Cubical approximation for directed topology I

TL;DR

The paper develops directed-topology tools by establishing simplicial and cubical approximation theorems for streams, enabling directed (co)homology computations via combinatorial models. It builds a bridge between topological directed spaces (streams) and combinatorial models (simplicial and cubical sets) through geometric realization and its adjoints, showing that realization induces a correspondence between weak directed diagram-homotopy theories with a right adjoint satisfying excision. It also proves that, for compact quadrangulable streams, two competing directed notions of homotopy coincide, and it proves directed approximation theorems up to subdivision in both the simplicial and cubical settings. Together, these results lay a foundation for tractable computation of directed invariants and set the stage for a forthcoming equivalence between stream realizations and cubical-simplicial homotopy theories.

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.

Figures (5)

• Figure 1: Failure of a homotopy through monotone maps to extend. A directed path on the illustrated square annulus $[0,3]\setminus[1,2]$ is a path monotone in both coordinates. The illustrated dotted homotopy of maps from $\{0,1\}$ to $[0,3]^2\setminus[1,2]^2$ fails to extend to a homotopy through directed paths from the illustrated solid directed path.
• Figure 2: Edgewise subdivision, a cubical analogue, and triangulation
• Figure 3: Weak (left) and strong (right) types of directed homotopies. The homotopies on both sides are identical up to reparametrization of paths. Only the right homotopy is monotone in the homotopy coordinate, traced by the dotted lines. For compact quadrangulable streams, the equivalence relations generated by weak and strong definitions of directed homotopy are equivalent [Theorem \ref{['thm:d']}].
• Figure 4: Simplicial subdivisions of $\Delta[2]$
• Figure 5: Lattice structure on a topological simplex. The corners of the simplex are totally ordered, starting from the bottom left all the way to the top right. The dotted lines connect two generic points to their infimum (bottom left of dotted square) and supremum (top right of dotted square).

Theorems & Definitions (99)

• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Example 3.4
• Definition 4.1
• Definition 4.2
• Proposition 4.3
• Proposition 4.4: krishnan:convenient
• Theorem 4.5: krishnan:convenient