Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
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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 ...
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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,...
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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 ...
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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 $(\infty,2)$-...
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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 $n$-...
user138661
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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 ...
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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 ...
user138661
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A compendium of weak factorization systems on $sSet$
A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying
Every morphism $f:x \to y \in \mathcal{C}$ can be factored (...
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A construction for the free $\omega$-category generated by a globular set
The forgetful functor from strict $\omega$-categories to globular sets has a left adjoint. Where can one find an explicit construction for this free functor?
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Are there universal homological functors?
There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?
That is, for each small abelian ...
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Where is it shown that homotopy sheaves form a higher stack?
Many references on infinity categories etc. advertise that one application is that it's an appropriate setting to glue (the appropriate replacement for) derived categories of sheaves.
What's the ...
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Counit map for compactly generated categories
Any compactly generated presentable stable $\infty$-category $C$ is known to be dualizable (with respect to Lurie's tensor product), so there is a coevaluation map:
$$Sp \to C \otimes C^{dual}.$$
...
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Examples of Lurie tensor product computations
I am interested in examples of computing the Lurie tensor product.
For example, if $A$ and $B$ are connective DG algebras (over $\mathbb{Z}$, say), then I think there is an equivalence $A\text{-mod} \...
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Derived $\infty$-category of sheaves and $\infty$-category of sheaves taking values in derived $\infty$-category
I am trying to understand the essential image of the following functor. Given a scheme $X$, we consider the corresponding small Zariski site $X_{zar}$. For a commutative ring $\Lambda$, let $\mathcal ...
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If the Tate construction vanishes for all trivial $G$-actions, then does it vanish for all $G$-actions?
Let $\mathcal{C}$ be a semiadditive $\infty$-category, complete and cocomplete, and let $G$ be a finite group. Then for any $X \in Fun(BG,\mathcal{C})$, there is a norm map $N_X: X_G \to X^G$. For ...
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Descent for the cotangent complex along faithfully flat SCRs
By Theorem 3.1 of Bhatt-Morrow-Scholze II (https://arxiv.org/pdf/1802.03261.pdf), we know that for $R$ a commutative ring, $\wedge^{i}L_{(-)/R}$ satisfies descent for faithfully flat maps $A \...
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A few questions while reading Higher Topos Theory
I am now reading Higher Topos Theory. I have recently met the following questions that I am not able to figure out and I am here to look for some answer or help.
First, in Lemma 2.2.3.6, while ...
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Does Ex^∞ send homotopy inverse limits of ∞-categories to homotopy inverse limits of spaces
Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the ...
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Visualization and new geometry in higher stacks
I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond ...
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Definition A.3.1.5 of Higher Topos Theory
I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model ...
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A characterization of maps that are homotopic relative to $A$ over $S$
Let $$\begin{array}{ccccccccc}
A & \rightarrow & X \\
i\downarrow & & \downarrow p \\
B & \xrightarrow{v} & S
\end{array} $$ be a commutative diagram of simplicial sets, with ...
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Weighted (co)limits as adjunctions
It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...
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Natural transformation of $A_\infty$-functors lifted from homology
Suppose you have two $A_\infty$-functors $\mathcal{F,G}:\mathcal{A}\longrightarrow \mathcal{B}$ which descend to $F,G:A \longrightarrow B$ in homology (here $A=H^0(\mathcal{A})$ and same for $\mathcal{...
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Understanding the proof of Proposition 1.2.13.8 in Lurie's Higher Topos Theory
In the first chapter of Lurie's HTT, Proposition 1.2.13.8 shows that the functors out of over (infinity)-categories reflect infinity-colimits.
For the life of me I cannot follow the proof.
Can ...
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Deligne's doubt about Voevodsky's Univalent Foundations
In a recent lecture at the Vladimir Voevodsky's Memorial Conference, Deligne wondered whether "the axioms used are strong enough for the intended purpose." (See his remark from roughly minute 40 in ...
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Higher $\infty$-categories
Is there a reason we consider $\infty$-categories to be the $\omega^{th}$ step in the 2-internalization inside Cat (or enrichment over Cat if you prefer)* process made invertible above some finite ...
