Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
961
questions
5
votes
1answer
244 views
Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself?
Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, ...
7
votes
0answers
97 views
Endofunctors of the surface category
Let $\mathrm{Cob}_2$ be the symmetric monoidal $(\infty,1)$-category whose objects are closed oriented $1$-manifolds and whose morphisms are compact oriented surface bordisms. (Higher morphisms are ...
6
votes
0answers
106 views
2-fibrations in terms of comma categories
Fibrations of 2-categories and bicategories were defined by Hermida and Bakovic, respectively, and their Grothendieck construction was studied by Buckley in Fibred 2-categories and bicategories. ...
4
votes
0answers
63 views
Braided monoidal categories as doubly degenerate tricategories
The fact that any tricategory with a single 0-cell and a single 1-cell is a braided monoidal category seems to be widely known. For instance, it appears in the "periodic table" of n-...
8
votes
1answer
171 views
Filtered 2-colimits commute with finite 2-limits
Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only ...
9
votes
0answers
186 views
Classification of derived formal group laws
Denote by $SCR$ the $\infty$-category of "simplicial commutative rings" (i.e. the nonabelian derived category of the category of finitely generated polynomial rings). Given $R \in SCR$, one ...
6
votes
0answers
174 views
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
0answers
59 views
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
1answer
243 views
$\infty$-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
0answers
453 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 ...
4
votes
0answers
167 views
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 ...
9
votes
1answer
347 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 ...
5
votes
1answer
195 views
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
1answer
105 views
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
votes
0answers
57 views
$2$-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\...
8
votes
2answers
633 views
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^*(-,...
21
votes
3answers
1k views
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
0answers
178 views
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
1answer
233 views
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
vote
0answers
64 views
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}}...
3
votes
1answer
109 views
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\...
13
votes
1answer
516 views
$\infty$-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
votes
0answers
67 views
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\...
1
vote
0answers
54 views
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
votes
0answers
167 views
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
1answer
305 views
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 \...
11
votes
1answer
154 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 fourteen of them:
Here $\mathsf{LaxCones}(\Delta_{X},D)\...
3
votes
1answer
255 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 $\...
5
votes
1answer
247 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 $\...
5
votes
1answer
371 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 ...
5
votes
1answer
416 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 ...
2
votes
0answers
116 views
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 ...
21
votes
0answers
444 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 ...
5
votes
2answers
408 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 ...
3
votes
1answer
266 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,...
3
votes
0answers
143 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 ...
9
votes
1answer
939 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 ...
2
votes
1answer
121 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)-...
9
votes
0answers
140 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 ...
4
votes
1answer
223 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 (=...
13
votes
1answer
517 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 ...
5
votes
1answer
163 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 ...
11
votes
2answers
353 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 ...
12
votes
0answers
176 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 ...
2
votes
1answer
71 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 ...
11
votes
3answers
663 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, ...
20
votes
1answer
401 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 ...
3
votes
1answer
108 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 ...
2
votes
0answers
100 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 ...
3
votes
1answer
166 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 $S$...
