4
votes
1answer
172 views

Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...
10
votes
1answer
425 views

The universal property of the Liouville $1$-form

I am not totally sure if this question is appropriate for MathOverflow, or if it more adeguate to MathStackexchange. As usual any feedback is welcome. Introduction Given an arbitrary smooth manifold ...
1
vote
0answers
93 views

The Abelian Group of Equivalence Classes of Gerbes

Are there any good references/explanations for understanding the "contracted product" (as Brylinski calls it) of a gerbe with abelian band? I am finding it difficult, using Brylisnki's definition. to ...
3
votes
3answers
291 views

What are the symmetries of a principal homogeneous bundle?

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where: $G$ is a Lie group, and $H$ ...
20
votes
4answers
1k views

What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...
1
vote
1answer
256 views

Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold?

Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a ...
7
votes
0answers
141 views

Immersions of Topoi

An immersion of smooth manifolds is a smooth map whose Jacobian has full rank at each point in the source manifold. Is there a notion of ``immersion'' for geometric morphisms of topoi which ...
3
votes
1answer
533 views

Convenient definition of “category of Riemannian manifolds”?

Has a notion of "category of Riemannian manifolds" been defined and used in the literature? For which reasons is it or would it (not) be a useful notion? I think the objects should be all (perhaps ...
2
votes
1answer
181 views

Natural Transformations from the Tensor product of tangent bundle into the second order tangent bundle

Short question with long title: Suppose $T$ is the tangent functor and $T^2:=T\circ T$ is the second order tangent functor. Are there natural transformations $T\otimes T \Rightarrow T^2$ ? I ...
0
votes
2answers
108 views

inclusions of linear colimits into smooth manifolds

Let $V$ be the category of finite dimensional vector spaces and $M$ the category of smooth finite dimensional Hausdorff manifolds. Now suppose any finite dimensional vector space is equipped with a ...
1
vote
1answer
168 views

gluing bundles as a 2-colimit

Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?
12
votes
2answers
888 views

Are all manifolds affine?

There is a classical result which says that the assignment $$M \mapsto C^{\infty}\left(M\right)$$ is an embedding of the category of (paracompact Hausdorff) smooth manifolds into the opposite category ...
5
votes
2answers
384 views

Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...
0
votes
0answers
232 views

On the universal pullback of fiber bundles

First suppose we have three smooth manifolds $M_1$, $M_2$ and $N$ with smooth transversal maps $p_1: M_1 \rightarrow N$ and $p_2: M_2 \rightarrow N$ then its a well known fact that the categoric ...
4
votes
2answers
1k views

Why SU(3) is not equal to SO(5)?

I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are ...
10
votes
1answer
344 views

Is every representable map a submersion?

Recall that a morphism $f:C \to D$ in a category $\mathscr{C}$ is representable if for all maps $g:E \to D$ in $\mathscr{C},$ the pullback $C \times_{D} E$ exists. Let now $\mathscr{C}$ be the ...
13
votes
4answers
1k views

The tangent bundle to an infinite-dimensional manifold

Suppose that $A,B$ are smooth ($\mathrm C^\infty$) manifolds, and denote by $\hom(A,B)$ the set of $\mathrm C^\infty$-maps $A \to B$. It is a perfectly well-defined set, but often one wants more. ...
5
votes
3answers
596 views

Generalized complex groupoid

Is there any nontrivial example of Generalized complex groupoid? By trivial, I mean all the classes of symplectic groupoids/ Abelian varieties as well as their products. What I mean is that, is ...
15
votes
5answers
1k views

Advantages of Diffeological Spaces over General Sheaves

I have been playing with/thinking about diffeological spaces a bit recently, and I would like understand something rather crucial before going further. First a little background: Diffeological spaces ...
13
votes
5answers
854 views

What abstract nonsense is necessary to say the word “submersion”?

This question is closely related to these two, but the former doesn't go far enough and the latter didn't attract much attention, and anyway I want to ask the question slightly differently. Recall ...
2
votes
1answer
336 views

Maps that admit local sections through each 'point' in the domain

In a recent MO question I asked about the relation between surjective submersions (in the category of smooth or otherwise manifolds) and maps that admit local sections. The latter, it turns out, are ...
6
votes
0answers
259 views

Generalized smooth spaces and infinite dimensional manifolds

There is a theorem due to Losik which shows that the category of Frechet manifolds embeds fully-faithfully into diffeological spaces. (Diffeological spaces are concrete sheaves on the site of ...
5
votes
1answer
336 views

Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to \left\[G\right]$, where $\left\[G\right]$ is the ...
8
votes
3answers
756 views

What's the “correct” smooth structure on the category of manifolds?

As will become clear, this is in some sense a follow up on my earlier question Why should I prefer bundles to (surjective) submersions?. As with that one, I hope that it's not too open-ended or ...
8
votes
2answers
958 views

What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy?

This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf. Let $M$ be a ...
6
votes
2answers
737 views

Is Lang's definition of a tensor bundle nonstandard?

Background Let $\mathfrak{A}$, $\mathfrak{B}$, and $\mathfrak{C}$ be subcategories of the category of Banach spaces (over $\mathbb{R}$). Suppose we have a functor ...
4
votes
4answers
736 views

A comprehensive functor of points approach for manifolds

This seems unrealistic, because the topology on a manifold doesn't have anything to do with the properties its structure sheaf, but I figured I might as well ask. This wouldn't be the first time I ...
11
votes
2answers
698 views

synthetic differential geometry and other alternative theories

There are models of differential geometry in which the intermediate value theorem is not true but every function is smooth. In fact I have a book sitting on my desk called "Models for Smooth ...
1
vote
1answer
262 views

Semiclassical explanation of “Structured” spaces [closed]

We have our general notions of manifolds, schemes, et cetera, and other geometric "spaces", and we realize that a lot of these look like topological spaces with structure sheaves i.e. structured ...
5
votes
3answers
740 views

Where does the generic triangle live?

This is a reformulation of my question Characterizing triangles unembeddedly. Motivation 1: There is such a thing as a generic group. In category theory this is done by constructing "theory" of ...
8
votes
4answers
712 views

How are invariants represented in category theory?

I'm trying to better understand how to think about invariance in the setting of category theory. In some cases it seems there's an obvious interpretation: for instance, the fundamental group $\pi_1$ ...
7
votes
2answers
815 views

“synthetic” reasoning applied to algebraic geometry

A hyperlinked and more detailed version of this question is at nLab:synthetic differential geometry applied to algebraic geometry. Repliers are kindly encouraged to copy-and-paste relevant bits ...