Manifold calculus beyond space-valued functors

Manifold calculus is a homotopy-theoretic technique to study presheaves on manifolds, which decomposes them into successive approximations called polynomial approximations. First invented by Weiss to study embedding spaces, it has become an important toolset for homotopical study of manifolds. 

Like ordinary calculus, manifold calculus has two "fundamental theorems," one which classifies polynomial presheaves, and the other that classifies homogeneous presheaves. Consistent with his goal to study embedding spaces, Weiss established these theorems for space-valued presheaves. 

From the perspective of studying manifold invariants, it is extremely natural to develop manifold calculus for presheaves with more general values, such as spectra and chain complexes. However, Weiss's proof of the fundamental theorems relies on ad-hoc constructions on spaces, which do not seem to generalize easily. 

In this talk, I will explain that the two fundamental theorems do not depend on space-level constructions. Consequently, they extend to presheaves valued in essentially any infinity category. This talk is based on my paper "A context for manifold calculus" (arXiv:2403.03321).