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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A doctrine is a categorification of a theory
So it says in the nlab page of doctrine.
Let's focus on first order theories for simplicity.
I have two questions, one regarding vertical categorification and another regarding horizontal ...
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Explicit description of straightening over a point
In Higher Topos Theory Lurie gives the following description of a cosimplicial simplicial space $Q^\bullet$ which geometric realization gives the straightening over a point:
For $n \geq 0$ let $P_{[n]}...
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Reconstruction of commutative differential graded algebras
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...
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Prove a space is contractible
I encountered this problem while reading the proof of Lemma 1.2.4.17 in Jacob Lurie's Higher Algebra
Recall that a topological simplex $|\Delta^n|$ can be identified with $\{0\leq x_1\leq\cdots\leq ...
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Is an epimorphism fibered in contractible kan complexes an acyclic kan fibration?
Suppose $f : X \to Y$ is a morphism of simplicial sets which is degreewise surjective and where the fiber over any vertex of $Y$ is an acyclic kan complex. Is $f$ necessarily an acyclic kan fibration? ...
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Conjugation Functor from a Groupoid to $\mathbf{Grp}$
Take a groupoid $\mathcal{C} \in \mathbf{Grpd}$. It's possible to construct a conjugation functor $F_{\mathcal{C} } : \mathcal{C} \to \mathbf{Grp}$ as follows:
For every object $x \in \text{ob}(\...
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Compact objects in the category of $n$-categories
An object in a category is called compact if the functor corepresented by it commutes with filtered colimits. Now what are the compact objects in the category of $n$-categories?
It was suggested in ...
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Is there more concise representation of simplicial sets than complete enumeration of its simplices and gluings - application to ∞-categories?
Is there more concise representation of simplicial sets than complete enumeration of its simplices and gluings. As special simplicial sets (e.Kan complexes) are models for the ∞-categories, then this ...
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Functor with two successive adjoints is a localization
Assume that we have a fully faithful functor $f_{!} \colon \mathcal{C} \to T^{-1}\mathcal{P}(\mathcal{G})$ where $\mathcal{G}$ is some small $\infty$-category and $T$ is a strongly saturated class of ...
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(Reference) Can every category be turned into a category of categories (monoids)?
Note: The following construction seems very straightforward, so a pointer to a reference discussing it in detail, or identifying a standard term used to describe the construction which would allow one ...
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Characterisation of terminal category in the 2-category sense
On nlab https://ncatlab.org/nlab/show/terminal+category, it is stated that a category is terminal in the 2-category sense precisely when it is inhabited and indiscrete. I wanted to try to prove this ...
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Geometric realization of a simplicial set depends functorially on the simplicial set
On Kerodon (subsection 1.1.8) the following definition is given for the geometric realization of a simplicial set:
Let $S_{\bullet}$ be a simplicial set and let $Y$ be a topological space. We will ...
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Mathematical efforts to embed any object into n-dimensional Euclidean space (or spaces)?
There are efforts to automate proof discovery using deep neural networks. There are varied approaches how the mathematical objects can be embedded into the layer of neural networks.
E.g. one can ...
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What can we say about the collection of sets $\{s_{ij}\}$ for some particular topos $T$?
Let $X$ be some space and let $T$ be topos on $X$ (e.g. Grohtendieck topos on the topological space).
Topos $T$ is the category of sheaves ${S_i}$, where each sheaf $S_i$ maps each open subset $O_j$ ...
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Category objects in $\infty$-groupoids vs Complete Segal Spaces
From my understanding, one way to motivate Complete Segal Spaces is to see them as $(\infty,1)$-category objects inside the $(\infty,1)$-category of spaces, or rather $\infty$-groupoids, represented ...
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Help me understand the diagram category
In category theory a diagram in a category $\textbf{C}$ is a functor $D:\textbf{J}\to\textbf{C}$, where $\textbf{J}$ is a small category. I want to understand categories of diagrams. The only ...
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Construction of 2-category of monoidal categories and (lax) monoidal functors as strict algebra category of a 2-monad
As motivation for 2-monads, I would like to understand an explicit construction of the 2-monad $T$ of which derived 2-category $T-\operatorname{Alg}_l$ of algebras as described in Lack's 2-categories ...
