Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,306
questions
2
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Interesting examples of large, accessible, non-presentable $\infty$-categories?
What are some interesting examples of accessible $\infty$-categories
which are not presentable and not small?
By interesting I mean a category which comes up naturally in a certain context and in a ...
1
vote
0
answers
67
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Underdetermined Polynomial $(\infty ,1)$-Functors
Is there any sense in which the full subcategory of n-excisive functors (on spectra or on pointed spaces) on those functors $F$ which satisfy a set of distinct (modulo homotopy equivalence) equations $...
6
votes
2
answers
567
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Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories
Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions ...
4
votes
1
answer
415
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Do coproducts of infinity-groupoids commute with pullbacks?
As stated in this question, coproducts commute with pullbacks in the category of sets.
Let $Grpd_{\infty}$ denote the $\infty$-category of $\infty$-groupoids. Do coproducts commute with pullbacks in $...
4
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0
answers
160
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Are there obstructions to refining partially coherent natural transformations up-to-homotopy between quasicategories?
EDIT: It looks like I didn't define the simplicial set $R^{\sim \Delta^1}$ clearly below. It's not simply $\mathrm{Ho}(R)^{\Delta^1}$. Rather, an $n$-simplex of $R^{\sim \Delta^1}$ is a 1-simplex of $\...
5
votes
1
answer
73
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Coherence laws when composing 2-monads
To have the composition of two monads be a monad itself, we need a
distributive law natural transformation satisfying certain coherence
laws.
I'm interested in the strict 2-monad case, i.e. a strict ...
7
votes
1
answer
744
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Colimits of cofibrations and homotopy colimits
Say C is a left proper model structure. I have a diagram where all maps are cofibrations. Is its colimit a homotopy colimit?
I know this is true for pushouts. Is it true for sequential colimits? ...
3
votes
1
answer
355
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The "Family" functor for infinity-categories
Background
Given a category $C$, one can consider the category $Fam(C)$ of set-indexed familiies of objects in $C$. Formally, the objects are pairs $(X,F)$ in which $X$ is a set and $F:X\rightarrow C$...
5
votes
1
answer
301
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Terminology for filtered $\infty$-categories
Often to prove that the Kanification of a simplicial set $X_\bullet$ is contractible, we instead prove that $X_\bullet$ is a contractible Kan complex (i.e. satisfies the extension property for ...
9
votes
2
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424
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Is there a model structure on (strict?) Monoidal Categories?
Basically, what the title says.
Presumably, one could use the fact that monoidal categories (resp. strict monoidal categories) are one-object bicategories (resp. 2-categories) and use the Lack model ...
2
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0
answers
197
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Connected families of objects in $(\infty,1)$-categories?
Background
Let $C$ be an (extensive) category. An object $X\in C$ is called connected if the functor $Hom(X,-):C\rightarrow Set$ preserves coproducts.
Given a category $C$, one can consider the ...
1
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0
answers
106
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Localizations and their underlying groupoids
Fix $\mathcal C$ an $\infty$-category and $W \subset \mathcal C$ the inclusion of an $\infty$-category containing all weak equivalences in $\mathcal C$.
The localization along $W$ is then (a Joyal-...
7
votes
0
answers
212
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$S^3$ partition function from the 1-category of $S^1$ in Chern Simons theory
I am trying to learn about some categorical aspects of topological quantum field theories. For concreteness, I am considering Chern Simons theory with gauge group $G$ in three dimensions. As I ...
2
votes
1
answer
287
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Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck construction?
Given a locally presentable ($\infty$,1)-category $C$, can the fibrewise stabilization of it's codomain fibration, also called its tangent category $TC$, be given in terms of the Grothendieck ...
2
votes
1
answer
789
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Colimits in the category of simplicial categories
A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (...
7
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2
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374
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Equivalent definitions of Cartesian Fibrations between Quasi-Categories
In the paper by Verity and Riehl "Fibrations and Yoneda lemma in an ${\infty}$-cosmos" (https://arxiv.org/abs/1506.05500), they prove a Yoneda lemma that holds in any ${\infty}$-cosmos (see Corollary ...
3
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0
answers
130
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Is there a construction capturing indexed families of adjunctions?
I'm sorry in advance if this question does not belong on this site. I am curious as to what is "really" going on when you have a family of functors indexed by elements in a base category, all of which ...
2
votes
1
answer
246
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Is the property of being a dg generator open?
