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4
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$\Omega$ 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}= ...
1
vote
1answer
101 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 ...
1
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0answers
32 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 ...
14
votes
2answers
567 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
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0answers
81 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
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0answers
133 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
203 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 ...
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vote
1answer
77 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 ...
4
votes
2answers
248 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
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0answers
94 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 ...
7
votes
1answer
224 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 ...
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0answers
165 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 ...
7
votes
1answer
115 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
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0answers
108 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
103 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 ...
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1answer
437 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
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1answer
150 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 $Fun( \Delta ,Cat )$?
...
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0answers
87 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
155 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 ...
9
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0answers
487 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
327 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 ...
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vote
1answer
190 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
110 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 ...
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0answers
94 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
89 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
363 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 ...
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0answers
155 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 ...
6
votes
1answer
262 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
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0answers
104 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
0answers
173 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
297 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
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1answer
205 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
226 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
157 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 ...
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1answer
194 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
351 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
207 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
329 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 ...
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2answers
550 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
153 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
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0answers
181 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
152 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
482 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 ...
10
votes
4answers
551 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 ...
4
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1answer
178 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 ...
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0answers
129 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 ...
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1answer
234 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.$$ ...
9
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1answer
484 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
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1answer
260 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
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0answers
178 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 ...
