Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,306
questions
6
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Does there exist an explicit formula for the underlying object of a coproduct of coherently associative algebras?
Lurie shows in "Higher Algebra", that the underlying object of a coproduct of commutative algebras in a symmetric monoidal $\infty$-category can be computed as the tensor product of the ...
2
votes
0
answers
36
views
Unitors in Morton's definition of a double bicategory
I am confused by the definition of a double bicategory by Morton in (Definition 3.1.1. in https://arxiv.org/abs/math/0611930), but I need it, so I want to make sure I understand it correctly before I ...
2
votes
1
answer
221
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Comparing the exit path category and the nerve of a stratified space
Let $P$ be a finite poset, and $X$ be a topological space stratified by $P$, in the sense that $X$ is equipped with a continuous map $X \to P$ in the Alexandroff topology (or equivalently with a ...
5
votes
2
answers
408
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Explicit description of the right adjoint
Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram.
Given the ...
3
votes
0
answers
112
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Abelianisation of Groupoids
I was wondering what a good source for the properties (or even the existence) of the abelianisation of a (2-) groupoid would be? A naive construction would certainly be to abelianise the automorphisms ...
7
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0
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123
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Is every compact sifted $\infty$-category trivial?
The paradigmatic sifted $\infty$-category to work with is the dual simplex category $\Delta^{op}$. Unfortunately (and unlike the analogous case of reflexive coequalizers in the 1-categorical setting), ...
2
votes
1
answer
80
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Strongly final vertex $Y$ in a simplicial set gives a retraction of $\mathcal{C}^{ \triangleright }$ onto $\mathcal{C}$
I have a question about a proof from Jacob Lurie's Higher Topos Theory (p 45):
Proposition 1.2.12.5. Let $\mathcal{C}$ be a simplicial set. Every strongly final object (see below what means) of
$\...
3
votes
0
answers
231
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Characterizing compactly assembled localizations of presheaf $\infty$-categories
After SAG Def 21.1.2.1, say that an $\infty$-category $\mathscr{C}$ is compactly assembled if there exists a small $\infty$-category $\mathscr{C}_0$ such that $\mathscr{C}$ is a retract of $\mathrm{...
2
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0
answers
131
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Does the category of stable infinity categories form a "subtractive Waldhausen" category?
In "The $K$-theory spectrum of varieties", Jonathan Campbell introduces the notion of a subtractive Waldhausen category, a slight generalization of the notion of Waldhausen category that ...
2
votes
1
answer
160
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Generalizing $n$ for $n$-categories
I understand that this is a very amateurish question, but perhaps I can be forgiven since higher category theory (or indeed any sort of category theory) is not my field.
I have been thinking about a ...
3
votes
0
answers
133
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Transporting $\mathbb E_n$-monoidal structures between categories
Suppose given an $\mathbb E_n$-monoidal presentable $\infty$-category $\mathcal C$ (wrt. the Lurie tensor product $\otimes$), and $\mathcal D$ a presentable $\infty$-category. Suppose given a pair of ...
5
votes
1
answer
246
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Prop 5.3.6.2 in higher topos theory
Here are the necessary notations and the statement of the proposition. I don't understand why the underlined sentence is true. Are the horizontal morphisms Cartesian fibration?
I would appreciate it ...
9
votes
2
answers
696
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Homotopy coherent generalization of classifying space theory
Classically, given a compact Lie group $G$, there is a topological space $BG$ which classifies principal $G$-bundles. This means that there is an equality of sets {principal $G$-bundles up to ...
2
votes
1
answer
629
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What is the dual of the stable infinity category of perfect complex on smooth proper variety?
Fix a commutative ring $R$.
Lurie proved that smooth proper $R$-linear stable infinity categories are dualizable in $\text{Cat}^\text{perf}_{R,\infty}$.
For a smooth proper variety $X$ over $R$, what ...
4
votes
1
answer
396
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$(n,1)$-dagger categories
In category theory, a dagger category is a precategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \...
8
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100
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Are the morphisms of a star-autonomous category superfluous?
Let $(C,\otimes,I,\ast)$ be a (symmetric, say) star-autonomous category. Then $C$ comes equipped with a lax symmetric monoidal functor $|-|_C := Hom_C(I,-) : C \to Set$. The general hom-sets of $C$ ...
5
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115
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What is the lax colimit of an identity 2-functor?
