Questions tagged [higher-category-theory]

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

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Different levels of isomorphism/equivalence/adjunction between bicategories

What are all the different levels of 'isomorphism/equivalence/adjunction' we can have between bicategories? Do any of them 'collapse' to one-another? When working with $1$-categories, we have four ...
Alec Rhea's user avatar
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Left anodyne is covariant equivalence

I have a question about the proof of left anodyne between two simplicial sets over $S$, where $S$ is a simplicial set, is covariant equivalence. In the proof (HTT 2.1.4.6 or https://ncatlab.org/nlab/...
Peter Liu's user avatar
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$\infty$-natural transformations and adjunctions

I'm having troubles proving these two related statements, which are immediate for 1-categories and should of course be true for $\infty$-categories: Given a natural transformation $\alpha: f \...
Giulio Lo Monaco's user avatar
7 votes
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585 views

Understanding the higher stack of perfect complexes

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff: We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero ...
Martin Hurtado's user avatar
4 votes
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Universal six-functor formalism on an $\infty$-category

In the article The Universal Six-Functor Formalism by Brad Drew and Martin Gallauer it is proved that for an ordinary category $S$ with a wide subcategory $P$ of 'smooth morphisms' containing all ...
Bastiaan Cnossen's user avatar
12 votes
1 answer
802 views

Modern proofs for simplicial localizations

I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the ...
Giulio Lo Monaco's user avatar
7 votes
1 answer
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Is the canonical model structure on strict $\infty$-Cat left proper?

Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in A folk model structure on omega-cat left proper ? All ...
Simon Henry's user avatar
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3 votes
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Coherence $2$-cells in braided monoidal bicategories

In a braided monoidal category $(\mathcal{C},\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}},\alpha,\lambda,\rho,\beta)$, we have $\beta_{\mathbf{1}_{\mathcal{C}},\mathbf{1}_{\mathcal{C}}}=\mathrm{id}_{...
Emily's user avatar
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3 votes
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$2$-dimensional adjunctions via co/Cartesian fibrations $\mathcal{M}\longrightarrow[1]$

Recall [HTT, Definition 5.2.2.1]: Definition 5.2.2.1. Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories. An adjunction between $\mathcal{C}$ and $\mathcal{D}$ is a map $q\colon\mathcal{M}\to\...
Emily's user avatar
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9 votes
2 answers
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Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?

I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning. Let $M$ be a manifold, and consider the presheaf $C^*(-,...
David Corwin's user avatar
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Is there a good general definition of "sheaves with values in a category"?

Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
Zhen Lin's user avatar
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Relation Hopf categories and categorified quantum groups

In the paper Hopf Categories Crane and Frenkel gave a definition of a Hopf category, which they considered as a categorification of a quantum group. Categorifications of quantum groups have later been ...
mtraube's user avatar
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1 answer
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Counterexamples concerning $\infty$-topoi with infinite homotopy dimension

In "Higher Topos Theory", Lurie introduces three different notions of dimension for an $\infty$-topos $\mathcal{X}$, namely: Homotopy dimension (henceforth h.dim.), which is $\leq n$ if $n$-...
Markus Zetto's user avatar
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Is there a bicategorical Yoneda lemma for marked lax transformations?

The bicategorical Yoneda lemma (see [Johnson–Yau, Chapter 8]) states that, given a bicategory $\mathcal{C}$ and a pseudopresheaf $\mathcal{F}\colon\mathcal{C}^{\mathsf{op}}\to\mathsf{Cats}_{\mathsf{2}}...
Emily's user avatar
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Explicit description of a pullback of $(2,1)$-categories

In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\...
EBP's user avatar
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1 answer
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$\infty$-topoi versus condensed anima

Let $ExDisc_\kappa$ denote the category of $\kappa$-small extremally disconnected topological spaces (for now fix a strong limit cardinal $\kappa$). There's a functor $ExDisc_\kappa \to \mathsf{RTop}$ ...
Maxime Ramzi's user avatar
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2 votes
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Can we simplify the definition of a biadjunction using coherence for bicategories?

In Biequivalences in Tricategories, Gurski defines a biadjunction of bicategories as a sextuple $(F,G,\eta,\epsilon,\Gamma,\Sigma)$ with $F\colon\mathcal{C}\longrightarrow\mathcal{D}$ and $G\colon\...
Emily's user avatar
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4 votes
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$2$-dimensional adjunctions via pre/post-composition

Recall that in the setting of $1$-categories, given functors $L\colon\mathcal{C}\longrightarrow\mathcal{D}$ and $R\colon\mathcal{D}\longrightarrow\mathcal{C}$, the following conditions are equivalent: ...
Emily's user avatar
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2 votes
1 answer
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full and faithful on mapping spaces

I need to prove or find a counterexample to the following: Let $C$ and $D$ be two $\infty$-categories. Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a full and faithful functor. Then for any $\infty$...
Wilson Forero's user avatar
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Higher categorical / operadic approach to homotopy associative, homotopy commutative, $H_\infty$ ring spectra?

Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $O$ be an operad (for example, $O$ could be an $A_m$ or $E_n$ operad or a tensor product thereof, and $\mathcal C$ could be spaces ...
Tim Campion's user avatar
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3 votes
1 answer
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Is the notion of a 2-category introduced to fix/forget the size issues in the definition of (an ordinary) category?

A category $\mathcal{C}$ consists of pair of classes $(\mathcal{C}_0, \mathcal{C}_1)$, along with maps $$\mathcal{C}_1\times_{\mathcal{C}_0}\mathcal{C}_1\rightarrow \mathcal{C}_1\rightrightarrows \...
Praphulla Koushik's user avatar
14 votes
1 answer
473 views

"Very lax" $2$-dimensional co/limits

In the setting of $1$-categories, there are two (unweighted) variant notions of limits, namely limits and colimits. For bicategories, there are sixteen of them: Each of these notions has an ...
Emily's user avatar
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3 votes
1 answer
362 views

Is every $\otimes$-invertible object "coherently sym-central"?

Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $L \in \mathcal C$ be a $\otimes$-invertible object. Then the braiding $L \otimes L \to L \otimes L$ is simply multiplication by $\...
Tim Campion's user avatar
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6 votes
1 answer
364 views

Lemma 5.4.5.11 of HTT

In Lemma 5.4.5.11 of HTT, the proof given relies on Lemma 5.4.5.10. However it seems that Lurie applies Lemma 5.4.5.10, which requires the given simplicial set to be contractable, to an arbitrary $\...
Sil Linskens's user avatar
5 votes
1 answer
474 views

Remark 5.4.2.15 in HTT

In Remark 5.4.2.15 in Higher Topos Theory, Lurie explains in what sense an accessible functor $F:\mathcal{C}\rightarrow \mathcal{D}$ between accessible $\infty$-categories is "determined by small ...
Sil Linskens's user avatar
9 votes
1 answer
1k views

How do $\infty$-categories allow us to do descent on the derived level?

I have heard that one application of $\infty$-categories is that they allow us to formulate a meaningful theory of descent for derived categories (say of sheaves on a scheme). While I'm sure the ...
Kim's user avatar
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On coalgebras and comodules in slice $\infty$-categories

Given a presentable Cartesian symmetric monoidal $\infty$-category $C$, every object is a cocommutative comonoid and for a fixed $Z\in C$ there is an equivalence $C_{/Z}\simeq LCoMod_{Z}(C)$ where the ...
Jonathan Beardsley's user avatar
25 votes
1 answer
1k views

What are Koszul dualities?

I am bewildered by the number of things I've heard referred to as "Koszul duality", and I would like to sort it out. At various different times, I believe I've seen any of the following ...
Tim Campion's user avatar
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5 votes
2 answers
582 views

Examples of categories cofibered in groupoids

In Chapter 2 of Lurie's Higher Topos Theory, the first main theorem establishes a connection between categories cofibered in groupoids and left fibrations and asserts the importance of studying left ...
afdsfasdf's user avatar
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1 answer
416 views

Do stalks see epimorphism of stacks?

Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $2$-sheaves. According to https://mathoverflow.net/q/307366, this is an epimorphism if and only if it is almost surjective,...
curious math guy's user avatar
3 votes
0 answers
162 views

Obstruction to delooping

Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this ...
Student's user avatar
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9 votes
1 answer
1k views

Cohesion relative to a pyknotic/condensed base

Something that usefully emerged for me from this discussion and follow-up MO question is that rather than see cohesiveness and condensedness/pyknoticity in rivalry with one another, as my initial ...
David Corfield's user avatar
3 votes
1 answer
159 views

Higher-dimensional paths as parametrizations of 1-dimensional paths

Sect. 6.7 of the HoTT Book establishes in the context of CW complexes that "we can obtain an n-dimensional path as a continuous family of 1-dimensional paths parametrized by an (n − 1)-...
Maximilian Doré's user avatar
9 votes
0 answers
195 views

A bicategorical representation theorem

The representation theorem in $1$-category theory states that a presheaf $\mathcal{F}$ is representable iff $\int_\mathcal{C}\mathcal{F}$ has a terminal object. When one tries to formulate a ...
Emily's user avatar
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4 votes
1 answer
349 views

Is $Set$ a tiny topos?

