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1
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1answer
177 views

gluing bundles as a 2-colimit

Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?
2
votes
3answers
1k views

Exact sequences in homotopy categories

I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology. How ...
3
votes
1answer
255 views

How can I see that the slice of a presheaf category is equivalent to the presheaf category of the category of elements?

Let $\mathcal{C}$ be a small category and $P$ a presheaf on $\mathcal{C}$. Then there is an equivalence $$\widehat{\mathcal{C}} / P \cong \widehat{\int_{\mathcal{C}}} P$$ Now there are (at least) two ...
8
votes
4answers
642 views

Higher categories in logic

I've read somewhere (probably in the nlab) that higher category theory has application in logic. By the way since now the only applications of higher category theory I've seen are in homotopy theory ...
6
votes
1answer
342 views

Detecting equivalences of (infinity) categories by nerves

I have two questions: Is there a way to tell if a functor $F:C \to D$ between two small categories is an equivalence in terms of the map $$N(F):N(C) \to N(D)$$ between simplicial sets? More ...
7
votes
3answers
601 views

A 2-category of chain complexes, chain maps, and chain homotopies?

First-time here... I hope my question isn't silly or anything... anyway... Consider the category of chain complexes and chain maps. We can also define chain homotopies between chain maps. Does this ...
6
votes
3answers
718 views

Segal's Original Definition of a Topological Category

Nowadays we can associate to a topological space $X$ a category called the fundamental (or Poincare) $\infty$-groupoid given by taking $Sing(X)$. There are many different categories that one can ...
4
votes
1answer
304 views

Writing an infinity groupoid as a colimit of sets

If we are given a simplicial set $X:\Delta^{op} \to Set$, we may regard it as a \emph{simplicial} simplicial set, i.e. a bisimplicial set by composing with the "constant" inclusion $Set \to ...
6
votes
1answer
352 views

Why are sheaves not preserved in this case?

Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi_{\mathscr{X}}:\int_{C} \mathscr{X}\to C$$ denote the associated fibered ...
9
votes
0answers
511 views

Cohesive ∞-toposes for analytic geometry

There is a class of big ∞-toposes that come with a good supply of intrinsic notions of differential geometry and differential cohomology: called cohesive ∞-toposes (after Lawvere's cohesive toposes). ...
5
votes
2answers
292 views

Fibration of Batanin/Leinster $\omega$-groupoids

Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster? I tried to search on Google and in Higher Operads, Higher Categories of Tom ...
8
votes
1answer
449 views

How do various notions of natural transformation relate to various notions of homotopy in $2Cat$?

In what follows, $2$-categories will be strict, and "$2$-functor" will mean "strict $2$-functor". (Please mention which terminological conventions you are using when answering.) I guess that the ...
4
votes
1answer
575 views

Why is the Street nerve of the Gray tensor product $[1]\otimes [1]$ isomorphic to $[1]\times [1]$

Recall that given two strict ω-categories $A$ and $B$, their lax Gray tensor product $A\otimes B$ is sent to the Verity-Gray tensor product of their associated complicial sets ...
10
votes
5answers
2k views

Computations in $\infty$-categories

Direct to the point. Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that ...
8
votes
0answers
435 views

Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title. In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are ...
7
votes
0answers
182 views

Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...
0
votes
1answer
387 views

Weak algebraic structures

The following question can be thought as a sequel of this one. Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...
2
votes
0answers
145 views

Universal polygraphic factorization of strict ω-categories relative to a cobase

Recall from 1 that a cofibration of strict ω-categories is a retract of relative $I$-cell complexes, where $I$ denotes the set of boundary inclusions $\partial D^n \hookrightarrow D^n$, where $D_n$ ...
6
votes
0answers
177 views

Canonical topology for big infinity topoi

It is well known that for $E$ a Grothendieck topos, (by appropriately making use of universes) $E$ carries the canonical Grothendieck topology generated by jointly surjective epimorphisms, say $J$, ...
8
votes
1answer
193 views

Flatness for infinity functors

It is well known that for ordinary categories, if $C$ has finite limits and $D$ is cocomplete, and $A:C \to D $ is left-exact (i.e. preserves finite limits) then the left-Kan extension of $F$ along ...
10
votes
1answer
528 views

Is the first differential Pontryagin class a morphism of stacks?

In Cech Cocycles for Characteristic Classes, Jean-Luc Brylinski and Dennis McLaughlin provide explicit formulas for Cech cocycles for characteristic classes of real and complex vector bundles, and ...
45
votes
7answers
3k views

Are higher categories useful?

