If you are an algebraist: general sheaves. If you are a geometer: diffeology.

I will try to explain what I mean. I may however still modify the following.

The philosophy of the theory of sheaves is summarised by this excerpt taken from François de Marçay in his lecture "Faisceaux":

Comment, à partir d’une collection d’objets qui sont définis seulement
sur des petits ouverts, peut-on construire un objet qui est défini sur
l’espace tout entier ?

That approach from local to global epitomises the theory of sheaves: knowing an object by knowing its details. The object of study appears as a puzzle of small parts. Indeed a diffeological space is associated with the sheaf of its plots. But is that all a diffeological space is? Is that able to satisfy the geometer?

According to Felix Klein, a *geometry* is defined in his "Erlangen program" as a group of transformations (the principal group) on some set (manifoldness):

Given a manifoldness and a group of transformations of the same; to
investigate the configurations belonging to the manifoldness with
regard to such properties as are not altered by the transformations of
the group.

Thus, the geometer starts with a manifoldness, however defined or chosen. He then chooses a group of transformations of this set, thus defining a geometry. The examples are numerous: from Euclidean geometry to projective geometry via spheric or hyperbolic geometry and so on. So:

The geometer knows an object not by its details, but in its unity, as a whole.
His vision of the object is sustained by the admissible global
transformations that define its geometry.

How, then, does "differential geometry" appear in this picture? Is it a geometry in Klein's sense? The short answer:

- Every diffeological space defines a (differential) geometry by the action of its group of diffeomorphisms. What is forbidden in classical differential geometry, because a triangle is not a manifold, is admitted in diffeology, because a triangle is a diffeological space whose diffeomorphisms can exchange only vertices with each other, like edges and preserve the interior.

Hence, diffeology enhances the concept of geometry by making each diffeological space a manifoldness à la Klein with principal group its group of diffeomorphisms. This could be a formal definition of *Differential Geometry*.

This is why the sheaf algebraic approach to diffeology is a contrario of its geometric approach, even if a diffeological space is a sheaf of plots.

BTW the global geometrical point of view of diffeology does not prevent from extending the group of diffeomorphisms to the germs of local diffeomorphisms to highlight the local structure and the singularities of the object. This is the point of view we used to revisit the theory of stratifications(*) in the paper :

P.I-Z with Serap Gürer. *Orbifolds as stratified diffeologies*. Differential Geometry and its Applications, Volume 86 (Feb. 2023).

I would add that looking at diffeology only from the point of view of sheaf theory constitutes the *passive approach*, whereas seeing diffeological spaces from the point of view of geometry, as the loci of the action of their group of diffeomorphisms, constitutes the *active approach*.

(*) What is defined above, the partition in orbits by the group of diffeomorphisms, is actually what we call the Klein stratification of a diffeological space.

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