Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
435
questions with no upvoted or accepted answers
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What is an $\infty\text{-}E_{\infty}$ morphism?
My question is essentially what the title says, but here is some background that I have gathered from skimming through the literature. Please feel free to correct me if my understanding is wrong at ...
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$(\infty,2)$-categories as colimits of orientals
Let $\mathcal{C}$ be an $\infty$-category represented by a fibrant simplicial set in the Joyal model structure. It is well known that $\mathcal{C}$ can be expressed as the (homotopy) colimit over its ...
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Cohomology of a countable directed union of groups
It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a ...
- 1,446
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A theory of higher limits of (1-)functors, after higher hochschild homology
$\newcommand{\Trans}{\mathrm{Trans}}\newcommand{\H}{\mathrm{H}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\Mod}{\mathrm{Mod}}$Recently I noticed that one may regard the co/...
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Relationship between closure under pushouts and closure under cobase-change?
Let $\mathcal C$ be a category with pushouts. Let $\mathcal L \subseteq Mor(\mathcal C)$ be a class of morphisms in $\mathcal C$. Consider the following conditions on $\mathcal L$:
Every identity ...
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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 $...
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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 \...
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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 $2$...
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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 ...
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A recursive attempt at $n$-dimensional coherence
For the purposes of this post we will use the one hom class definition of a category.
Note that a functor $F:\mathcal{C}\to\mathcal{D}$ between categories is a pair of functions $F_0:{\bf Ob}_\mathcal{...
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Constructing lax limits from lax limits
Let $K$ be a 2-category. It's well-known that if $K$ has all PIE limits, then $K$ also has all lax limits. But I don't know a general "limit-decomposition" result which works "...
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Counit functor associated to a bicartesian fibration
I would like to understand $\infty$ categorical adjunctions better. I am far from an expert, and so I would greatly appreciate published references (with no unproven foundational assumptions) ...
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When is the fat join a monoidal structure?
This question is about the following general construction.
Definition:
Let $(\mathcal C, \otimes)$ be a cocomplete, monoidal biclosed category whose unit $\ast$ is terminal. Let $I$ be an "...
- 61.6k
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Factorization homology and topological conformal field theories
My question concerns some of the results of Costello's "Topological conformal field theories and Calabi-Yau categories" and how they are related/ can be rederived via the description of (...
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What are common examples for Paré's list of double categories?
In Robert Paré's 2018 presentation "Double Categories: The best thing since slice categories", the slides non-exhaustively give the following listing of interesting situations where double ...
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Why is the definition of a Dwyer map so particular?
Recall that a functor $F : A \to B$ is said to be a Dwyer map if the following two conditions are satisfied:
$F$ is a sieve inclusion.
There exists a factorization of $F$ as $A \to W \to B$ where $W ...
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Bi/tricategorical coherence in terms of surface diagrams
Is there a typed-up version of the coherence theorem for bicategories in terms of surface diagrams? What about the GPS tricategorical coherence theorem in terms of 'volume diagrams'?
I'm aware of ...
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Tensor product of t-structures compatible with filtered colimits
Let $C,D$ be two stable presentable $(\infty,1)$-categories, equipped with accessible t-structures. Then you can define an accessible t-structure on $C\otimes D$ by having $(C\otimes D)_{\geq 0}$ be ...
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Definition of $E_{n}$-operad in dgCat
In "Derived Algebraic Geometry and Deformation Quantization" Toën defines in 5.1.2 an $E_{n}$-monoidal A-linear dg-category as an $E_{n}$-monoid in the symmetric monoidal $\infty$-category $...
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On how Simpson's model structure on Tamsamani $n$-prenerves is cofibrantly generated
I was reading through "A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert Van-Kampen" by Simpson and was struggling to piece together the ...
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Étale stack on $\text{Spec}(k)$
Let $\mathcal{F}:\text{Ét}(\text{Spec}(k))^{\mathrm{op}}\rightarrow \text{Set}$ be an étale presheaf. The étale sheaf condition for the cover $\text{Spec}(l)\rightarrow \text{Spec}(k)$ is
$$\mathcal{F}...
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Surprising examples of functors which preserve cofiltered limits but not all limits?
Question: What are some "surprising" examples of functors (resp. $\infty$-functors) $F$ which preserve cofiltered limits?
I'm not quite sure what "surprising" means, but I think ...
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Simplicial matrices and the nerves of weak n-categories II, III, and IV
Duskin introduced his nerve functor (see the nLab or Kerodon) in the paper
Duskin, John W. Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. [Link].
While three ...
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Is there a n-category structure on algebras for $e_n$-like operads?
I'm fishing in troubled waters here and therefore the question is vague and meant to be as general as possible. In particular "$e_n$-like operad" can be an algebraic or topological $e_n$ operad, as ...
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$\mathbb Z \otimes_\mathbb S \mathbb Z$ is concentrated in degree $0$ : mistake in the argument
I'm not sure this is research level so if this is not appropriate, feel free to move the question to StackExchange. However, I post it here since my "fake proof" is based on a (recent) paper and I'm ...
