Large Scale Open Subsets of Configuration Spaces and the Foundations of Factorization Homology

This project (joint with Andrew Blumberg) aims to adapt the foundations of factorization homology to be more amenable to equivariant generalizations for actions of positive dimensional compact Lie groups. This requires finding replacements for arguments that discretize configuration spaces (e.g., arguments in terms of quasi-categories) or that use a local-to-global approach (arguments in terms of a small neighborhood of a point in a configuration space).  In practice, each such theorem reduces (by a Quillen Theorem A argument) to showing that a comparison map from a certain homotopy coend (that depends on the specifics of  the statement) to a configuration space (or related space) is a weak equivalence; the replacement strategy is to construct a cover of the configuration space by "large scale" open subsets whose intersection combinatorics and homotopy types mirror those of the composition combinatorics and homotopy types in a bar construction for the homotopy coend.  ("Large scale" is descriptive rather than technical: the game to is to describe open subsets of configuration spaces C(n,M) that will be G-stable when M=G/H is an orbit space for a positive dimensional compact Lie group G.)  In other words, the project is to deduce the properties of factorization of homology directly from "large scale" open covers of configuration spaces.