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3
votes
0answers
110 views

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 ...
5
votes
0answers
100 views

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 ...
13
votes
1answer
349 views

Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory

Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...
8
votes
1answer
201 views

Localizing 2-categories about a single morphism

This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and ...
2
votes
0answers
144 views

Axioms for a symmetric monoidal bicategory

I start reading the axioms for a symmetric monoidal bicategory. The axioms include so many diagrams to be satisfied. I am wondering if people really use these axioms directly to check a given data is ...
5
votes
2answers
366 views

Homotopy factorization of morphisms of chain complexes

This is a sort of follow up to this MO question. Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
9
votes
1answer
783 views

Learning higher differential geometry

I have read parts of the motivation on nlab and all the posts on MO I could find on the subject, and by now there are a few questions on my mind. If they trivial for someone who understands the ...
4
votes
0answers
125 views

Stable $\infty$ categories as a 2-category

Is there a treatment in the literature of stable $\infty$ categories as a 2-category? I.e. with non invertible 2-morphisms. Mostly I am interested in the behavior of the tensor product with respect ...
0
votes
1answer
365 views

What is the delooping of a looping?

What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$? The ...
9
votes
2answers
474 views

Relation between fully-extended TQFT and a “topless” TQFT

Consider 3-dimensional TQFTs for example. One version of them is the 3-2-1-0 fully extended TQFT. Do we have another version: 2-1-0 extended "TQFT"? If yes, do we have an example of 2-1-0 extended ...
1
vote
2answers
208 views

Defining degeneracies for semi-simplicial sets with inner Kan conditions

Suppose we are given a category enriched over semi-simplicial sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this ...
2
votes
0answers
192 views

Banach space interpolation theory in terms of categories

I have recently learned a bit about higher category theory. And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory? Is it ...
2
votes
0answers
37 views

Connection on 3-bundle given as triplet of forms

A connection on a bundle is given locally by a Lie algebra-valued 1-form. Gauge transformations act in the usual way on the forms, and form a groupoid. A connection on a 2-bundle is given locally by ...
1
vote
0answers
127 views

Continuity of Kan extension along the Yoneda embedding

Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a ...
6
votes
2answers
258 views

Weakening simplicial identities

The generators $d_i, s_i$ for morphisms of the simplicial category satisfy simplicial identities: $d_jd_i = d_id_{j−1}$ for $i < j$ $s_jd_i = d_is_{j−1}$ for $i < j$ $s_jd_i = id$ for $i = ...
11
votes
0answers
253 views

Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...
10
votes
1answer
637 views

I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category?

I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need ...
7
votes
1answer
238 views

$\Omega$ and $B$ as adjoints between symmetric monoidal $(\infty,n)$- and $(\infty,n-1)$-categories

Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= ...
2
votes
1answer
249 views

Follow up: Is continuity preserved when taking comma object?

Here, I asked wether taking lax pullback preserves continuity, but got no precise answer. However, I have found this recent article by Riehl and Verity which proves something very similar, but I ...
2
votes
0answers
57 views

Pseudofunctors of 2-variables and Gray tensor product of bicategories

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be bicategories. We say a 'pseudofunctors of 2-variables' consists of two families of pseufunctors Fix $A\in obj\mathcal{A}$, we have a pseudofunctor ...
17
votes
3answers
1k views

What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a ...
3
votes
0answers
90 views

Bicategorical limits with parameters

(This question was asked in http://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.) Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a ...
2
votes
0answers
172 views

Is continuity of a functor stable under pullback?

Let $p:C\rightarrow D$, $i:F\rightarrow D$ be functors of 2-categories, and we form the lax pullback of $p$ along $i$ $$ \bar{p}:C\times_D^{lax} F\rightarrow F$$ Q1: Is it true that if $p$ ...
3
votes
1answer
384 views

A question on the Grothendieck construction

The usual Grothendieck construction is for pseudofunctors $I^{op}\to Cat$, where $I$ is a 1-category and $Cat$ is the 2-category of 1-categories. The Grothendieck construction produces a category with ...
1
vote
2answers
174 views

DG categories - pre-triangulated versus small limits

A DG category can be considered as an infinity category, say by taking Dold-Kan of the coconnective part of Hom spaces, thus obtaining a simplicial category. My question is, are the following ...
5
votes
2answers
375 views

What's an initial object in a poset-enriched category?

