The higher-category-theory tag has no wiki summary.
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A definition of ($\infty$-)cat(egorie)s [on hold]
Yesterday I uploaded an updated version of my paper on "Cats". It will be available on the arXiv as of monday; until then, you can download a copy here.
In that paper, I take Simpson's category ...
5
votes
1answer
95 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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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 ...
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0answers
40 views
Is it possible to express the pentagon condition by associativity? [migrated]
Let me start by considering associativity. Let $C$ be a set with a binary operation (multiplication) $C\times C\to C$. The associativity law is $(ab)c=a(bc)$. It can be expressed by a commutativity ...
2
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0answers
93 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
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1answer
317 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 ...
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1answer
142 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
75 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
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1answer
148 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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452 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
308 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
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1answer
187 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
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0answers
96 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
84 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
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0answers
66 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
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2answers
354 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
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0answers
142 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
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1answer
254 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 ...
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0answers
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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
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0answers
169 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
293 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
194 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
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3answers
219 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
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1answer
155 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
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1answer
191 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
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2answers
344 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 ...
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1answer
200 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
326 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 ...
21
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2answers
528 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
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1answer
152 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
178 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
149 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
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1answer
480 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 ...
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4answers
532 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
175 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 ...
2
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123 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
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1answer
226 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.$$ ...
9
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1answer
457 views
Lower Algebra: Modules over the monoidal category of abelian groups
Proposition 6.3.2.18 of Higher Algebra identifies $Mod_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable ...
8
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1answer
254 views
The state of the art in the rectification of homotopy-coherent structures
My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of:
Cordier and Porter proved a ...
3
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0answers
174 views
For an $(\infty,1)$-topos, is the object functor from groupoid objects to the topos a fibration, cofibration?
This question is kind of suggested by the question Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?
Question: If $\mathcal C$ in an ...
5
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1answer
289 views
Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?
Let $C$ be an $(\infty,1)$-topos. The $(\infty,1)$-category of group objects in $C$ is a full sub-$(\infty,1)$-category of groupoid objects in $C$:
$${\mathsf{Grp}}(C) \hookrightarrow ...
7
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1answer
225 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 ...
17
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0answers
456 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
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0answers
231 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
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0answers
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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 ...
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0answers
211 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 ...
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1answer
260 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
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0answers
159 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
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2answers
423 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
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2answers
248 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 ...
