# Questions tagged [higher-category-theory]

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

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### Proposition A.2.6.15 in HTT

This is a cross-post of a question in MSE.
I am reading Lurie's Higher Topos Theory and I need some help to understand a part of the proof of Proposition A.2.6.15. (A.2.6.13 in the published version)...

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### Strictness of two operations on proarrow equipments

There are several equivalent definitions of a profunctor between categories $C$ and $D$. I'm interested in the following two:
A functor $C\times D^o \to \text{Set}$
A co-continuous functor between ...

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### What is higher representation theory?

Can anyone please introduce higher representation theory?
By Yoneda embedding, we know that global dimension of finitely generated category $\bmod\Lambda$ of Artin algebra $\Lambda$ is no more than $2$...

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### Natural numbers object in a double category

I've been playing with double categories. I'm having trouble figuring out appropriate laws for induction squares in a double category.
Assume an object $\mathbb{N}$.
Assume horizontal arrows
zero $0\...

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### TR2 for homotopy category of stable $\infty$-category

I’m trying to understand Lurie’s proof that the homotopy category of a stable $\infty$-category is triangulated. In showing TR2, he constructs a diagram
$$\require{AMScd}
\begin{CD}
X @>f>> Y ...

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### Is every folk cofibration of strict $\omega$-categories a monomorphism?

In the folk model structure on the category $sCat_\omega$ of strict $\omega$-categories, the cofibrations are generated by the boundary inclusions $\{\partial \mathbb G_n \to \mathbb G_n \mid n \in \...

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### Unifying the $n$-truncation factorization system in a topos with the $n$-truncation factorization system of a t-structure

(This is a corrected and more detailed version of an earlier question.)
In good circumstances, e.g. the setting of an $\infty$-topos, we have for each $n\geq -1$ an orthogonal factorization system ...

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### Is there a Dold-Kan theorem for circle actions?

There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 ...

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### When is the category of connective objects a topos?

There is only one stable $\infty$-topos, namely the trivial category. However, the theory of stable $\infty$-categories with $t$-structure is strikingly reminiscent of the theory of topoi, as both ...

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### Is there a "duality involution" on presentable categories?

$\newcommand\Psh{\mathit{Psh}}\newcommand\Pr{\mathit{Pr}}$Let $\Psh$ be the category of presheaf categories and cocontinuous functors which preserve tiny objects. There is a functor $(-)^\ast : \Psh \...

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### Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?

$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and ...

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### What are the algebras for the laxification 2-monad?

Let $C$ be a small 2-category. Let $[C , Cat]$ denote the 2-category of strict functors to $Cat$, 2-natural transformations, and modifications. Let $[[ C, Cat ]]$ denote the 2-category with the same ...

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### Is Koszul duality a deformation theory when not over a field?

Let $k$ be a field. Then Thm 15.3.3.1 of Lurie's SAG says that Koszul duality, regarded as a contravariant endofunctor $\bar D$ of augmented $E_n$-algebras over $k$, is a deformation theory in the ...

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### Riemann-Hilbert-type correspondence for locally constant factorization algebras

This is related to a previous post, but a bit softer and should probably stand on its own.
In Appendix A of "Higher Algebra", Lurie shows that for a reasonably good topological space, there ...

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### $\mathbb{E}_M$ as colimit of little cubes operads

In Lurie's "Higher Algebra", Remark 5.4.5.2 towards the end, there is the following statement: "It follows that $\mathbb{E}_M$ can be identified with the colimit of a diagram of $\infty$...

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### A recursive attempt at $n$-dimensional coherence

For the purposes of this post we will use the one hom class definition of a category.
Note that a functor $F:\mathcal{C}\to\mathcal{D}$ between categories is a pair of functions $F_0:{\bf Ob}_\mathcal{...

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### Summary of different types of TQFT?

For the purposes of this question, a TQFT comprises the following data:
An "upper dimenison" $n \in \mathbb N$.
A "lower dimension" $0 \leq l \leq n$.
A choice of structure ...

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### Formula for the left adjoint of the nerve functor?

