# Tagged Questions

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

In this post on the n-Category CafĂ©, 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### gluing bundles as a 2-colimit

Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?
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### 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 ...
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### 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 - ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $\pi_1$ ...