Absolutely homotopy-cartesian squares

TL;DR

This paper introduces the concept of absolutely cartesian diagrams, where applying any homotopy functor preserves the cartesian property, and provides a classification theorem for such squares in spaces.

Contribution

It defines the notion of absolutely cartesian diagrams and proves a classification theorem for squares, extending the understanding of homotopy limits in this context.

Findings

Classification theorem for absolutely cartesian squares of spaces

Framework for analyzing diagrams preserved under all homotopy functors

Conjecture for classification of higher-dimensional cubes

Abstract

We call a diagram D absolutely cartesian if F(D) is homotopy cartesian for all homotopy functors F. This is a sensible notion for diagrams in categories C where Goodwillie's calculus of functors may be set up for functors with domain C. We prove a classification theorem for absolutely cartesian squares of spaces and state a conjecture of the classification for higher dimensional cubes.