I recently stumbled upon a formula for the left adjoint of the nerve functor. Let $X$ and $Y$ be simplicial sets, then:
\begin{equation}
\mathbf{sSet}(X,Y)
\cong\mathbf{sSet}(\varinjlim_{\Delta^n\...

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### 2-categories for the working algebraic geometer

I study algebraic geometry / number theory and from time to time I stumble upon 2-categorical (co)limits. I have two main examples in mind:
Example 1) In étale cohomology, the (triangulated) derived ...

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### "Approximating" functors by Hom/Tensor product

Consider two dg-algebras $A,B$ and their respective derived categories $D(A),D(B)$. A natural way to give a covariant functor is to take an $(A,B)$-bimodule $X$ and to tensor with it, that is
$$D(A)\...

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### Subcategory of coherent objects in an $\infty$-topos forming a local $\infty$-pretopos

My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following:
Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the ...

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### Constructing sections of a cocartesian fibration

Suppose $\mathcal{E} \to \mathcal{C}$ is a cocartesian fibration over (the nerve of) a classical category, and there is a section on zero simplices that sends $C$ to $s(C)$ such that, for every edge $...

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### Questions about HTT lemma 4.1.3.2

Set $C\rightarrow D\in sSet_{/S}$ a categorical equivalence over $S$, $E\rightarrow S$ a categorical fibration. Why is $Map_S(D,E)\rightarrow Map_S(C,E)$ still a categorical equivalence?
When $S=*$, ...

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### Free $2$-category on a $2$-quiver

The construction of the free category on a quiver is standard in category theory.
Is there some $2$-dimensional analog of a quiver such that each $2$-quiver freely gives rise to a $2$-category? How ...

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### What is the union of subobjects in a 2-category?

I'm trying to understand how to take unions and intersections of subobjects in a 2-category (actually, I'm not 100% sure what subobjects are, but in the case I'm thinking about I have an idea). I ...

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### HTT 3.3.1.3.Reduce the class of inner anodyne morphisms to generators

This is the first part of Lurie's HTT 3.3.1.3:
Proposition 3.3.1.3. Let $p: X \rightarrow S$ be a Cartesian fibration and let $T \rightarrow$ $S$ be a categorical equivalence. Then the induced map $X ...

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### Factorization algebras as factorizable cosheaves on the (extended) Ran Space

A basic fact in the theory of factorization algebras is that, to state it in a rough way, the exit path category of the Ran space of a topological manifold $M$ is equivalent to the category consisting ...

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### Defining the bivariant mapping space functor via the twisted arrow $\infty$-category

I have what is surely a very silly question involving the construction of the bivariant mapping space functor $map_C(y,x):C^{op}\times C\to C$ starting from an $\infty$-category C. I’m using Land’s ...

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### Horizontal closure of 'almost $2$-categories'

Is there a reference discussing the notion of 'free horizontal closure' for an 'almost $2$-category', where all that's missing are some horizontal composites of $2$-cells?
The motivation for this ...

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### Graphical intuition of infinity-categories

I am new to $\infty$-categories and I am aware of how higher cells are represented as the unique fillers of inner horn. However, I'm struggling to gain the necessary graphical intuition beyond trivial ...

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### Are monomorphisms of strict $\omega$-categories stable under pushout along folk cofibrations?

Let $f : A \to B$ be a monomorphism of strict $\omega$-categories, and let $d : \partial \mathbb G_n \to A$ be an attaching map. There is an induced map $g : A \cup_{\partial \mathbb G_n} \mathbb G_n \...

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### Constructing lax limits from lax limits

Let $K$ be a 2-category. It's well-known that if $K$ has all PIE limits, then $K$ also has all lax limits. But I don't know a general "limit-decomposition" result which works "...

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### How to define the $\infty$-category of left fibrations?

In his book Introduction to Infinity-Categories, Land in his Theorem 3.3.16 asserts an equivalence of $\infty$-categories where one of the categories $\mathrm{LFib}(\mathcal C)$ is the full ...

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### Lurie's applications of $\infty$-topoi in topology

Lurie's book Higher Topos Theory is extremely interesting, but pretty overwhelming. I don't have the time to read it at the moment. However, the last chapter (7) gives applications of $\infty$-topoi ...

