# Questions tagged [generalized-smooth-spaces]

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20
questions

**4**

votes

**2**answers

239 views

### Is there any Lie groupoid structure on $Hom(\mathcal{G}, \mathcal{H})$ where $\mathcal{G}$ and $\mathcal{H}$ are Lie groupoids?

We know that in general, there is no smooth manifold structure on $Hom(X,\, Y)$ where $X$ and $Y$ are smooth manifolds, but under certain nice conditions (see https://ncatlab.org/nlab/show/manifold+...

**2**

votes

**0**answers

93 views

### Construction of differential of a smooth map between diffeological spaces with respect to internal tangent spaces

According to page 5 , definition 3.1 of https://arxiv.org/pdf/1411.5425.pdf, there is a notion of Internal Tangent Space of a Diffeological space $X$ at a point $x \in X$.
Basically if $x \in X$, they ...

**4**

votes

**3**answers

397 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**

votes

**1**answer

214 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**

votes

**0**answers

85 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**

vote

**0**answers

65 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 ...

**2**

votes

**0**answers

369 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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107 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**

votes

**2**answers

727 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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151 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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723 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:
...

**1**

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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**

votes

**0**answers

184 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**

votes

**0**answers

227 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**

votes

**0**answers

141 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 ...

**0**

votes

**2**answers

472 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 ...

**3**

votes

**2**answers

356 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**

votes

**0**answers

300 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**

votes

**1**answer

285 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 ...

**20**

votes

**2**answers

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 ...