Suppose $\mathcal{C}$ is a dg category (over some base) with all colimits. We say that $X\in \mathcal{C}$ is a generator if $\mathcal{C}$ is equivalent to $\operatorname{End}_\mathcal{C}X$-modules (...
19
votes
1
answer
784
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Is there an analog of Kan's $Ex^\infty$ functor for quasicategories?
Is there a fibrant replacement functor in the Joyal model structure which can be described non-recursively, like $Ex^\infty$ for the Quillen model structure? I believe another way to put this is to ...
3
votes
1
answer
155
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Could $RHom(A,-)$ distinguish non-quasi-equivalent dg-categories?
One of the remarkable results in Toen's paper is the existence of internal homs of dg-categories. Actually for two dg-categories $A$ and $B$, there exists a dg-category $RHom(A,B)$ such that for any ...
2
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0
answers
150
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Why is the stabilization of augmented $\mathbb{E}_\infty$-algebras equivalent to $k$-module spectra?
(I have already asked this on Math.SE, but it didn't draw much attention there, so I am reposting it here.)
Example 1.1.4 of Jacob Lurie's DAGX says that the stabilization $\operatorname{Stab}((\...
2
votes
0
answers
111
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Pullback along a Lax Functor
Consider a 2-category of bicategories, lax functors and transformations (I'm actually using double categories but the result should be the same).
Given a cospan $\mathbb{C} \to \mathbb{E} \leftarrow \...
0
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2
answers
292
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Higher category theory (invertible morphism, Topological categories, monoidal n-category) [closed]
I'm Looking at a paper using higher category theory. Would you please answer these questions ?
1) A weak n-category (or $\infty$-category) has objects and k-morphisms. What is an invertible morphism ?...
2
votes
0
answers
141
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homotopy quotient categories [closed]
(Trying to rephrase an earlier question)
In topology, a continuous map $f: X \to Y$ has a ``homotopy cofiber" $Cf$ included in a cofiber sequence
$$
X \to Y \to Cf \to \Sigma X \to \Sigma Y \to \...
8
votes
0
answers
236
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Framed higher Hochschild cohomology
Given an $E_n$-algebra $A$, one can define its $E_n$-Hochschild complex $CH_{E_n}(A,A)$ by the formula $$Ch_{E_n}(A,A)=RHom_{Mod_A^{E_n}}(A,A)$$ where $Mod_A^{E_n}$ is the category of $A$-modules over ...
1
vote
0
answers
170
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Posets as (0,1)-categories
I am reading on the nLab that a poset can be seen as a (0,1)-category.
I was assuming all along that an ($n$,$r$)-category were a category where all morphisms of order larger than $n$ are trivial. ...
0
votes
1
answer
130
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Equality of lax natural transformations in the constructive approach
In the constructive approach to category theory, a category comes equipped with an equality (an equivalence relation) between its morphisms but not between its objects.
Let C and D be such categories,...
6
votes
0
answers
206
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Can representable presheaves be made injectively fibrant?
I suspect that the answer to my question is no, but let me give it a shot anyway.
If $\mathcal{A}$ is a small simplicially enriched category, then the category of simplicial presheaves $\mathsf{sSet}^...
4
votes
0
answers
173
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Homotopy invariance of the moduli stack of smooth $G$-bundles?
This question ought to have a straightforward (perhaps even glaringly obvious) answer, but so far I've already wasted a few hours trying to untangle this web of inconsistent identifications. I'm sure ...
3
votes
2
answers
730
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The homotopy pullback of a point along itself is the loop space
I have seen on the nLab that we can view the loop space as a particular homotopy pullback, and that it is even the way a "loop space object" is defined in general (when it exists).
Can someone give ...
9
votes
1
answer
559
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$\Theta$-Sets and Higher-QuasiCategories
In his well-known paper "Disks, Duality and $\Theta$-Categories, Joyal defines at the very end a $\Theta$-Category to be a cellular set suitably "fibrant".
I was wondering whether someone has worked ...
1
vote
1
answer
253
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internal logic from higher categories
nLab has an article on internal logic. Homotopy type theory also discusses its application in logic. Any one knows references to the correspondence study of the internal logic induced from higher ...
1
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0
answers
96
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Frobenius Monoids as Collapsed 2-Categories
Let $\mathbf{COB}_2$ denote the 2-category given by
$\bullet$ objects are finite sets of points
$\bullet$ 1-morphisms between these are 1d cobordisms
$\bullet$ 2-morphisms are 2d cobordisms with ...
