This is the condition of analytic manifold (e.g., Stiefel andinfold and Grassmann manifolds), and we need its smoothness (i.e., to be imbedded in Euclidean ) simply for practicability. Other manifolds are too many, however., the Smooth Riemannian manifolds (Stiefel manifolds and Grassmann manifolds) are revolutional in complex structural data analysis.

Dear Muhammad Ali

Thank you for your answer.

If I understand you well, you are speaking of embedded manifolds. Any embedded manifold inherits the differentiable structure of the affine outer space. So an atlas exists.

In case the space is symmetric or homogeneous an atlas does not have to be made explicit. One knows that it exists in this case also. In the case it is extrinsic symmetric both conditions are met.

A classical condition, as I wrote, for abstract manifolds is to require the existence of local charts on one hand, but also that the chart-changes are smooth diffeomorphisms. Not every geometric space has this property. (e.g. orbifolds do not).

My question is: Can one give examples of geometric spaces that are differentiable or smooth in some sense, without being locally isomorphic to an affine space.

I imagine it is also possible to give examples of spaces that are locally diffeomorphic to an affine space, but where charts cannot be glued together in a smooth way.

I am very agree with you. Also, In such a case its theoretical strength of (if and only if) / or existence & uniqueness condition do not satisfy. Accordingly, such weak formulations may be okay for academic-theoretic studies, however., for practical purposes we may need analytic/differentiable/smooth/locally-Euclidean/symmetric-positive-definite etc., manifolds.

You can have a look at the book

Diffeology by Patrick Iglesias-Zemmour.

Thank you for your valuable comments and suggestions.

Muhammad Ali : You are right that one can privilege the study of euclidean geometry, semi-simple Lie-groups, etc. These build the most natural, may be most useful and most often met spaces. Students should know them. The role of charts in geometry is an heritage of a Cartesian vision of geometry. The project to emphasize the role of algebras over a (commutative) monad can be valuable also because linear algebra is may-be not the only access to geometry and monads are much more universal.

Yuri Ximenes Martins

Jean-Pierre Magnot : I own this book and have a little idea of its basic ideas. It is a fact that a diffeology is a good candidate for the monads I consider in my study. One should develop this aspect.

The role of 1-dimensional spaces in this context can be easily replaced by 2-dimensional ones. I think something essential happens with respect to differentiability between dimension 1 and 2.

In any case I would like to better understand what it means to differentiate in terms of categorical notions (and independently of the set of reals for instance), this is why I am not completely happy with the definitions of smoothness we have so far.

Dear Yuri Ximenes Martins

(In the new version I removed some notational ambiguities also.)

Join my project ;-)

Your work is extremely interesting since it reveals the nature of notions discovered in differential geometry for smooth manifolds.

I think however that in any Topos with a natural numbers object building manifolds by gluing together open sets of Euclidean spaces remains a fundamental tool, at least in order to prove existence of categorical constructions which you study.

I am very interested in developments of your work.

Dear Tommaso Boccellari ,

Thank you for your suggestions and encouragements.

Adding examples, and also non classical ones, is important to prove that the construction has some interest.

I have thought about some alternative starting points for finding such examples :

PS I think category theorists have know a lot about this for a long time, but differential geometers are sometimes not aware of them... I think it useful to present DG in a different way.

I would like to give a partly answer (from the viewpoint of category theory) to the initial question :

First, why using charts : open subsets of R^n have many good properties that allow to prove classical theorems of differential geometry. I cite : Theorem of constant rank, theorem of local inversion, etc.

Using these I suppose (without having checked details) that one can deduce that the category of open subsets of R^n (with smooth constant rank mappings as morphisms) has pullbacks.

The existence of an atlas, supposedly, allows to prove that the category of smooth manifolds has also pullbacks, which might be not given otherwise ?

Secondly, in this category of smooth manifolds ( or more basically in the category of open subsets of R^n and smooth constant rank mappings ) I wondered if the tangent functor preserves pullbacks . This is indeed a necessary condition for being a right adjoint functor. (I need this because I would like that the tangent functor has a left adjoint. )

(The intuition behind all this is that I am interested in the question under which conditions differentiation and integration can be seen as adjoint functorial operations. Think also of Green theorems.)

Dear Yuri Ximenes Martins

Thank you for these hints ! The idea was to use Freyds adjoint functor theorem to find a left-adjoint to the tangent functor. I have to check this.

Do you know about a canonical way to add limits resp co-limits to a category ?

I might be a standard result, but I don't have it in mind.

Sounds good =), I don't know if I understand everything, but I will try to take advantage of it.

What do you mean by localization in this context ? Is it like building the "fraction category" from the category of smooth spaces ?

Yuri Ximenes Martins

I can't remember, but I suppose it is true : The category of Frölicher spaces also admits pullbacks ? At least I am sure to remember that it is cartesian closed.

Now these good properties might not have been guaranteed if a space was simply a set of points together with a set of curves, preserved by morphisms. ( Indeed for every space there is a set of curves and a set of functions and both are saturated with respect to a fixed function set inside R->R, e.g. (the affine functions, or the polynomial functions, or the smooth functions, etc.))

You said : "The immediate step is to consider localizations, which implies looking at sheaf categories on the category of smooth manifolds."

How do Frölicher spaces fit into this scheme ?

Wow !

(Impressed by your encyclopedic knowledge !)

Dear Yuri Ximenes Martins

I would like to make the following, somehow artistic, contribution :

Thinking about interesting function spaces to consider for Frölicher monoids I add the following to the list before :

I mean monoids M included in R->R, or lets say in C^\infty(R,R).

etc.

All these should give rise to a category of "smooth spaces". Some might turn out to be well known categories, some might be new. I wonder if 3. gives rise to the category of compact spaces (?). Anyhow all these should give cartesian closed categories, giving an alternative way to prove cartesian closedness.

Since nobody seems to have mentioned it yet, Andrew Stace has several preprints discussing categories of generalised manifolds and their relation to each other in a very broad sense.

Most relevant to the question is perhaps

Article Comparative Smootheology

Sorry, just noticed that it was one of the TAC references from above (but there are also other papers by Andrew on the subject you might want to check out)