Multivariable manifold calculus of functors

TL;DR

This paper extends manifold calculus of functors to a multivariable setting, enabling the analysis of functors from products of open set categories, with applications to link maps and derivatives.

Contribution

It introduces multivariable manifold calculus, constructs Taylor approximations, classifies homogeneous functors, and relates single-variable and multivariable theories.

Findings

Constructed multivariable Taylor approximations.

Classified multivariable homogeneous functors.

Applied to compute derivatives of functors and analyze link maps.

Abstract

Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce "multivariable" manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. We construct multivariable Taylor approximations to such functors, classify multivariable homogeneous functors, apply this classification to compute the derivatives of a functor, and show what this gives for the space of link maps. We also relate Taylor approximations in single variable calculus to our multivariable ones.