I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the ...
3
votes
0answers
113 views

Recognition principle for 2-categories (2-groupoids)

Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...
8
votes
1answer
287 views

Adjoining adjoints in a 2-category

For a given 2-category $C$, does there exist a faithful and locally faithful 2-functor $C \to C^*$, such that the image of every 1-morphism of $C$ has a right adjoint in $C^*$? Below are some of my ...
9
votes
0answers
215 views

Are n-truncated quasicategories a model for n-categories?

In Higher Topos Theory, Lurie gives a description of $n$-categories (section 2.3.4) in terms of quasicategories. In other words, he gives a definition (2.3.4.1) of an $\infty$-categorical notion of ...
8
votes
1answer
156 views

coends of stable infinity categories

Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote ...
6
votes
0answers
156 views

Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that: $\mathcal C$ is ...
2
votes
0answers
126 views

Homotopy limits of weak diagrams

This question is somehow related to my question at http://math.stackexchange.com/questions/683915/derived-pseudo-functor and to the question here: A homotopy commutative diagram that cannot be ...
2
votes
1answer
543 views

On the foundations for large categories

There are some basic discussions on the motivations of large categories and small categories [here] (On the large cardinals foundations of categories) and [here] (Large cardinal axioms and ...
0
votes
1answer
183 views

Homotopy limit of a cosimplicial category

Consider the usual model structure on Cat (category of small categories). Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$? ...
1
vote
0answers
93 views

A definition of a version of n-category with linear structure and tensor product

Let us assume that the $m$-categories with $0\leq m <n$ are already defined. 1) A $n$-category has a set of $p$-morphisms, labeled by $i^{[p]}$, for $0\leq p <n$, and each set of the ...
2
votes
1answer
202 views

n-limits and the Descent Category

Probably, this question could be at http://math.stackexchange.com/, but, for some reason, I can't access that site right now. So, I apologize in advance. At this article ...
11
votes
0answers
792 views

Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...
2
votes
1answer
378 views

A simple definition of n-category [closed]

Maybe there is no simple definition of $n$-category understandable for a physicist. Then I would like to know what are the trivial $0$-category, trivial $1$-category, trivial $2$-category, etc. How ...
1
vote
1answer
198 views

What is this name of this 2-category without very much structure?

I was wondering if there is a name for this 2-category which is like the 2-category of natural transformations, but does not actually require the 1-morphisms to be functors or the 2-morphisms to be ...
2
votes
0answers
138 views

Cloven Kan fibrations

For each groupoid $G$ there is a monad such that the cloven fibred groupoids over $G$ are algebras for this monad. I am looking for an infinite dimensional counterpart: a monad on simplicial sets ...
1
vote
0answers
159 views

The bar construction or the quotient for monoidal category action on a category

Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category, $$ N_\bullet (M\rtimes C)= \cdots M\times ...
4
votes
1answer
227 views

Simple-minded coherence of tricategories

Recall Mac Lane's version of coherence for monoidal categories, which one can state informally as follows: "Simple-minded" coherence for monoidal categories Let $A$, $A^\prime$ be two ...
4
votes
2answers
428 views

Small objects in categories

I would like to pick out small objects from a category. I would like to find such a notion which Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least ...
1
vote
0answers
205 views

A potential definition of weak $\omega$-categories

This question was inspired by the Homotopy Type Theory Book. Might we define a weak $\omega$-category as described below? Is any similar approach already considered in the literature? Let ...
8
votes
1answer
374 views

Are $(\infty,1)$-categories $A_\infty$ categories?

Let $X$ be a set. One can define a non-symmetric colored version of the non-unital $A_\infty$ operad as follows. The set of colors is the set of ordered pairs in $X$. Let ...
2
votes
0answers
136 views

Model structure on stacks

does anyone know if there is a model structure on stacks where the cofibrations are the monomorphisms ? As far as I know usually to get a model structure on stacks one localizes a model structure on ...
5
votes
3answers
419 views

What is a Homotopy between $L_\infty$-algebra morphisms II

I would like to continue on question I asking, what is a homotopy between Lie infinity algebras, since I'm not satisfied in two directions: 1.) The naive approach to define a homotopy would be ...
4
votes
2answers
342 views

2 and 3 pullbacks

If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times_C C^I \times_C B$, where $I$ is the ...
3
votes
1answer
288 views

Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology

Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this ...
3
votes
3answers
272 views

Higher dimensional pasting diagram of cubes

A pasting diagram is the analogue of composition of arrows in higher categories. The nlab page gives an example of a two dimensional diagram and how to compute it as a composite of two morphisms. I am ...