Questions tagged [higher-category-theory]

For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

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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 ...
Saal Hardali's user avatar
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1 vote
0 answers
67 views

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 $...
Mathemologist's user avatar
6 votes
2 answers
567 views

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 ...
Saal Hardali's user avatar
  • 7,549
4 votes
1 answer
415 views

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 $...
Karthik Yegnesh's user avatar
4 votes
0 answers
160 views

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 $\...
Kevin Arlin's user avatar
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5 votes
1 answer
73 views

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 ...
Mike Stay's user avatar
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7 votes
1 answer
744 views

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? ...
naive-theorist's user avatar
3 votes
1 answer
355 views

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 $F:X\rightarrow C$...
Karthik Yegnesh's user avatar
5 votes
1 answer
301 views

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 ...
John Pardon's user avatar
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9 votes
2 answers
424 views

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 ...
Maxime Lucas's user avatar
2 votes
0 answers
197 views

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 ...
Karthik Yegnesh's user avatar
1 vote
0 answers
106 views

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-...
homotopy-theorist's user avatar
7 votes
0 answers
212 views

$S^3$ 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 ...
nio's user avatar
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2 votes
1 answer
287 views

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 ...
Mathemologist's user avatar
2 votes
1 answer
789 views

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 (...
O-Ren Ishii's user avatar
7 votes
2 answers
374 views

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 ...
Edoardo Lanari's user avatar
3 votes
0 answers
130 views

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 ...
Mathemologist's user avatar
2 votes
1 answer
246 views

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 (...
Dmitry Vaintrob's user avatar
19 votes
1 answer
784 views

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 ...
Tim Campion's user avatar
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3 votes
1 answer
155 views

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 ...
Zhaoting Wei's user avatar
  • 8,657
2 votes
0 answers
150 views

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}((\...
a-w's user avatar
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2 votes
0 answers
111 views

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 \...
Max New's user avatar
  • 785
0 votes
2 answers
292 views

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 ?...
user100851's user avatar
2 votes
0 answers
141 views

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 \...
Vladimir Baranovsky's user avatar
8 votes
0 answers
236 views

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 ...
Sinan Yalin's user avatar
  • 1,589
1 vote
0 answers
170 views

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. ...
geodude's user avatar
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0 votes
1 answer
130 views

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,...
Bob's user avatar
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6 votes
0 answers
206 views

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}^...
Karol Szumiło's user avatar
4 votes
0 answers
173 views

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 ...
Daniel Grady's user avatar
3 votes
2 answers
730 views

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 ...
geodude's user avatar
  • 2,129
9 votes
1 answer
559 views

$\Theta$-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 ...
Edoardo Lanari's user avatar
1 vote
1 answer
253 views

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 ...
Tom's user avatar
  • 179
1 vote
0 answers
96 views

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 ...
Dmitry V's user avatar
  • 433
19 votes
3 answers
2k views

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 ...
Anette's user avatar
  • 585
13 votes
1 answer
464 views

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/...
Yonatan Harpaz's user avatar
1 vote
0 answers
143 views

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 "...
SWV's user avatar
  • 143
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{...
Ben G's user avatar
  • 403
8 votes
1 answer
335 views

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 ...
Manuel Bärenz's user avatar
1 vote
0 answers
66 views

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....
user84563's user avatar
  • 915
7 votes
2 answers
444 views

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 ...
Valery Isaev's user avatar
  • 4,410
3 votes
1 answer
108 views

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 ...
Lukas's user avatar
  • 33
3 votes
1 answer
222 views

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 (...
user84563's user avatar
  • 915
16 votes
1 answer
1k views

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 $\infty$-...
Yuri Sulyma's user avatar
  • 1,513
2 votes
0 answers
120 views

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 ...
Tim Campion's user avatar
  • 60.6k
2 votes
1 answer
177 views

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 ...
Daniel Grady's user avatar
8 votes
2 answers
375 views

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 "...
Pedro's user avatar
  • 733
5 votes
0 answers
104 views

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 ...
user84563's user avatar
  • 915
3 votes
0 answers
235 views

"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 ...
user84563's user avatar
  • 915
6 votes
1 answer
199 views

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 ...
Valery Isaev's user avatar
  • 4,410
7 votes
0 answers
241 views

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 ...
Tim Campion's user avatar
  • 60.6k

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