Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,306
questions
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Monomorphisms in $\mathcal{C}\!at_\infty$
I'm trying to work through what the $(-1)$-truncated morphisms are in $\def\Catinf{\mathcal{C}\!at_\infty} \Catinf$.
BLUF: The correct characterization is that $F : C \to D$ is a (-1)-truncated map ...
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2
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What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
My question is as in the title:
Does anyone have an example (supposing one exists) of an
$\infty$-topos which is known not to be equivalent to sheaves on a
Grothendieck site?
An $\infty$-topos is as ...
4
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0
answers
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Generators and colimit-closures in higher categories
In ordinary categories $\mathcal{C}$, there are nice conditions under which a generator is also a colimit-generator for $\mathcal{C}$. In other words, under suitable conditions, if $\mathcal{C}$ ...
8
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1
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Is the simplicial nerve a localization?
Given a simplicial category $\mathcal{C}_{\ast}$ (if necessary, you may assume it's fibrant), denote as $\mathcal{C}$ its underlying ordinary category, and as $\mathcal{W}$ the class of all ...
2
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0
answers
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Grothendieck groupoid associated to bicategory
Given a finite abelian category $\mathcal{C}$, we can associate to $\mathcal{C}$ its Grothendieck group $\mathsf{Gr}(\mathcal{C})$, which is the free abelian group generated by isomorphism classes of ...
5
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0
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Are any formal systems based upon the idea of "iterated characterization pushing" currently in existence? If not, is anyone working on them?
I had an idea in regards to the design of formal systems with foundational aspirations.
To convey the idea, let's talk a bit about the second-order Peano axioms. The way these axioms work, we have a ...
1
vote
1
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198
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Computing units in a dg-algebra
Let $\mathbb{G}_m= Spec(k[z,z^{-1}])$ be the usual multiplicative group over a field $k$ viewed as a discrete commutative dg-algebra, and let $A$ be some arbitrary commutative dg-algebra concentrated ...
5
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Uniqueness of Witten-Dijkgraaf 2D TQFT at 0th dimension
If I understand correctly, Dijkgraaf-Witten TQFT in dimension 2 is the
following. Fix any finite group $G$, we define a field over a closed
2-manifold to be a principle $G$ bundle (it's automatically ...
3
votes
0
answers
101
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Asymmetric universal property for tensor products
Cartesian products have the following asymmetric description: $X\times Y=\coprod_{x\in X} Y$. I am looking for a similar description of tensor products (of abelian groups, for example).
If $A$ is an ...
4
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0
answers
149
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Adjunctions in a weak $2$-category
Is the notion of an adjunction well defined in an arbitrary weak $2$-category?
In particular, it seems like we need at least invertible unitors in the ambient $2$-category to write down the triangle ...
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4
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How should I think about presentable $\infty$-categories?
Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been ...
8
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3
answers
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Meaning of A-infinity relations
I am learning A-infinity category with Fukaya category in mind, and would like to understand the meaning of A-infinity relations.
In particular, as $N=1$, it means $dd=0$. As $N=2$, it means that $d$ ...
17
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2
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Correspondence between classes of model categories and classes of $\infty$-categories
We know by Karol Szumiło's thesis (https://arxiv.org/pdf/1411.0303.pdf) that there is an equivalence between the two fibration categories of cofibration categories on one side and cocomplete $\infty$-...
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Model structures on groupoids
Let me preface by saying that I'm very inexperienced with model categories. I was thinking about the following example, and I'm now wondering whether it fits into the theory of model stuctures:
...
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On 2 crossed modules
Let $(G,H,\alpha,\tau)$ and $(H,J,\alpha',\tau')$ be two crossed modules of groups. It is given that $Kernel(\tau) \cap Image(\tau')$ =$e$. For every $h\in H$ does there always exist a $j_h \in J$ ...
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Is simplicial localization part of a Quillen equivalence between relative categories and simplicial categories?
There are many models for $\infty$-categories. One of them is relative categories – AKA categories with weak equivalences – which have a model structure due to Barwick–Kan. Another one is simplicial ...
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Is it possible to use Feynman diagrams to represent a dot product $a \cdot b$?
Feynman diagrams are topological entities, but they describe linear
operators
It has been observed that Feynman diagrams are in particular string diagrams (morphisms in monoidal categories)) in ...
14
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1
answer
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Are locally presentable categories determined by their objects?
Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable categories. Assume that $f$ induces an equivalence between the groupoids underlying $\mathcal{C}...
5
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1
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Are constant $\infty$-sheaves constant on connected components?
Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty\text{Sh}(C, J)$. There is a
natural geometric morphism to $\infty\text{Grpd}$ whose ...
6
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1
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Higher categorical analogue of the equivalence between the category of representations of a monoid and the category of the monoid ring of the monoid
In classical algebra, there is a notion of "monoid rings" such that the functor taking monoids to the monoid rings is the left adjoint to the forgetful functor from the category of commutative rings ...
3
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1
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Rigidification of marked simplicial sets
It is well known that there exists a Quillen equivalence,
$$\mathfrak{C}: Set_{\Delta} \rightleftarrows Cat_{\Delta}: N, \enspace \mathfrak{C} \dashv N$$
between Joyal's model structure on ...
