2
votes
1answer
120 views

what is the stabilization of pointed sets?

Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab). The famous example is ...
3
votes
0answers
67 views

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 ...
2
votes
0answers
92 views

When do limits and colimits of infinity-categories commute?

This is a question for someone who read (or wrote) enough of Lurie's HTT to know a reference. Suppose $D,E$ are small "diagram" $(\infty,1)$-categories, and $\mathcal{C}$ is a stable infinity-category ...
3
votes
0answers
122 views

Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category

Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$. In HigherAlgebra, the derived category ...
0
votes
0answers
77 views

(Homotopy) limits and colimits in a dg-category

It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of ...
1
vote
0answers
35 views

Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones

This question is strictly related to this other one. Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let be a diagram in $Z^0(\mathcal A)$, where the rows are ...
3
votes
1answer
92 views

Reconstructing a morphism of exact triangles in the homotopy cat. of a dg-cat. using a “functorial cone map”

I set this problem in the framework of (pretriangulated) dg-categories; everything can probably be translated in the world of stable $(\infty,1)$-categories. Let $\mathcal A$ be a pretriangulated ...
5
votes
1answer
202 views

Constructing a “geometric” model structure on Cat by localizing the “categorical” model structure

Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the ...
4
votes
1answer
98 views

Good properties of the $H^0$ functor (from quasi-functors to ordinary functors)

Let $\mathcal A, \mathcal B$ be dg-categories over a field $k$. I denote by $\mathcal{RHom}(\mathcal A,\mathcal B)$ the dg-category (defined up to quasi-equivalence) which gives the internal hom in ...
5
votes
1answer
105 views

Do non-subcanonical Grothendieck topologies always induce a category of fractions?

Suppose that $\mathscr{C}$ is a (possibly higher) category and $J$ is a Grothendieck topology which is not subcanonical. Denote the composite $$\mathscr{C} \hookrightarrow ...
3
votes
1answer
123 views

Exponential objects in a category of abstract automata

I'm working with a more or less standard definition of the category Aut(C) of automata over a category C (where C has finite products) which has tuples $$ A=\langle I_{A},O_{A},S_{A},\sigma_{A}, ...
1
vote
0answers
61 views

When can I compute the simplicial mapping space from a presheaf to a simplicial presheaf naively?

Suppose that $\mathscr{C}$ is is a small category, $Y$ is a presheaf (of sets) on $\mathscr{C},$ and $X_\bullet$ is a simplicial presheaf. There is a spectrum of simplicial model structures on ...
3
votes
0answers
108 views

When does prolongation preserve sheaves?

Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will ...
1
vote
0answers
97 views

Retracts of 2-categories

Let $\mathbf{C}$ be a strict $2$-category and let $M = \{(x_\alpha,y_\alpha)\}$ be a collection of object-pairs so that each hom-category $\mathbf{C}(x_\alpha,y_\alpha)$ has an initial element ...
3
votes
0answers
114 views

When do localizations of presentable (infinity) categories commute?

Suppose that $\mathscr{P}$ is a (locally) presentable $\left(\infty,1\right)$ category (which we can assume WLOG is infinity presheaves on some small $\left(\infty,1\right)$ category) , and $R$ and ...
6
votes
0answers
112 views

When does a sheaf of categories represent a homotopy sheaf?

Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...
3
votes
0answers
97 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
83 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
282 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 ...
7
votes
1answer
179 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 ...
1
vote
0answers
128 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
333 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). ...
3
votes
0answers
98 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
210 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 ...
2
votes
0answers
161 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 ...
0
votes
0answers
98 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
240 views

Weakening simplicial identities

The generators $d_i, s_i$ for morphisms of the simplicial category satisfy simplicial identities: $d_jd_i = d_id_{jāˆ’1}$ for $i < j$ $s_jd_i = d_is_{jāˆ’1}$ for $i < j$ $s_jd_i = id$ for $i = ...
11
votes
0answers
188 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 ...
9
votes
1answer
580 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 ...
6
votes
1answer
219 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
200 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 ...
3
votes
0answers
86 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
144 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
255 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 ...
4
votes
2answers
290 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
104 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
260 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 ...
7
votes
1answer
127 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 ...
4
votes
0answers
120 views

Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category ā€” 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
117 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 ...
1
vote
1answer
502 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
161 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 $Fun( \Delta ,Cat )$? ...
2
votes
1answer
174 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 ...
1
vote
1answer
193 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 ...
4
votes
2answers
383 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 ...
0
votes
0answers
188 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 ...
7
votes
1answer
296 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 ...
3
votes
2answers
311 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
3answers
241 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 ...
3
votes
1answer
158 views

What is the suitable setting for supercoherence with value in a bicategory?

It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103ā€“194. The result can be roughly stated as ...