Manifold calculus and homotopy sheaves

TL;DR

This paper develops an enriched version of manifold calculus that better handles natural enrichments and connects the Taylor tower to operad theory, specifically framed little discs operads.

Contribution

It introduces an enriched manifold calculus framework and relates the Taylor tower to homotopy sheafifications and operad modules.

Findings

Enriched manifold calculus extends the original framework.

The Taylor tower corresponds to homotopy sheafifications.

Limit of the Taylor tower relates to operad module maps.

Abstract

Manifold calculus is a form of functor calculus concerned with functors from some category of manifolds to spaces. A weakness in the original formulation is that it is not continuous in the sense that it does not handle well the natural enrichments. In this paper, we correct this by defining an enriched version of manifold calculus which essentially extends the discrete setting. Along the way, we recast the Taylor tower as a tower of homotopy sheafifications. As a spin-off we obtain a natural connection to operads: the limit of the Taylor tower is a certain (derived) space of right module maps over the framed little discs operad.