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Questions tagged [higher-category-theory]

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

7
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1answer
285 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 ...
1
vote
1answer
197 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 ...
1
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0answers
84 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 ...
12
votes
3answers
979 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 ...
11
votes
0answers
294 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/...
1
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0answers
85 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 "...
5
votes
0answers
245 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{...
6
votes
1answer
216 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 ...
1
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0answers
61 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....
6
votes
2answers
267 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 ...
2
votes
1answer
79 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 ...
2
votes
1answer
137 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 (...
15
votes
1answer
562 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 $\infty$-...
2
votes
0answers
107 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 ...
2
votes
1answer
156 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 ...
4
votes
2answers
273 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 "...
4
votes
0answers
86 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 ...
3
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0answers
184 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 ...
6
votes
1answer
178 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 ...
7
votes
0answers
191 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 ...
4
votes
1answer
288 views

When does every $\infty$-localization correspond to a Bousfield localization?

Let $\mathcal{M}$ be a model category presenting an $\infty$-category $\mathcal{C}$. I believe that every left Bousfield localization $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ corresponds to a ...
4
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0answers
139 views

Is there a schema category for hyperstructures?

I am completely fascinated by Niels Baas' notion of hyperstructures, chiefly because I can see how such gadgets could be used in modeling both biological and social systems, or other evolutionary ...
6
votes
1answer
293 views

The naive approach to deriving profunctors - What's wrong with it?

Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ ...
6
votes
1answer
259 views

Are accessible $\infty$-categories closed under accessible localizations?

The defining problem of homotopy theory is that often when one localizes a nice category at a reasonable class of morphisms, the result is a very bad category. Does passing to the $\infty$ world fix ...
5
votes
1answer
252 views

Methods for defining/calculating homotopy limits of quasicategories

I am working on a project which requires that I calculate homotopy limits of homotopy theories (i.e. $(\infty,1)$-categories). It may be relevant that the homotopy limits which interest me are in the ...
15
votes
1answer
364 views

Why is every object cofibrant in an excellent model category?

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...
6
votes
1answer
381 views

Interaction of Grothendieck Construction with Coherent Nerve

There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper here) and one for quasicategories (see the Unstraightening and ...
17
votes
1answer
674 views

Errata on Rezk's paper

I am reading this paper of Rezk's http://arxiv.org/abs/0901.3602 A cartesian presentation of weak n-categories, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in ...
5
votes
3answers
154 views

Set of functions is not a bifunctor on Rel

Let Rel be the category whose objects are sets and whose morphisms are binary relations, with composition defined by $x (S \circ R) z \Leftrightarrow (\exists y : x R y \wedge y S z)$, and identity ...
12
votes
2answers
274 views

Which simplicial objects are Čech nerves?

In 1-categories, a regular epimorphism is a coequalizer of some parallel pair. An effective epimorphism is one which coequalizes its kernel pair. In the presence of kernel pairs, regular and ...
2
votes
1answer
168 views

Appropriate morphisms and 2-morphisms in Ind(C)

As I was trying to understand the category $Ind(C)$ of diagrams of the form $I \to C$, where $I$ is a small filtered $(0,1)$-category, I wondered whether it is possible to define morphisms directly, ...
10
votes
3answers
712 views

“Spatial (geometrical)” realization of Elementary topos?

It is well known that (Grothendieck) Topos (in fact, Model topos too) has many good geometrical properties. In many senses reflects general forms of generic geometry. Note: Grothendieck view of Topos ...
7
votes
1answer
134 views

Can a weak fibration category be non saturated?

A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories $$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$ containing all the isomorphisms, such that the following ...
13
votes
1answer
666 views

Difficulties with descent data as homotopy limit of image of Čech nerve

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ...
5
votes
1answer
332 views

What do globes (used to construct globular sets, $\omega$-categories, etc.) actually look like?

Nlab introduces the globular category as a geometrical model to construct certain higher categorical structures (e. g. strict $\omega$-categories), just as quasi-categories, for example, are modelled ...
9
votes
1answer
452 views

Coherence and rewriting

In category theory there are numerous coherence theorems (https://ncatlab.org/nlab/show/coherence+theorem). One example is the Mac Lane's coherence theorem for monoidal categories. This and probably ...
8
votes
1answer
326 views

Relative version of Quillen's theorem A

Quillen's Theorem A is formulated as follows: Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category $F/y$ is contractible. Then $F$ induces a weak equivalence ...
1
vote
0answers
47 views

Pseudopullback of dimension three

What is the name of the appropriate analogue of the pseudopullback for dimension three? That is to say, a pseudonatural equivalence $fg\simeq hj $ which is universal in the obvious sense... Thank ...
82
votes
3answers
7k views

What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...
2
votes
1answer
78 views

Why does vertical multiplication in 2-groups not follow the same order as horizontal one when constructed from crossed modules?

Consider a 2-group (seen as a 2-category with only one object $\star$) constructed from a crossed module $(G,H,t,\rhd)$ ($\circ$ will denote 1-morphisms composition, $\circ_{h}$ 2-morphisms horizontal ...
21
votes
2answers
1k views

How to stop worrying about enriched categories?

Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...
15
votes
1answer
290 views

Descent of Higher categorical structures along geometric morphisms

Let $f: \mathcal{E} \rightarrow \mathcal{T}$ be a geometric morphism between two (Grothendieck) toposes (or maybe more generally a bounded geometric morphism between elementary toposes). It is well ...
4
votes
0answers
110 views

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 ...
3
votes
0answers
105 views

What do you call the coherence cells in a lax morphism?

The original question a friend asked me is what to call the coherence cells in a lax monoidal functor. After looking around, I was surprised to realize that when it comes to monoidal functors, ...
5
votes
1answer
156 views

Uniqueness of $\infty$-adjoints

Adjoints in a 2-category are essentially unique, in the following strong sense. If $\mathbf{2}$ denotes the "walking arrow" category $(\cdot \to \cdot)$, then there is a 2-category $\mathrm{Adj}_1$ ...
6
votes
1answer
225 views

Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors $$ ...
11
votes
1answer
395 views

Is the hom-simplicial set in the hammock localization a nerve?

Let $(C,w)$ be a relative category. Then associated to it we have its hammock localization, $L^H(C,w)$, which is a simplicially enriched category. If $X,Y\in C$, the description of the simplicial set ...
2
votes
2answers
202 views

The source-side-opposite of the arrow category

Given a category $C$, is there a name for the following category: $\mathrm{Obj}(D) = \left\{ (x, y, f) \middle| x, y \in \mathrm{Obj}(C), f \in C(x, y) \right\}$ $D((x, y, f), (x', y', f')) = \left\{ ...
2
votes
0answers
57 views

Holonomy 2-functor transformation by transition functions

The holonomy 2-functor on a $\mathcal{G}$-principal 2-bundle associates a bigon: $$\mathsf{hol}_i(\Sigma):\mathsf{hol}_i(\gamma)\Rightarrow \mathsf{hol}_i(\gamma')$$ in $\mathcal{G}$ to each bigon: $$\...
5
votes
2answers
261 views

Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined $$ RHom(C,...