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Shapes of cores of symmetric monoidal $(\infty,n)$-categories (with duals)
According to the cobordism hypothesis, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-categories with duals, then framed fully extended TQFTs with target $\mathcal{C}$ are an $\infty$-groupoid, ...
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Why do we need straightening?
The method of straightening and unstraightening produces, for instance, an equivalence between left fibration and copresheaves. The construction seems very intricate. But what is the need for this ...
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The notion of $\infty$-Cooperads for which Bar-Cobar duality is an equivalence
In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. ...
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Are any formal systems based upon the idea of "iterated characterization pushing" currently in existence? If not, is anyone working on them?
I had an idea in regards to the design of formal systems with foundational aspirations.
To convey the idea, let's talk a bit about the second-order Peano axioms. The way these axioms work, we have a ...
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Uniqueness of Witten-Dijkgraaf 2D TQFT at 0th dimension
If I understand correctly, Dijkgraaf-Witten TQFT in dimension 2 is the
following. Fix any finite group $G$, we define a field over a closed
2-manifold to be a principle $G$ bundle (it's automatically ...
- 5,038
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Model structures on groupoids
Let me preface by saying that I'm very inexperienced with model categories. I was thinking about the following example, and I'm now wondering whether it fits into the theory of model stuctures:
...
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CoCartesian vs. locally CoCartesian fibrations
Say $\pi: C\to J$ is an inner fibration of $\infty$-categories. Then "morally", $\pi$ corresponds to a diagram indexed by $J$ in the "category of categories with correspondences", and if $\pi$ is ...
- 7,902
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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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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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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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Internal hom as 2-Kan lift of pseudofunctor
Consider a situation where there is a pseudofunctor from some category $C$.
Due to it's geometrical properties, it might induce some sort of an internal hom on the original category (making it a ...
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Question about $(\infty, 1)$-category of $(\infty, 1)$-categories
I would like to realize $QCat:=(sSet)_{\text{Joyal}}$ as an $(∞,1)-category$ in the sense of Lurie (i.e. weak Kan complex). I would like to do this in the following way:
1) First since $QCat$ is a ...
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"Characteristics" (thick subcategories) in $n$-groupoids
$
\newcommand{\Ab}{\mathbf{Ab}}
\newcommand{\Sp}{\mathbf{Sp}}
$In 0-groupoids (sets), the thick subcategories of the category of abelian groups $\Ab$ are given by the primes $p$ and $0$, which we can ...
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DG model of A-infinity category
Given a $k$-linear dg category $\mathcal{C}_{dg},$ I can produce an ($A_\infty$-quasi-equivalent) $k$-linear $A_\infty$ category $\mathcal{C}_{A_\infty}$ by taking the homotopy category $H^0(\mathcal{...
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Model Category presentation of Pointed and Connected $\infty$-categories
Let $(\infty,1)Cat^{*/}_{con}$ denote the $(\infty,1)$-category of pointed and connected $(\infty,1)$-categories (that is, $(\infty,1)$-categories with a distinguished object such that there exists a ...
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What terminology surrounds "involutive" double categories?
Write $\mathbf{Cat}$ for the world of categories. Then $\mathbf{Cat}$ has:
objects (namely cateories)
arrows (namely, functors)
proarrows (namely, bimodules)
squares (namely, functors between pairs ...
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Lifting commutative diagrams of functors from the homotopy level to the "higher" level
Let $\mathcal A$ and $\mathcal B$ be differential graded categories over a field. Let $F, G, K : \mathcal A \to \mathcal B$ be quasi-functors (see here for definitions). Assume you have morphisms of ...
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How do you compute a homotopy colimit in a category of fibrant objects?
This question may be a bit vague, (so if suggested, can I make it community wiki), but I was wondering what techniques there exists for computing homotopy colimits in a category of fibrant objects. A ...
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On the preservations of certain colimits (by covers) under simplicial localization (of a category of fibrant objects)
Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of ...
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Non-degenerate limits of topoi
Let $\mathcal{E}$ be an elementary topos with natural numbers object $\mathbb{N}$, and let $\mathbb{C}$ be a cofiltered internal category in $\mathcal{E}$. Suppose we have a diagram of shape $\mathbb{...
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(∞,n)-category of triangulated cobordisms
What is an accepted definition of a (∞,n)-category of triangulated cobordisms?
Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
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Chain/Hierarchy of Monoids
Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:...
- 1,852
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Localizations that are endofunctors
Usually when $F: C \rightarrow D$ is a localization functor, the categories $C$ and $D$ are not equivalent. My question is when is it possible for $C, D$ to be abstractly equivalent but $F$ is not an ...
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Cocartesian fibration classifying $\mathrm{Fun}(F,G)$
Throughout this question we consider $\infty$-categories.
Fix a cartesian fibration $p : \mathcal{F} \to \mathcal{C}$ and a cocartesian fibration $q : \mathcal{G} \to \mathcal{C}$ which straighten to $...
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An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?
Past this question in MO have raised the following questions for me.
Question
In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra.
However, do ...
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