minus compact subsets.

Let $M$ be a manifold and $U \subseteq M$ be a relative compact open set of $M$. I run into an equivalence $$\partial U \cong \operatorname{holim}_{K \subseteq U} U \setminus K $$ where the inverse ...
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Equivalence and isomorphism of $\hom$s in $2$-categories/bicategories

In a usual $(1)$-category $C$, if $A$ is an object and $B\cong B'$ are isomorphic objects, then the hom-sets $\hom(A,B)\cong \hom(A,B')$ are in bijective correspondence, because composition with the ...
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What would be a polynomial profunctor?

A polynomial endofunctor can be defined as a tuple $(S, F)$ of a set $S$ and an index $F: S \rightarrow \text{Set}$. It's extension or semantics is an endofunctor $\text{Set} \rightarrow \text{Set}$ $$...
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On the lifting property of canonical map between inner Hom

This question arises from the lecture notes. The notation $\perp$ means the thing on the left has the left lifting property w.r.t the thing on the right and, at the same time the right has right ...
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What's the free indexed category/ free indexed monad in the bicategory of endospans?

In Coq a loose approximation to the free monad over a polynomial endofunctor might be. ...
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How do you show that Lurie's straightening preserves colimits?

I am struggling to show that the straightening construction (2.2.1 in Higher Topos Theory) preserves colimits. More specifically let $M_X:=\mathfrak{C}X^\triangleright\sqcup_{\mathfrak{C}X}C^{op}$ for ...
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Local systems are the same as modules over chains of based loop space?

Let $M$ be a "good" topological space such as a manifold or a CW complex and assume it's connected. We use $\operatorname{Loc}(M)$ to denote the category of local systems either as a dg ...
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4 votes
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Natural transformation between sheaves in homotopy theory

Firstly a small disclaimer. I am not an expert in the theory of higher sheaves and their presentation in the model categories, so please feel free to correct all inaccuracies in the question itself! ...
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1 vote
1 answer
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Why Lie integration is called "integration"?

According to nlab on Lie integration, The Lie integration of $\mathfrak{a}$ is essentially the simplicial object whose k-cells are the d-paths in $\mathfrak{a}$, where $\mathfrak{a}$ is a $\infty$-Lie ...
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