Let $Topos$ be the $(2,1)$-category of Grothendieck toposes and geometric morphisms. This is a $V$-sized, locally $V$-sized, locally locally small $(2,1)$-category with all small (2,1)-colimits (=...
Tim Campion's user avatar
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15 votes
1 answer
1k views

Is there a condensed / pyknotic refinement of the shape of an $\infty$-topos?

Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow ...
Tim Campion's user avatar
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6 votes
1 answer
574 views

Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$?

In Proposition 5.5.7.6 of Lurie's Higher Topos Theory, Lurie states that the $\infty$-category $\operatorname{Pr}^R_{\omega}$ of compactly generated $\infty$-categories and filtered-colimit-preserving ...
Bastiaan Cnossen's user avatar
14 votes
2 answers
729 views

A construction of the universal cocartesian fibration

Usually I see the universal (small) cocartesian fibration $\mathcal{Z} \to Cat_\infty$ constructed in a relatively opaque fashion, such as via the unstraightening construction. I've stumbled on what ...
UniversalCocartesianFibration's user avatar
14 votes
0 answers
289 views

Which limits distribute over which colimits in $Set$? How about in $Spaces$?

I've never really thought much about distributivity of limits and colimits -- I tend to think more about commutativity of limits and colimits. This question makes me want to change that. The question ...
Tim Campion's user avatar
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2 votes
1 answer
105 views

Reference for proof that vertical composition of pseudonatural transformations is a pseudonatural transformation

Is there a standard reference for the fact that the vertical composition of two pseudonatural transformations between pseudofunctors between bicategories is a pseudonatural transformation? Recall that ...
Alec Rhea's user avatar
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12 votes
3 answers
1k views

Elementary theory of the category of groupoids?

One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, ...
user avatar
27 votes
1 answer
935 views

"Non-categorical" examples of $(\infty, \infty)$-categories

This title probably seems strange, so let me explain. Out of the several different ways of modeling $(\infty, n)$-categories, complicial sets and comical sets allow $n = \infty$, providing ...
5 votes
1 answer
232 views

Categorical Morita equivalence implies equivalence of module categories?

Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true ($R$-Mod) $\simeq$ ($S$-Mod). $S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective ...
Student's user avatar
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2 votes
0 answers
157 views

The derived $\infty$-category of sheaves on a site is closed symmetric monoidal

Let $X$ be a quasicompact semiseparated scheme. I am trying to recover the (closed) symmetric monoidal structure on $\mathcal{D}(\mathrm{QCoh}(X))$, the derived $\infty$-category of quasicoherent ...
Adrien MORIN's user avatar
3 votes
1 answer
234 views

Hom-space between Picard stacks

This is from Deligne, La formule de dualité globale, SGA 4, tome 3, Expose XVIII, and I am confused about how the hom-space between Picard stacks is again a Picard stack. A quick rewind. For a site $S$...
Frid Fu's user avatar
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2 votes
1 answer
276 views

question about notation in HTT of J.Lurie

In page 27 in HTT of J.Lurie, the expression $$\text{Map}_S(X,Y):=Y^X\times_{S^X}\{\phi\}\in \text{Set}_\Delta$$ appears for simplicial set $X,Y,S$ in Warning 1.2.2.2. However, I couldn't understand ...
afdsfasdf's user avatar
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6 votes
2 answers
856 views

Multicategories vs Categories

One of the initial motivating factors for learning category theory, besides needing it for my work, was the idea that almost all mathematical notions I would encounter could be understood using ...
Alec Rhea's user avatar
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2 votes
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About adding strict unity to bicategories and the strictification functor

I use the terminology of https://ncatlab.org and: [CT] Coherence for tricategories - R. Gordon, A. J. Power, Ross Street . [EK] Closed Categories - Samuel EilenbergG. Max Kelly . Strict unity I ...
Buschi Sergio's user avatar
6 votes
1 answer
373 views

What are some "good" examples of Kan simplicial manifolds?

According to the definition 1.1 of the paper Kan Replacement of simplicial manifolds by Chenchang Zhu https://arxiv.org/pdf/0812.4150.pdf, A Kan simplicial manifold is a simplicial manifold $X$ such ...
Adittya Chaudhuri's user avatar
4 votes
0 answers
73 views

Coherence for monoidal 2-categories vs coherence for braided monoidal categories

Gordon, Power and Street have proven that every monoidal 2-category is equivalent to a Gray monoid. This means that the only coherence 2-isomorphisms we have to be concerned about are the ...
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