Recall that if $C$ is a category, then the identity functor $Id_C : C \to C$ has a colimit if and only if $C$ has a terminal object $1_C$, and in this case $\varinjlim Id_C = 1_C$.
Question: Now let $...
5
votes
0
answers
243
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Do topoi have injective hulls?
Recall that a topos $\mathcal I$ is injective (with respect to embeddings) if and only if $\mathcal I$ is a retract of $Psh(C)$ for some finitely-complete $C$. Say that an embedding $f : \mathcal X \...
2
votes
0
answers
173
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Is there an interesting model structure on sSet whose fibered objects are exactly contractible Kan Complexes?
For the Kan-Quillen model structure, the fibered objects are exactly the Kan Complexes and for the Joyal model structure, the fibered objects are exactly the $\infty$-categories. This follows from the ...
6
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0
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152
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Does $Pr^L_\kappa \in Pr^L$ behave like an "object classifier" or "universe"?
Let $Pr^L$ denote the $\infty$-category of presentable $\infty$-categories and left adjoint functors. Let $Pr^L_\kappa \subset Pr^L$ denote the locally-full subcategory of $\kappa$-compactly-generated ...
0
votes
0
answers
83
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(Non-)Completeness for connected pointed $\infty$-groupoids
In an example in Lurie's HA, it is implied that the $\left(\infty,1\right)$-category of connected pointed $\infty$-groupoids is presentable. But it is not closed under homotopy pullbacks (e.g., $\...
8
votes
2
answers
326
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Given a Lie $2$-group $G$ does every principal $G$ $2$-bundle admit a $2$-connection?
The statement is true for Lie groups and principal bundles, with every principal bundle admitting a connection and I see no reason for the analogue result not to hold in the Lie $2$-group case but I ...
2
votes
0
answers
168
views
When does Tate spectral sequence degenerate at $E_2$?
For a spectrum $M \in \text{Sp}^{B\mathbb{S}^1}$ with a circle group action, there is Tate spectral sequence
$$
E_2^{ij}=\pi_{-i}(H(\pi_{-j}M))^{t\mathbb{S}^1} \Longrightarrow \pi_{-i-j}(M^{t\mathbb{S}...
14
votes
2
answers
656
views
When is a stable $\infty$-category the stabilization of an $\infty$-topos?
Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ ...
1
vote
0
answers
184
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Is there a (Grothendieck) $\infty$-topos for which Whitehead's theorem only holds for maps between truncated objects?
We know that non-hypercomplete $\infty$-toposes exist. Is there such a topos $\mathcal{E}$ with the following property?
For any $X, Y \in \mathcal{E}$, if all weak homotopy
equivalences (or $\infty$-...
2
votes
0
answers
209
views
$2$-fiber products
Let $\cal{C}$ be a $2$-category. Then there is a notion of a $2$-fiber product (see ncatlab or Stacksproject). This notion is quite elaborate and lengthy to define. However, one can try to give a more ...
4
votes
0
answers
150
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Fibrations of $n$-groupoids in the folk model structure on $n$-categories
Define a strict $n$-groupoid to be a strict $n$-category all of whose morphisms are weakly invertible.
[For $1\leq k < n$ a $k$-morphism $f:x\to y$ is weakly invertible if there exists $g:y\to x$ ...
9
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answers
147
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How does one classify monoidal biclosed structures on $Cat$?
Foltz, Kelly, and Lair assert that there are exactly two monoidal biclosed structures on the 1-category $Cat$ of small categories. But most of the proof is left as an "exercise" (see Prop 4)....
3
votes
0
answers
161
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Symmetrical monoidal $2$-category of cohomological correspondences
My question is whether a symmetric monoidal $2$-category of ``cohomological correspondences'' has been been rigorously constructed anywhere in the literature.
Let me be more precise about what I mean.
...
3
votes
0
answers
86
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What is the free lax-idempotent adjunction?
Let $Adj$ be the free adjunction, i.e. the 2-category such that for any 2-category $K$, the functor 2-category $2Fun(Adj, K)$ is the 2-category of adjunctions in $K$ (naturally in $K$). Note that $Adj$...
2
votes
1
answer
237
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$\infty$-groupoid iff Kan condition
I'm going through Chapter 4 of "Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects" which begins with an overview of $\infty$-categories. Theorem 2 states that a ...
15
votes
2
answers
383
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Complexity of coherence diagrams in an $n$-category
As we proceed from categories to bicategories to tricategories to tetracategories, the coherence diagrams expand at an alarming rate, taking up a page, then 5 pages, then 51 pages. There is a shared ...
