Questions tagged [generalized-smooth-spaces]
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29 questions
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What happens if we add an initial object to a Lawvere theory?
Motivation
There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects ...
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Differential equations where we are tryng to find a smooth space(maybe with additional structure)
Is there literature on the differential equation $TM=N$ where $M$ and $N$ are two smooth spaces (maybe with additional structure) and $T$ stands for the tangent functor?
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Is there a notion of a complex/analytic diffeological space?
I have a bit of a general question. This seems like something you can do, but I can't seem to find much reference for this.. Perhaps something like this already exists in a different guise. But, is ...
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Applications of diffeological spaces to ordinary differential geometry
Recently I've been learning more about differential geometry, and I came upon the notion of a diffeological space, which encompasses a number of already known extensions of smooth manifolds or related ...
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Are there textbooks on differential geometry in the language of smooth sets or smooth derived stacks?
In differential geometry it is often natural to speak of infinite-dimensional manifolds (e.g., the manifold of mappings between two smooth manifolds). Different versions of generalized smooth spaces ...
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Can one make sense of de Rham cohomology for the complement of a (dense) irrational flow on the torus?
Recent work has led me to consider whether one could define consider the complement of a dense irrational flow on the torus $P_\alpha \subset T^2$ as some kind of generalized smooth space, and ...
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Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds
This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following ...
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Is the category of diffeological spaces a full subcategory of locally ringed spaces?
It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar ...
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A-spaces vs generalized spaces
$\newcommand\Set{\mathit{Set}}$I've been searching about the notions of smooth generalized spaces and I come across with two definitions that seem very good ones.
John Baez and Alexander Hoffunung ...
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Do colimits of manifolds coincide with underlying colimits as topological spaces?
Categories of manifolds (possibly with extra structure) tend not to have all colimits.
Other questions have addressed when colimits of manifolds exist.
I'd like to know what we can say in general ...
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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+...
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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 ...
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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 ...
- 1,215
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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 ...
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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 ...
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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/(...
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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?
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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 \...
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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 ...
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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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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 $M^N$?
user80247
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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 ...
user80247
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(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 ...
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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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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 ...
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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 ...
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Regular maps between Fréchet 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 Fréchet ...
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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 ...
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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 ...