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Smash product and the integers in a Grothendieck $(\infty, 1)$-topos
Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to this page in nlab, for any $X \in \mathcal{H}$, the suspension object $\Sigma X$ is homotopy equivalent to the smash product $B \...
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Inverting a suspension object in a stable monoidal category
Suppose we are given a cocomplete closed symmetric monoidal stable $(\infty,1)$-category $\mathcal{C}$ with suspension $\Sigma$, and let $X \in \mathcal{C}$ be dualizable. I'd like to create a new ...
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Homotopy groups of Diffeomorphisms of punctured d-dim ball
Let $\mathbb{D}_n^d$ be the $d$-dimensional unit ball with $n$ punctures. I am interested in the groups of orientation-preserving diffeomorphisms $Diff(\mathbb{D}_n^d)$ that fix the punctures. In ...
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Infinity categorical analogue of 2-dimensional monad theory
I'm wondering whether there is an infinity categorical analogue to the results of Two-dimensional monad theory. For the most part, I'm interested in the relation between strict functors of infinity ...
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Compact Generation of Co-Module Categories
Let $\mathcal{C}$ be a compactly generated stable $\infty$-category, linear over a field of characteristic $0$ (i.e., so that it is in particular a dg-category). Let $A$ be a co-monad acting on $\...
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Why is the straightening functor the analogue of the Grothendieck construction?
In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between pseudo functors into the category of ...
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Does a cartesian transformation induce a cartesian transformation on absolute limit cones?
Fix a category or $\infty$-category $C$ with all small limits.
We call a natural transformation $\alpha\colon f\to g$ between two functors $f,g\colon K\to C$ Cartesian, if for every arrow $k\colon a\...
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Definition of $Fun^G( \mathcal C, \mathcal D)$ in the setting of quasicategories
Our research group is currently going through the paper by Thomas Nikolaus and Peter Scholze On Topological Cyclic Homology and in the Appendix B, Proposition B.5., they use the notation $Fun^{B \...
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$\omega$-categorical algebra
Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be ...
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Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category
On the abstract of a paper by Emily Riehl and Dominic Verity, it is stated that
Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence.
Where can ...
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Reference request: levelwise detection of a morphism of $\infty$-functors being an isomorphism
Is there a reference for the following?
Consider quasi-categories $I,C$. Suppose that a morphism between functors $\alpha : \Delta^1 \to Fun(I,C)$ is given. Suppose that for every $i \in I$, denoting ...
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Sheaves over a sheaf
Everything I write I mean in the in the sense of Lurie's HTT.
Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C}...
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A multicategory is a ... with one object?
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly ...
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Need help understanding comment in Higher Topos Theory
I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
Lemma 2.4.4.1. Let $p : \mathcal{C} \rightarrow \mathcal{...
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Stability of accessible $\infty$-categories under some operations
I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.
In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...
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Factorization of a map from a contractible Kan complex through a Kan complex
Suppose we are given a contractible Kan complex $S$ and a map of simplicial set $f : S \rightarrow T$. Under what conditions can we say that $f$ factors through the largest Kan complex $Z$ contained ...
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Could we form the homotopy category of a dg-category by inverting homotopic invertible morphisms?
Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but ...
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Remark 2.4.1.4 Higher Topos Theory
In HTT, given a inner fibration $p : X \rightarrow S$ of simplicial, an edge $f : x \rightarrow y$ of the simplicial set $X$ is said to be a $p$-Cartesian if the induced map
$$ X_{/f} \rightarrow ...
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Higher Topos Theory Theorem 2.2.5.3
The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.
We have a trivial Kan ...
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Does the Dwyer-Kan model structure make dgCat a model $2$-category?
Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
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Universal property of sheaf category
Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...
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Functoriality of (co)limits in $\infty$-categories
I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) ...
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Comonad for normalized pseudofunctors for strict higher categories
Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{...
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Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?
In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$
Joyal's isofibration theorem says precisely
An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model ...
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