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Building Ab from Set just as we build Spectra from Top
In the land of $\infty$-categories, we construct Spectra via starting from pointed topological space, and stabilizing via inverting loopspace.
In the hell of $1$-categories, the analog of pointed ...
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Motivating the definition of adjoint equivalence
Recently I have been trying to convince myself that the most natural definition for equivalence between categories is the notion of adjoint equivalence rather than simply equivalence. Of course, every ...
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Standard $n$-simplex visualization
For $n \in \mathbb{N}$, the standard simplicial $n$-simplex $\Delta[n]$ is the simplicial set which is represented (as a presheaf) by the object $[n]$ in the simplex category, so $\Delta[n]= \Delta(-,[...
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Computing the homotopy limit of a constant diagram.
Let $X$ be a nice space, and view it as an $\infty$-groupoid via its singular simplicial set. Consider the constant functor $\mathbb{S}$ valued functor to Spectra, mapping all simplices to the sphere ...
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Is there some $Ext$ group that classifies extensions in the **Derived Category**?
Let $\mathcal{A}$ be an abelian category, we have $D(\mathcal{A})$
For $A,C \in \mathcal{A}$, we know that $Ext^1(A,C)$ classifies extensions $0 \to C \to B \to A \to 0$
I want an analog for $Der(\...
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Characterization of boundary in simplicial sets
I am reading kerodon, and I am stuck on exercise 1.1.2.8., which characterizes the simplicial maps of $\partial \Delta^{n}$ on a simplicial set $S_{\cdot}$ as lists $(\sigma_{1},...,\sigma_{n})$ of $n-...
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Can higher dimensional categories be represented as hypergraph and if not why?
As the diagrams of categories can be represented as graphs of objects and morphisms, I was wondering if (the diagrams of) higher-dimensional categories could be represented as hypergraphs, and if not ...
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are comma-objects semi-coflexible
I need someone to read through my proof, because I feel very uncertain about 2-categorical limits. A strict indexed category $C:\mathscr S^{op}\to Cat$ is semi coflexible when every pseudo-...
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Shortest path from undergrad to the (co)tangent complex?
After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being ...
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Motivation for Excisive Functors
In Goodwillie's Calculus I he defines an excisive functor to be one, which maps homotopy pushouts to homotopy pullbacks.
Why the change of universal properties?
It is surprisingly hard to find ...
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Natural transformations of functors $\rm Cat\to Cat$
For any two categories $C$, $D$ denote by $[C,D]$ the category of functors $C\to D$. Fixed two categories $C$, $C'$, there are two functors $\rm Cat\to Cat$, namely $[C,-]$ and $[C',-]$; suppose to ...
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Cat-enriched right adjoint
In Streicher's text "Fibered categories" it is described how to construct a 2-functor $Sp:Fib_\mathscr S\to 2Cat_s(\mathscr S^{op},Cat)$ such that $R = Sp\circ \int$ is right 2-adjoint to ...
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Definition of a contractible category
A simplicial set can be interpreted both as a generalized space as well as a generalized category. While reading about bisimplicial sets I came across the statement that the spatial $\Delta^{1}$ is ...
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Interpretation of HoTT in the Reedy model structure on bisimplicial sets [closed]
I was trying to understand the interpretation of HoTT in the Reedy model structure on bisimplicial sets. While going through, it suggests to think of bisimplicial sets as having a "spatial" ...
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How are the horizontal maps in the push out square defined? [closed]
Let A be an Eilenberg-Zilber category and $X\subset Y$ be presheaves over A. For any non-negative integer n, there is a canonical push out square $\require{AMScd}$
\begin{CD}
\sqcup_{y\in \sum}\...
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How do you calculate the partition function on a manifold-with-corners in extended TQFT?
I'm a physicist trying to study Topological Quantum Field Theory (TQFT), so apologies if the following has some basic mistakes or misuse of terminology. When answering please bear in mind that I'm not ...