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### The $(\infty, 1)$-category of all topological spaces, including the bad ones

[Edit: Corrected some false claims and modified questions accordingly.]
Let $\mathcal{S}$ be the cocomplete $(\infty, 1)$-category generated by a point.
This is conventionally known as the $(\infty, 1)...

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### Counit functor associated to a bicartesian fibration

I would like to understand $\infty$ categorical adjunctions better. I am far from an expert, and so I would greatly appreciate published references (with no unproven foundational assumptions) ...

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### Is freeness of strict $n$-categories preserved by deleting a dimension?

Let $C$ be an $n$-dimensional globular set. Then for $0 \leq i \leq n$ there is a globular set $C^{(i)}$ obtained from $C$ by forgetting the $i$-cells (so that for $j > i$, the $j$-cells of $C$ ...

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### Does the Hochschild cohomology of an $A_{\infty}$-algebra have an algebra structure?

For an algebra $A$ we can define its Hochschild cohomology (see this Wikipedia page) $HH^{\cdot}(A,A)$. It is well-known that the cup product makes $HH^{\cdot}(A,A)$ a (graded-commutative) algebra.
...

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### nPOV: Cohomology and derived functors

In the nPOV, cohomology is realized as the connected component of the derived hom space [1]. Namely,
$$H^n(X,Y) = \pi_0 \mathbb{H}(X,B^nY),$$
where $\mathbb{H}$ is an $(\infty,1)$-topos and $B$ is the ...

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### When is the fat join a monoidal structure?

This question is about the following general construction.
Definition:
Let $(\mathcal C, \otimes)$ be a cocomplete, monoidal biclosed category whose unit $\ast$ is terminal. Let $I$ be an "...

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### Augmented algebras over $\infty$-operads via the envelope

Let $\mathcal{O}^\otimes$ be an $\infty$-operad and $\mathcal{C}^\otimes$ a symmetric monoidal $\infty$-category, both in the sense of Lurie's Higher Algebra.
By augmented $\mathcal{O}^\otimes$-...

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### Why are quasi-categories better than simplicial categories?

There are many models for $(\infty,1)$-categories: simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.
Doubtlessly the model most used to do higher category theory in ...

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### Is Joyal's category $\Theta_n$ the "only" Reedy category which is dense in $n$-categories?

Let $Cat_n$ denote the category of $n$-categories. Then Joyal's category $\Theta_n$ is (1) a full dense subcategory of $Cat_n$ which (2) is also a Reedy category.
Question: Is Joyal's category $\...

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### Lurie's HTT 2.4.1.8

I have some questions about how to apply proposition 2.4.1.8 in HTT which says:
In several places, variant forms of this proposition have been used without explanation such as in the proof of Lemma 2....

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### How to formulate the univalence axiom without universes?

The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot ...

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### Calculation of simplicial nerves

I know the general definition for simplicial nerve $\mathfrak{C}:Set_{\Delta}\rightarrow Cat_{\Delta}$.We define what are $\mathfrak{C}[\Delta^n]$ and $\mathfrak{C}[f]$ for morphisms between those $\...

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### Is there an analogue of Kan's $\operatorname{Ex}^\infty$ functor for $(\infty,n)$-categories?

$\DeclareMathOperator\Ex{Ex}$Kan's $\Ex^\infty$ functor (see Why is Kan's $Ex^\infty$ functor useful?) $\Ex^\infty\colon\mathsf{sSets}\to\mathsf{Kan}$ produces a Kan complex from a simplicial set ...

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### Does the 2-category of topoi have exponential objects?

Does the 2-category of Grothendieck topoi have exponential objects?
There are size issues: Since Grothendieck topoi are supposed to have a small set of generators, the collection of objects of a ...

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### Pseudomonoids versus monoidal pseudofunctors from $\Delta$

I have been trying to find some literature (if there is any) on the relationship between pseudomonoids and monoidal pseudofunctors from the monoidal theory of a monoid, $\Delta$ (I am interested in ...

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### What is the correct statement of Theorem 4.2 in Street's "Parity Complexes"?

Ross Street's 1991 paper Parity Complexes (apologies; I don't know how to find DOI links for Cahiers papers) develops some very useful tools for working with free strict $\omega$-categories. There is ...