19
votes
3
answers
2k
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Is it always possible to write a scheme as a colimit of affine schemes?
My question is: Is it possible to write any scheme as a (1-categorical) colimit of a diagram of affines? If no, what are some examples?
I ask this question because I have read that one can write any ...
13
votes
1
answer
464
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Is the operadic nerve functor an equivalence of ∞-categories?
It is now known that the $\infty$-category of $\infty$-operads as defined by Lurie is equivalent to the underlying $\infty$-category of the model category of simplicial operads, see http://arxiv.org/...
1
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0
answers
143
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Informal description of symmetric monoidal $(\infty,n)$-categories
I know the question of what is a symmetric monoidal category has shown up here.
I was wondering if there was a more informal way of describing a symmetric monoidal $(\infty, n)$-category as a "...
5
votes
0
answers
390
views
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{...
8
votes
1
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335
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What's the (monoidal) image of a monoidal functor?
For an ordinary functor $F\colon \mathcal{C} \to \mathcal{D}$ of categories, there is a construction $\operatorname{im} F$, the image of $F$, which is again a category, and $F$ factors through that ...
1
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0
answers
66
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Morphism of 2-sites
Let $X$ and $Y$ be Grothendieck sites. A $\textit{morphism}$ between $X$ and $Y$ is a functor on the underlying categories that is covering-flat and preserves covering families. See https://ncatlab....
7
votes
2
answers
444
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Homotopy function complex for quasi-categories
The model structure on sSet for quasi-categories (the Joyal model structure) is not enriched over sSet with the Quillen model structure, so ordinary internal Hom of simplicial sets is not a correct ...
3
votes
1
answer
108
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Reference for the definition of epi-1-morphisms in bicategories
I am looking for a reference concering the definition of epi-1-morphisms and mono-1-morphisms in an arbitrary bicategory. These concepts should be defined somewhere, but I am not able to find a ...
3
votes
1
answer
222
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Adjunction between locally presentable and ordinary categories?
Let $Pres$ denote the 2-category of locally presentable categories (categories that are accessible/cocomplete), cocontinuous functors, and natural transformations. Let $Cat$ denote the 2-category of (...
16
votes
1
answer
1k
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What's the stabilization of the $\infty$-category of $\infty$-categories?
$\require{AMScd}$One nice thing about $\infty$-categories is that spaces are themselves $\infty$-categories. What's the analogue for spectra? Presumably this would be the stabilization of the $\infty$-...
2
votes
0
answers
120
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Left-inducing a model structure from categories to relative categories
One can left-induce a model structure from $\mathsf{Cat}$ to $\mathsf{RelCat}$ along the "homotopy category" functor. The weak equivalences are the relative functors which induce equivalences of ...
2
votes
1
answer
177
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Symmetric spectra for simplicial sheaves
Let $C$ be a site. Localizing either the projective or injective model structure on simplicial presheaves at the Cech nerves of covers in $C$ gives a new model structure on simplicial presheaves which ...
8
votes
2
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375
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Can $\mathsf{Set}$ be seen as a (non-trivial) 2-category?
I know that $\mathsf{Rel}$ can be seen as a 2-category with inclusion of relations as 2-morphisms, when passing to $\mathsf{Set}$, relations become functions and inclusions are constrained to be "...
5
votes
0
answers
104
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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 ...
3
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0
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235
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"2-Sheafification" with Values in non $Cat$ categories?
Let $X$ be a 2-site and consider the category of 2-presheaves over $X$, which will be denoted as $Pshv(X;Cat)$. These are $Cat$-valued 2-functors, where $Cat$ is the 2-category of categories. There is ...
6
votes
1
answer
199
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Do homotopy categories of finitely (co)complete quasicategories determine categorical equivalences?
Let $F : C \to D$ be an exact functor between (co)fibration categories such that $Ho(F) : Ho(C) \to Ho(D)$ is an equivalence of homotopy categories. Cisinski proved that in this case $F$ is an ...
7
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0
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241
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Is the theory of weak $n$-categories a cofibrant replacement of the theory of strict ones?
I have algebraic models of $n$-categories in mind. By "theory of (weak) $n$-categories", I mean "[monad / operad / whatever] whose algebras are (weak) $n$-categories".
To be more precise: fix an ...