7
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1
answer
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The model category structure on $\mathbf{TMon}$
I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer.
I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...
3
votes
1
answer
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Proving a Kan-like condition for functors to model categories?
I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...
3
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0
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105
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Concerning the definition of a 2-crossed module
Question:
Is there some generalization of the definition of crossed-module which appropriately fits into the holonomy-considerations I am interested in and has, as an example, the generalization of ...
3
votes
1
answer
345
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Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers
Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
4
votes
1
answer
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Existence of pointwise Kan extensions in $\infty$-categories
This answer by Emily Riehl mentions a to-be-published proof of the fact that $\infty$-categorical (pointwise) Kan extensions exist when the target category admits (co)limits indexed by the relevant ...
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0
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Classifying Objects for Fibrations Defined by a Lifting Property
I have been studying weak factorization systems for their use in model categories. I am trying to use these to abstract away from a common phenomenon underlying fibrations. In brief, it seems as ...
2
votes
0
answers
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Simplicial sets of categories as models for $(\infty,1)$-categories [duplicate]
Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise.
In my understanding, there are several models for $(\infty,...
4
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Derived weight filtration on motivic Galois representations
Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for ...
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3
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Definition of $E_n$-modules for an $E_n$-algebra
The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more ...
7
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1
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Categorical Significance of Fibrations
It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
2
votes
2
answers
348
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Tensor schemes "with relations"
In "The geometry of tensor calculus I," Joyal and Street introduce the notion of a tensor scheme, a set of abstract objects, together with formal morphisms between words in this set. They then prove ...
5
votes
0
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CoCartesian vs. locally CoCartesian fibrations
Say $\pi: C\to J$ is an inner fibration of $\infty$-categories. Then "morally", $\pi$ corresponds to a diagram indexed by $J$ in the "category of categories with correspondences", and if $\pi$ is ...
8
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1
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Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie
I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of ...
6
votes
1
answer
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Tricategorical coherence
Why does coherence begin to matter at the tricategorical level?
It is well known that every weak $2$-category is equivalent to a strict $2$-category, with the equivalence essentially given by (unless ...
21
votes
1
answer
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Natural examples of $(\infty,n)$-categories for large $n$
In Higher Topos Theory, Lurie argues that the coherence diagrams for fully weak $n$-categories with $n>2$ are 'so complicated as to be essentially unusable', before mentioning that several dramatic ...
4
votes
1
answer
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Coherence theorem for tetracategories, weak $n$-categories
Is there a coherence theorem/conjecture for tetracategories (weak $4$-categories)?
Todd Trimble mentions in his notes on tetracategories that his pasting definitions are essentially unambiguous due ...
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votes
2
answers
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Composition map in $\infty$-categories
Let $\mathcal{C}$ be an $\infty$-category, and let $u:x\rightarrow y$ be an edge. It seems reasonable to say that: The map $Map_{\mathbb{C}[\mathcal{C}]}(a,x)\rightarrow Map_{\mathbb{C}[\mathcal{C}]}(...
3
votes
1
answer
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On the Group Structure of Morphism Set of a Strict 2-Group
The standard definition of a strict 2-group says that it is a strict monoidal category in which every morphism is invertible and each object has a strict inverse.
Also it is a well known fact that a ...
11
votes
1
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What does the homotopy coherent nerve do to spaces of enriched functors?
Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a ...
2
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0
answers
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Comparing cohomology using homotopy fibre
I have a question, which might be very basic, but I don't know enough topology to answer.
Suppose you have a map of topological spaces (or homotopy types) $f : X \to Y$, with homotopy fibre given by ...
3
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On cofibrations of simplicially enriched categories
Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells.
We have a canonical inclusion functor ,
$$i: C \...
14
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2
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Describing fiber products in stable $\infty$-categories
Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times_Z Y$ ...
9
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What is an example of a quasicategory with an outer 4-horn which has no filler?
A quasicategory has fillers for all inner horns $\Lambda^i[n]$ where $n\geq 2$ and $0<i<n$, but it need not have fillers for $i=0$ (or $i=n)$. In particular, for $n=2$ and $n=3$ there are easy ...
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When is Fun(X,C) comonadic over C with respect to the colimit functor?
Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular,...
5
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Are semisimplicial hypercoverings in a hypercomplete $\infty$-topos effective?
A hypercovering in an $\infty$-topos $E$ is a simplicial object $U \in {\rm Fun}(\Delta^{\rm op},E)$ such that each map $U_n \to ({\rm cosk}_{n-1} U)_n$ is an effective epimorphism. Theorem 6.5.3.12 ...
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2-categorical Yoneda embedding and limits
Does the 2-categorical Yoneda embedding defined in A.2 Section 5 of `A study in derived algebraic geometry' (Gaitsgory and Rozenblyum, version of 2018-11-14) preserve limits for any $(\infty,2)$-...
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Results relying on higher derived algebraic geometry
Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-...
9
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1
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Anabelian geometry ~ higher category theory
Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of ...
3
votes
1
answer
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Geometry of 2-arrows
It is frequently said that one of the contributions of Grothendieck to geometry was to systematically think about the properties of morphisms, as opposed to the properties of spaces themselves (the ...