20
votes
2
answers
2k
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Mark Hovey's open problems in the theory of model categories
Mark Hovey maintains a list of open problems in model category theory. I think this list is quite old, and I don't know if Hovey is still updating it or not.
My question is:
i) which of the 13 ...
3
votes
1
answer
371
views
In Lurie's "Higher topos theory" lemma 4.3.2.7
In Lurie's "Higher topos theory" lemma 4.3.2.7, I’m trying to understand “In particular, we may identify $p$-colimits of $F$ with $p$-colimits of $F_0$”:
Lemma 4.3.2.7. Suppose we are ...
2
votes
1
answer
213
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Simple reference for $\infty$ categorical mate correspondence
I am looking for a reference for the the following statement that is related to the mate correspondence for natural transformations.
Let $\mathcal C$ and $\mathcal D$ be $\infty$-categories, and let $...
2
votes
1
answer
191
views
Strictification of $\mathcal{V}$-pseudofunctors
Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a ...
10
votes
2
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525
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Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?
Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor
$$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$
I don't know much about ...
9
votes
1
answer
522
views
Proposition A.2.6.15 in HTT
This is a cross-post of a question in MSE.
I am reading Lurie's Higher Topos Theory and I need some help to understand a part of the proof of Proposition A.2.6.15. (A.2.6.13 in the published version)...
6
votes
1
answer
143
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Strictness of two operations on proarrow equipments
There are several equivalent definitions of a profunctor between categories $C$ and $D$. I'm interested in the following two:
A functor $C\times D^o \to \text{Set}$
A co-continuous functor between ...
5
votes
0
answers
548
views
What is higher representation theory?
Can anyone please introduce higher representation theory?
By Yoneda embedding, we know that global dimension of finitely generated category $\bmod\Lambda$ of Artin algebra $\Lambda$ is no more than $2$...
4
votes
0
answers
114
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Natural numbers object in a double category
I've been playing with double categories. I'm having trouble figuring out appropriate laws for induction squares in a double category.
Assume an object $\mathbb{N}$.
Assume horizontal arrows
zero $0\...
2
votes
2
answers
567
views
TR2 for homotopy category of stable $\infty$-category
I’m trying to understand Lurie’s proof that the homotopy category of a stable $\infty$-category is triangulated. In showing TR2, he constructs a diagram
$$\require{AMScd}
\begin{CD}
X @>f>> Y ...
5
votes
1
answer
152
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Is every folk cofibration of strict $\omega$-categories a monomorphism?
In the folk model structure on the category $sCat_\omega$ of strict $\omega$-categories, the cofibrations are generated by the boundary inclusions $\{\partial \mathbb G_n \to \mathbb G_n \mid n \in \...
2
votes
0
answers
129
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Unifying the $n$-truncation factorization system in a topos with the $n$-truncation factorization system of a t-structure
(This is a corrected and more detailed version of an earlier question.)
In good circumstances, e.g. the setting of an $\infty$-topos, we have for each $n\geq -1$ an orthogonal factorization system ...
8
votes
1
answer
451
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Is there a Dold-Kan theorem for circle actions?
There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 ...
2
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0
answers
130
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When is the category of connective objects a topos?
There is only one stable $\infty$-topos, namely the trivial category. However, the theory of stable $\infty$-categories with $t$-structure is strikingly reminiscent of the theory of topoi, as both ...
2
votes
2
answers
280
views
Is there a "duality involution" on presentable categories?
$\newcommand\Psh{\mathit{Psh}}\newcommand\Pr{\mathit{Pr}}$Let $\Psh$ be the category of presheaf categories and cocontinuous functors which preserve tiny objects. There is a functor $(-)^\ast : \Psh \...
8
votes
1
answer
268
views
Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?
$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and ...
4
votes
1
answer
156
views
What are the algebras for the laxification 2-monad?
Let $C$ be a small 2-category. Let $[C , Cat]$ denote the 2-category of strict functors to $Cat$, 2-natural transformations, and modifications. Let $[[ C, Cat ]]$ denote the 2-category with the same ...
5
votes
0
answers
241
views
Is Koszul duality a deformation theory when not over a field?
Let $k$ be a field. Then Thm 15.3.3.1 of Lurie's SAG says that Koszul duality, regarded as a contravariant endofunctor $\bar D$ of augmented $E_n$-algebras over $k$, is a deformation theory in the ...