The tag has no wiki summary.

learn more… | top users | synonyms

7
votes
1answer
126 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 — 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
116 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
500 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
159 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
90 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
172 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 ...
10
votes
0answers
564 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
355 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
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 ...
2
votes
0answers
125 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
116 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
143 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
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
187 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
292 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 ...
1
vote
0answers
117 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 ...
3
votes
1answer
242 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 ...
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
1answer
240 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
240 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 103–194. The result can be roughly stated as ...
7
votes
1answer
208 views

Are topoi and etale geometric morphisms locally small?

This question is related to this one: Local smallness and (higher) topoi which has not yet been answered. The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in ...
9
votes
2answers
365 views

Do Homotopy Fully Faithful Functors Push-out?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces. Are homotopy fully faithful functors preserved under (homotopy) pushout? More ...
6
votes
1answer
210 views

Joins of, and limits in, $(\infty,1)$-categories via profunctors

I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind. In the classical setting it is almost a triviality to express the join ...
9
votes
1answer
332 views

Local smallness and (higher) topoi

The $2$-category of topoi and geometric morphisms is not locally small. For example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is ...
22
votes
2answers
658 views

How do you define (infinity,1) categories in Homotopy Type Theory?

One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...
2
votes
1answer
155 views

Create 2-category from a cartesian category

I have a way of constructing (something like) a 2-category from a category with products $\mathcal{C}$. My question: Is this a correct construction, and if so, does it have a name? Define ...
7
votes
0answers
187 views

Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...
1
vote
1answer
169 views

Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?

Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
9
votes
1answer
498 views

What interesting homotopy invariants can I write down using the universal property of homotopy types?

I've recently been led to believe some version of the following statement: Weak homotopy types, or equivalently $\infty$-groupoids (let me not commit myself to a particular model of these), are ...
9
votes
4answers
581 views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
5
votes
1answer
190 views

What's the link between topological spaces as locales and topological spaces as infinity-groupoids?

I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...
3
votes
0answers
142 views

About a Double-pseudo-category generalization of the module bicategory construction

To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is ...
8
votes
1answer
250 views

Loops and suspensions of higher categories

Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ ...
10
votes
1answer
580 views

Lower Algebra: Modules over the monoidal category of abelian groups

Proposition 6.3.2.18 of Higher Algebra identifies $Mod_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable ...
8
votes
1answer
281 views

The state of the art in the rectification of homotopy-coherent structures

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a ...
3
votes
0answers
196 views

For an $(\infty,1)$-topos, is the object functor from groupoid objects to the topos a fibration, cofibration?

This question is kind of suggested by the question Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects? Question: If $\mathcal C$ in an ...
5
votes
1answer
308 views

Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?

Let $C$ be an $(\infty,1)$-topos. The $(\infty,1)$-category of group objects in $C$ is a full sub-$(\infty,1)$-category of groupoid objects in $C$: $${\mathsf{Grp}}(C) \hookrightarrow ...
7
votes
1answer
233 views

Is there a Wall finiteness obstruction in other settings?

Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW ...
21
votes
0answers
622 views

categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result? Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...
3
votes
0answers
249 views

In the topos-theoretic interpretation of Physics by Isham & Doering what role does intuitionistic logic play? [closed]

I've asked this question on Physics.SE but was advised to ask it here. Isham & Doering have written a series of papers exploring how to ground physics in topoi. Now the internal logic of topoi is ...
6
votes
0answers
152 views

Compact objects in undercategories and filtered colimits

Let $\mathcal{C}$ be a compactly generated presentable $(\infty, 1)$-category. Consider the functor $$ \Phi: \mathcal{C} \to \mathrm{Cat}_\infty, \quad x \mapsto (\mathcal{C}_{x/})^\omega,$$ that ...
2
votes
0answers
239 views

Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry. Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...
7
votes
1answer
266 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
3
votes
0answers
176 views

(∞,n)-category of triangulated cobordisms

What is an accepted definition of a (∞,n)-category of triangulated cobordisms? Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
16
votes
2answers
468 views

Homotopy-theoretic derived Morita equivalences

Recall that two $k$-algebras $A, B$ are Morita equivalent iff their categories of left modules are equivalent. However, this relation turns out to be rather fine and one introduces a coarser ...
5
votes
2answers
275 views

How does one Segal-subdivide a 2-category?

Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...
3
votes
2answers
217 views

Localisation in a quasi-category

Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction. Now consider ...
32
votes
4answers
2k views

Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...