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](https://ncatlab.org/nlab/show/generalized+smooth+space) are proposed for this purpose. I find the most natural and preferable approach: categories of sheaves on suitable sites (among other things, such categories are automatically Grothendieck topoi with all the properties that follow). Ideally, I'm looking for a textbook on differential geometry from scratch that actively uses, wherever appropriate, generalized smooth spaces, which are defined as the category of sheaves on some site (this is my only requirement). If such textbooks do not exist, then any literature on generalized smooth spaces of this kind, where some definitions are given and some theorems are proved, would be useful to me.



Are there textbooks on differential geometry in the language of smooth sets or smooth derived stacks?
“[Diffeology](http://math.huji.ac.il/%7Epiz/Site/TheBook-rep.html)” by Patrick Iglesias-Zemmour is probably the closest match.


He develops differential forms and de Rham cohomology, fiber bundles, connections, and symplectic geometry in the language of diffeological spaces, i.e., concrete sheaves of sets on the site of smooth manifolds. This book is closest in style to a conventional differential geometry textbook.


Another book is “[Synthetic geometry of manifolds](https://users-math.au.dk/kock/SGM-final.pdf)” by Anders Kock,
which treats differential forms, Lie groups and algebras, principal bundles with connections, jets and differential operators. It has a somewhat different focus (e.g., infinitesimals and the internal language of toposes) than the previous book.


In relation to this one can also mention “Models for smooth infinitesimal analysis” by Ieke Moerdijk and Gonzalo Reyes, which covers some foundational topics in differential geometry, like differential forms.