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Reedy fibrant replacement for Segal categories
Suppose $C$ is a Segal category, that is a functor $X: \Delta^{op} \rightarrow Spaces$ so that $X_0$ is discrete and satisfies the Segal condition $X_k \cong X_1 \times_{X_0} \cdots \times_{X_0} X_1$. ...
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$\mathrm{Top}$ as a double category
In notes of Susan Niefields about double categories (http://www.tac.mta.ca/tac/volumes/26/26/26-26.pdf) one of the first examples of a double category is a double category structure on $\mathrm{Top}$, ...
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Natural transformation between functors induces homotopy between maps on the nerve
I'm learning what is the nerve of a category and I saw that if we have a natural transformation between two functors F and G, then we have a homotopy between the maps N(F) and N(G) where N(-) denotes ...
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State-sum construction of the Drinfeld center of a fusion 2-category
If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-...
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$\mathfrak{h}(\mathcal{G}^{\Delta^1})$ is the category of commutative squares in $\mathfrak{h}\mathcal{G}$
This is related to the question Homotopy category of a symmetric monoidal $(\infty,1)$-category is symmetric monoidal.
Let us fix an $(\infty,0)$-category $\mathcal{G}$ (a Kan complex). We want to ...
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Is there the notion of a partial mapping in category theory?
In category theory we are largely concerned with mappings between objects. These could be the mappings between the objects within a category (e.g. connections between members of a set that makeup the ...
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Realization of a limit as a set of natural transformations
Consider the slice category associated to the Yoneda embedding $\mathcal{C}$ $\to$ $\mathcal{PSh(C)}$. Now the lemma says, every presheaf is a colimit of representables. So we resort to Yoneda lemma ...
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What are E-modules from the point of view of derived algebraic geometry?
For a complex manifold $X$, there exists a ring of formal differential operators on $T^* M$, normally denoted as $\mathscr{E}_X$. Locally, a section of $\mathscr{E}_X$ is given by a formal sum
$$P = \...
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Homotopy category of a symmetric monoidal $(\infty,1)$-category is symmetric monoidal
Let $\mathcal{C}$ be symmetric monoidal $(\infty,1)$-category, that is, a functor $\mathcal{C} \colon \Delta^{\text{op}} \times \Gamma^\text{op} \to \text{sSet}$, where $\Delta$ is the simplex ...
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Hochschild homology of stable categories as topological chiral homology
Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$.
Its Ind-completion $\mathscr{C} := \operatorname{Ind}(\mathscr{C}_0)$ ...
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Understanding the Beck-Chevalley condition (II)
The older question here on the site asks about the intuitive meaning of the Beck-Chevalley condition. Accidentally one of the answers has caught my eye.
It made me wonder if it is possible to describe ...
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Where does one learn how to apply categorical algebra and higher abstractions to algebraic topology?
Tl;Dr: I know higher category theory and algebra is used ubiquitously in advanced algebraic topology. However, every time I ask someone, or try to find out, how one actually learns to apply the higher ...
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Ambiguity in usage of $\tau_{\leq k}$
In HTT, 5.5.6.19, the possible ambiguity of the notation $\tau_{\leq k} \mathcal C$ is discussed. By the first definition, it denotes the subcategory on the $k$-truncated objects but it can also be ...
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Notation for the data of Double Category
I have some problems to understand a notation introduced in Ronald Brown's Crossed complexes and higher homotopy groupoids
as non commutative tools for higher dimensional
local-to-global problems (p ...
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An infinity-category is a functor, but is it a category in some way and is the simplex notions so universal?
For some time I had the impression that infinity-categories are just a generalization of higher order categories: categories whose arrows have arrows among them, and they have further arrows etc. ...
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Is there algebra (algebraic manipulation) of simplicial sets?
Is there algebra of simplicial sets? For example, symbolic representation of simplicial sets and operations on those representations that allow to construct new simplicial set from existing one – join,...
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When is the infinite suspension functor fully faithful?
Given a presentable $\infty$--category $\mathscr{C}$, one has a left adjoint $\Sigma^\infty_+ : \mathscr{C} \to \text{Sp}(\mathscr{C})$ to the infinite loop functor $\Omega^\infty : \text{Sp}(\mathscr{...
