Where to find about the category theoretic study of manifolds?

Question

I'm looking for a resource about a category theoretic study of manifolds. What do you think is a good start?

Hint: Not after very advanced resources. So no worries (indeed, preferred) if it's an elementary one!

Thanks.

Answer 1

The closest I have come across is the book Natural Operations in Differential Geometry by Kolár, Michor, and Slovák. From the preface

Third in the beginning of this book we try to give an introduction to the fundamentals of differential geometry (manifolds, flows, Lie groups, differential forms, bundles and connections) which stresses naturality and functoriality from the beginning and is as coordinate free as possible.

The book isn't focused on exploring the category theory of the category of (smooth) manifolds, but instead uses the language of category theory, when appropriate, to clarify aspects of differential geometry.

If you are looking for the category theory of the category of (smooth) manifolds, I at least am only aware of some recent work by J.R.B. Cockett and G.S.H. Cruttwell which investigates the notion of a tangent structure on a category and how one can recover some classical notions such as the Lie bracket, vector bundles, etc. even (and especially) when the category is NOT the classical category of smooth manifolds. Some papers in this direction are linked to at Cruttwell's website.