Elmendorf's Theorem for Diagrams

TL;DR

This paper generalizes Elmendorf's Theorem to diagrams indexed by small categories, showing that the homotopical data of a D-space can be fully understood through its orbits, extending classical G-space actions.

Contribution

It provides a generalized version of Elmendorf's Theorem for D-spaces, broadening the understanding of orbit theory in the context of category actions.

Findings

Proves a generalized Elmendorf's Theorem for D-spaces.

Shows the homotopical data of D-spaces is captured by their orbits.

Extends classical orbit theory to diagram categories.

Abstract

The notion of a continuous $G$-action on a topological space readily generalizes to that of a continuous $D$-action, where $D$ is any small category. Dror Farjoun and Zabrodsky introduced a generalized notion of orbit, which is key to understanding spaces with continuous $D$-action. We give an overview of the theory of orbits and then prove a generalization of "Elmendorf's Theorem,'' which roughly states that the homotopical data of of a $D$-space is precisely captured by the homotopical data of its orbits.