Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,305
questions
6
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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 ...
2
votes
0
answers
76
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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/...
6
votes
1
answer
412
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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 \...
7
votes
0
answers
585
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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 ...
4
votes
0
answers
308
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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 ...
12
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1
answer
802
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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 ...
7
votes
1
answer
288
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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 ...
3
votes
1
answer
152
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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}_{...
3
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0
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82
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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\...
9
votes
2
answers
726
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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^*(-,...
28
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3
answers
3k
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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 ...
2
votes
0
answers
507
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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 ...
2
votes
1
answer
440
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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 $n$-...
1
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0
answers
117
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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}}...
4
votes
1
answer
246
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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 $F:C\...
15
votes
1
answer
2k
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$\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 $ExDisc_\kappa \to \mathsf{RTop}$ ...
2
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0
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134
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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\...
4
votes
0
answers
103
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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:
...
2
votes
1
answer
138
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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 $\infty$...
7
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0
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199
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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 ...
3
votes
1
answer
337
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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 \...
14
votes
1
answer
473
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"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 ...
3
votes
1
answer
362
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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 $\...
6
votes
1
answer
364
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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 $\...
5
votes
1
answer
474
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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 ...
9
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1
answer
1k
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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 ...
3
votes
0
answers
155
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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 ...
25
votes
1
answer
1k
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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 ...
5
votes
2
answers
582
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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 ...
4
votes
1
answer
416
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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,...
3
votes
0
answers
162
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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 ...
9
votes
1
answer
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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 ...
3
votes
1
answer
159
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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)-...
9
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0
answers
195
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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 ...
4
votes
1
answer
349
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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 (=...
15
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1
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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 ...
6
votes
1
answer
574
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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 ...
14
votes
2
answers
729
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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 ...
14
votes
0
answers
289
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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 ...
2
votes
1
answer
105
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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 ...
12
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3
answers
1k
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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, ...
27
votes
1
answer
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"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
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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 ...
2
votes
0
answers
157
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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 ...
3
votes
1
answer
234
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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$...
2
votes
1
answer
276
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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 ...
6
votes
2
answers
856
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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 ...
2
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0
answers
76
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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 ...
6
votes
1
answer
373
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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 ...
4
votes
0
answers
73
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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 ...