Questions tagged [higher-category-theory]

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

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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 ...
Questioner's user avatar
51 votes
2 answers
4k views

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 ...
Charles Rezk's user avatar
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4 votes
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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 $\mathcal{C}$ ...
Exit path's user avatar
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8 votes
1 answer
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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 ...
Giulio Lo Monaco's user avatar
2 votes
0 answers
119 views

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 ...
Bipolar Minds's user avatar
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0 answers
216 views

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 ...
goblin GONE's user avatar
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1 vote
1 answer
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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 ...
Exit path's user avatar
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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 ...
Student's user avatar
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3 votes
0 answers
101 views

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 ...
John Berman's user avatar
4 votes
0 answers
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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 ...
Alec Rhea's user avatar
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30 votes
4 answers
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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 ...
Patriot's user avatar
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8 votes
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 $d$ ...
Student's user avatar
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17 votes
2 answers
445 views

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 $\infty$-...
Giulio Lo Monaco's user avatar
5 votes
0 answers
231 views

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: ...
unknownymous's user avatar
2 votes
0 answers
118 views

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 $j_h \in J$ ...
Adittya Chaudhuri's user avatar
4 votes
0 answers
157 views

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 ...
Najib Idrissi's user avatar
6 votes
1 answer
318 views

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 ...
Jack's user avatar
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14 votes
1 answer
599 views

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}...
G. Stefanich's user avatar
5 votes
1 answer
292 views

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 ...
merle's user avatar
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6 votes
1 answer
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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 ...
kenta kobayashi's user avatar
3 votes
1 answer
243 views

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 ...
F.Abellan's user avatar
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7 votes
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 ...
Matt's user avatar
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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 ...
Julian Chaidez's user avatar
3 votes
0 answers
105 views

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 ...
cheyne's user avatar
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3 votes
1 answer
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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 ...
Nicky's user avatar
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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 ...
Robin Stoll's user avatar
1 vote
0 answers
116 views

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 ...
Ronald J. Zallman's user avatar
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,...
Julian Chaidez's user avatar
4 votes
0 answers
165 views

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 ...
Dmitry Vaintrob's user avatar
9 votes
3 answers
1k views

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 ...
Exit path's user avatar
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7 votes
1 answer
477 views

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}]...
Ronald J. Zallman's user avatar
2 votes
2 answers
348 views

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 ...
Ben Webster's user avatar
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5 votes
0 answers
221 views

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 ...
Dmitry Vaintrob's user avatar
8 votes
1 answer
715 views

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 ...
Robin Stoll's user avatar
6 votes
1 answer
247 views

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 ...
Alec Rhea's user avatar
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21 votes
1 answer
843 views

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
164 views

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 ...
Alec Rhea's user avatar
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3 votes
2 answers
572 views

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}]}(...
user09127's user avatar
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3 votes
1 answer
181 views

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 ...
Adittya Chaudhuri's user avatar
11 votes
1 answer
802 views

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 ...
Jonathan Beardsley's user avatar
2 votes
0 answers
291 views

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 ...
unknownymous's user avatar
3 votes
0 answers
72 views

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 \...
F.Abellan's user avatar
  • 447
14 votes
2 answers
1k views

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 $X \times_Z Y$ ...
Question Machine's user avatar
9 votes
2 answers
452 views

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 ...
Matt Feller's user avatar
10 votes
0 answers
325 views

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,...
Jonathan Beardsley's user avatar
5 votes
1 answer
340 views

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 ...
Mike Shulman's user avatar
6 votes
0 answers
162 views

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 $(\infty,2)$-...
Wonderfield's user avatar
12 votes
2 answers
2k views

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 $n$-...
user avatar
9 votes
1 answer
2k views

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 ...
prdnr's user avatar
  • 111
3 votes
1 answer
229 views

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 ...
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