# 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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1 vote
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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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### 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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1 vote
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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 $C, D$ ...
1 vote
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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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### 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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1 vote
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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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