Questions tagged [generalized-smooth-spaces]

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3
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2answers
221 views

What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?

Let $G$ and $H$ be Lie groupoids. We know that there are two notions of equivalences of Lie groupoids: Strongly equivalent Lie groupoids: (My terminology) A homomorphism $\phi:G \rightarrow H$ of ...
2
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1answer
194 views

On the proof of “Mapping space is a Chen space”

According to the page 5 in the paper Convenient Categories of Smooth Spaces https://arxiv.org/pdf/0807.1704.pdf by Baez and Hoffnung, Chen space is defined as follows: (Note:I used different ...
3
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0answers
58 views

When is the quotient of a manifold by a discrete group of diffeomorphisms a diffeological covering space?

I was reading An Introduction to Diffeology by Patrick Iglesias-Zemmour and he defines a diffeological covering space as a diffeological fiber bundle with discrete fiber. My question: Consider a ...
1
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0answers
64 views

If all length metrics are strong equivalent on a closed connected topology manifold?

Let $M$ be a connected closed topology manifold and $d$ is a length metric (or an inner metric) on it , i.e. the distance between every pair of points is equal to the infimum of the length of ...
3
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0answers
322 views

Geometry of the irrational torus

One of the motivations of diffeology is to study singular spaces such as the irrational torus. The irrational torus $T_α$ of slope $α∈R∖Q $ as a diffeological space is given by the quotient space $ R/(...
4
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0answers
105 views

Maps between simplicial manifolds

Is it true that every continuous simplicial map between smooth simplicial manifolds can be approximate (in some adequate sense) by a smooth simplicial map?
6
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2answers
684 views

Differential forms as a sheaf on the site of all manifolds vs. sheaf on an individual manifold

The de Rham complex can be viewed as a sheaf $\Omega$ on the entire site $\mathsf{Man}$ of smooth manifolds via the usual pullback of differential forms $(f: M \to N) \mapsto (f^*: \Omega(N) \to \...
4
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0answers
145 views

Homotopy invariance of the moduli stack of smooth $G$-bundles?

This question ought to have a straightforward (perhaps even glaringly obvious) answer, but so far I've already wasted a few hours trying to untangle this web of inconsistent identifications. I'm sure ...
8
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0answers
607 views

Topologies on compactly supported functions

Let M be a (non-compact) smooth manifold and consider the set $C^\infty_c(M)$ of smooth real-valued functions with compact support. We can give this function space several topologies. Here are four: ...
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0answers
109 views

Cohomology theory for generalized smooth spaces

Is there a cohomology theory for generalized smooth spaces, for example, smooth topos? In particular, if we know the cohomology of $M$ and $N$, can we calculate that of $M^N$?
4
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0answers
178 views

smooth topos as generalized smooth space

I'm interested in generalized smooth spaces. I know there are several spaces such as Deffeological space, Frölicher space, Chen space, etc... and there are some papers compare them. However, I haven't ...
5
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0answers
212 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
2
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0answers
137 views

Relationship between tangent spaces and tangent categories for smooth topoi

Let $\mathscr E$ be a smooth topos, where I am using the terminology of nLab. In particular that imples the existence of a line object $R$ and an "infinitesimally thickened point", which is an object ...
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2answers
462 views

Are exotic spheres still exotic in generalised smooth spaces? [closed]

This is really more of a philosophical question, and the title is somewhat rhetorical: Exotic spheres are a feature of smooth manifold theory, where certain spheres can have more than one ...
2
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2answers
312 views

are immersions/submersions captured in generalised smooth spaces by some universal property?

Immersions & sumersions are important in differential manifolds. They rely on their definition of the construction of the tangent bundle. I realise that generalised smooth spaces do not have a ...
4
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0answers
294 views

Regular maps between Frechet manifolds and pullbacks

An oft-used example of a regular map of finite-dimensional smooth manifolds is a submersion. We have the well-known result that the pullback of a submersion exists and is a submersion. For Frechet ...
4
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1answer
272 views

Can ordinary schemes be described as sheves of algebras/models for a certain Lawvere theory?

Consider the definition of a $\mathcal{C}^{\infty}$-scheme given in Dominique Joyce's "Algebraic geometry over C-infty rings". As far as I uderstand (not being an expert either in C-infty rings nor in ...
19
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2answers
1k views

Nice application of generalized smooth spaces

I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Chen spaces, Frolicher spaces ...) form much nicer categories of ...