Cubical Approximation for Directed Topology II

TL;DR

The paper addresses the challenge of comparing directed topological spaces with combinatorial models by establishing a fundamental equivalence between the directed homotopy category of cubical sets and that of directed topological spaces, extending the classical equivalence between cubical sets and topological spaces to diagrams over a small indexing category ${\mathscr{G}}$. The approach centers on cubcats, a robust cubical-operations framework built from a comonadic/pro-completion strategy with endofunctors like $\mathbf{R}$ and $\mathbf{S}$, enabling a derived cubical approximation that respects directionality. This framework yields concrete computations: directed homotopy monoids, directed singular $1$-cohomology monoids, and a combinatorial characterization of isomorphisms between small categories up to zig-zags of natural transformations as directed homotopy equivalences between directed classifying spaces. The results provide new tools for directed topology with potential applications to directed type theory and semantics of concurrent or hybrid systems, offering a principled way to encode and manipulate directed higher-categorical information inside a tractable cubical framework.

Abstract

The paper establishes an equivalence between directed homotopy categories of (diagrams of) cubical sets and (diagrams of) directed topological spaces. This equivalence both lifts and extends an equivalence between classical homotopy categories of cubical sets and topological spaces. Some simple applications include combinatorial descriptions and subsequent calculations of directed homotopy monoids and directed singular 1-cohomology monoids. Another application is a characterization of isomorphisms between small categories up to zig-zags of natural transformations as directed homotopy equivalences between directed classifying spaces. Cubical sets throughout the paper are taken to mean presheaves over the minimal symmetric monoidal variant of the cube category. Along the way, the paper characterizes morphisms in this variant as the interval-preserving lattice homomorphisms between finite Boolean lattices.

Figures (4)

• Figure 1: Directed Homotopy Inextendability Depicted above are Minkowski $(1+1)$-spacetimes. Thin dotted diagonal lines represent the speed of light. Curves represent causal paths, smooth paths whose derivative always lies in the future-facing part of the closed light cone. In the left picture, a dotted homotopy of the restriction of the causal path to its endpoints extends to a classical homotopy from the causal path, but not to another causal path. In the right picture, a dotted homotopy of the restriction of the left causal path to its endpoints extends to a classical homotopy from one causal path to another causal path, but not through such causal paths; a homotopy between such causal paths must go through a path that travels faster than the speed of light. A directed homotopy is, in particular, a homotopy through directed maps, such as causal paths.
• Figure 2: Equivalence as different categorical structures. The directed graphs above freely generate equivalent groupoids but freely generate mutually inequivalent categories, some of which are nonetheless directed homotopy equivalent to one another. After passage to free categories, the left two directed graphs are directed homotopy equivalent to one another, the right two directed graphs are directed homotopy equivalent to one another, but the left two and the right two are not directed homotopy equivalent to one another. Intuitively, classical equivalences ignore the structure of time in state spaces while categorical equivalences are sensitive to arbitrary subdivisions of time. Directed homotopy sidesteps some of the combinatorial explosion that bedevils geometric models of state spaces sensitive to arbitrary subdivisions in time. Section §\ref{['sec:categorical.homotopy']} formalizes the different notions of equivalence between small categories.
• Figure 3: Order-theoretic Subdivision. The left square and right squares represent the Hasse diagrams for the respective posets $[1]^{2}$ and $[2]^{2}$. Ordered pairs of elements in the left poset, or equivalently monotone functions $[1]\rightarrow[1]^{2}$, are in 1-1 correspondence with elements in $[2]^{2}$. Along this correspondence, for example, the ordered pair of larger points on the left corresponds to the large point on the right.
• Figure 4: Conal manifoldsConal manifolds, smooth manifolds whose tangent spaces are all equipped with convex cones, naturally encode state spaces of processes under some causal constraints. The convex cones define partial orders on an open basis of charts that uniquely extend to circulations on the entire manifold. The time-oriented Klein bottle (left) and time-oriented torus (right) depicted above are examples of conal manifolds that arise as directed realizations of cubical sets. Over cancellative commutative monoid coefficients $\tau$, their directed $1$-cohomologies are $\tau\times_{2\tau}\tau$ (left) and $\tau^2$ (right) by a simple application of cubical approximation [Examples \ref{['eg:klein.calculation']} and \ref{['eg:torus.calculation']}].

Theorems & Definitions (134)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Example 3.6