Cubical approximation for directed topology I

TL;DR

This paper develops cubical and simplicial approximation theorems for directed spaces, establishing an equivalence between cubical sets and directed spaces, which aids in calculating directed homotopy invariants.

Contribution

It introduces new approximation theorems and an equivalence between cubical sets and directed spaces, facilitating computations in directed homotopy theory.

Findings

Geometric realization induces an equivalence between cubical sets and directed spaces.

The right adjoint satisfies an excision theorem.

Criteria established for when different homotopy relations coincide.

Abstract

Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than classical homotopy invariants on underlying spaces because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with temporal structure encoding the orientations of simplices and 1-cubes. In an attempt to develop calculational tools for directed homotopy theory, we prove appropriate simplicial and cubical approximation theorems. We consequently show that geometric realization induces an equivalence between weak homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies an excision theorem. Along the way, we give criteria for two different homotopy relations on directed maps in the literature to coincide.