Of course, personally, I think the answer is a big Yes! However once, a while ago, while giving a talk about higher category theory, I was asked a question about whether higher category theory was ...
2
votes
1answer
282 views

Classification of principal G-bundles over a differentiable stack

According to "Notes on differentiable stacks" by Heinloth, the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13) (Here $G$ is a Lie group.) My questions are: (1) What ...
5
votes
1answer
368 views

Presheaves on a complete Segal space

Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which ...
3
votes
2answers
510 views

Functorial choice of pullbacks in a locally cartesian closed $(\infty,1)$-category

In a locally cartesian closed category $\mathcal C$, for every map $f:A\to B$, there is an associated pullback functor $f^* : \mathcal C/B \to\mathcal C/A$. Moreover, if $g:B\to C$, the two functors ...
4
votes
0answers
194 views

Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

Recall: Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we ...
0
votes
0answers
248 views

cokernel for $L_\infty$-algebra morphisms

As I have asked a wrong question previously, I edited a bit. It is obvious that the cokernel construction does not work well for the category of Lie algebras. The cokernel exists only for a normal ...
4
votes
4answers
615 views

What is the intuition of connections for cubical sets?

I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, ...
9
votes
0answers
446 views

Is there any elementary text unravelling the definitions of 2-category, lax functor and lax transformation, allowing people who do not know in the first place what these things are to really understand the definitions?

The question is in the title. My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people ...
2
votes
4answers
298 views

Composition of composable 2-cells in a 2-category is unambiguously defined?

I believe the following nice statement is true, but I cannot find a reference or proof it myself. In a 2-category(i.e., bicategory), the composition of composable 2-cells is unambiguously defined. ...
1
vote
0answers
187 views

Proof for a simplicial morphism

Sorry for the title but I can't come up with another one... ... Suppose X and Y are simplicial sets and $f : X \rightarrow Y$ is a map such that: 1.) f maps dimensions the right way, that is ...
5
votes
1answer
357 views

Compare three 2-categories of (Lie) groupoids

Lie groupoids are groupoids with smooth structures. There is a nature 2-category of Lie groupoids: Lie groupoids, smooth functors of Lie groupoids, smooth natural transformations of smooth functors. ...
11
votes
2answers
1k views

What is a symmetric monoidal $(\infty,n)$-category?

This question arose from reading Jacob Lurie's "Classification of topological field theories" paper. In that paper, he uses complete $n$-fold Segal spaces as a model for $(\infty,n)$-categories, but ...
0
votes
0answers
347 views

[]-infinity algebra and Projective representation

This is a very vague question. We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...
7
votes
0answers
250 views

Completion of n-fold Segal spaces

During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for ...
1
vote
1answer
475 views

What is exact sequence in higher categories?

What is the higher categorical generalization of exact sequence (3 terms or $\mathbb{Z}$ terms)? In particularly, consider the simplest cases: chain complexes, $L_\infty$-algebras.
8
votes
4answers
709 views

Limits in an $(\infty,1)$-category

In ordinary category theory, the notion of limit in a category $C$ is usually formulated with a category (of indices) $J$ and a functor $F:J\to C$ (a diagram in $C$), and a limit of this diagram is ...
19
votes
2answers
849 views

Homotopy groups of spheres in a $(\infty, 1)$-topos

Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces). You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, ...
2
votes
2answers
228 views

Automorphisms and Bicategories

Sorry if this isn't the right place for this, it hasn't gotten any answers on ME. I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are ...
18
votes
3answers
2k views

Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
33
votes
4answers
5k views

Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
2
votes
1answer
270 views

What are $n$-poset?

Yesterday I was wandering for the $n$-lab and I've found the definition of $n$-poset. Following this post it seems that a $n$-poset should be a $(n,n+1)$-category. Now an $(n,r)$-category should be a ...
5
votes
1answer
179 views

2-completeness analog of completeness theorem

It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations". I ...
4
votes
2answers
1k views

Mathematics needed for higher dimensional category theory? [closed]

I'm a undergrad(third year, Manchester uni and want to do a PhD) that is thinking of doing a PhD in this area or category theory in general.(Sorry for asking it here, Maths exchange stack didn't help ...
6
votes
1answer
664 views

Ordinal category theory?

Just out of curiosity: Is there a notion of $\alpha$-category for an ordinal number $\alpha$, extending the given notions for $\alpha \leq \omega$? If there is none, which one would you propose? Feel ...
9
votes
2answers
1k views

Semi-simplicial versus simplicial sets (and simplicial categories)

Hi, Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving ...
6
votes
1answer
549 views

How should I think of the $\infty$-category of spectra?

I've seen a bunch of definitions of spectra in the literature, and the fanciest seems to be the $(\infty, 1)$-category of spectra obtaining by "stablizing" the higher category of spaces, as in DAG I. ...
13
votes
2answers
568 views

Homotopy Fixed Points of SO(2) on Fully Dualizable Algebras

Note: by fixed points, I always mean homotopy fixed points. As explained in Jacob Lurie's paper on the cobordism hypothesis, we have an action of O(2) on the $\infty $-groupoid $X$ given by ...
11
votes
1answer
404 views

The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that: The cofibrations are the ...
19
votes
6answers
2k views

Concrete example of $\infty$-categories.

I've seen many different notion of $\infty$-categories, actual I've seen the operadic-globular ones of Batanin and Leinster and the opetopic too and eventually I'll see the simplicial ones too. ...