Combinatorial manifolds and Kleene's theorem, homotopically

TL;DR

This paper introduces a method to construct categories of combinatorial manifolds using unique factorization systems, with applications to precubical sets and an abstract proof of Kleene's theorem.

Contribution

It develops a general framework for building categories of combinatorial manifolds as coreflective subcategories, linking topology, automata theory, and category theory.

Findings

Constructed a category of euclidean precubical sets as coreflective in relational precubical sets.

Provided an abstract proof of Kleene's theorem using manifold automata.

Linked combinatorial topology with automata theory through a categorical approach.

Abstract

We give a general method to build categories of combinatorial manifolds, i.e. categories of combinatorial objects satisfying some local property at every "point", as coreflective subcategories of categories of relational presheaves. To do this, we crucially rely on unique factorization systems, and we can interpet our technique as a way of building a model category whose cofibrant objects are exactly the combinatorial manifolds. We then illustrate the usefulness of this point of view by two applications. First we build a category of euclidean precubical sets, i.e. precubical sets that locally look like a grid (of some fixed dimension), and show that it is coreflective in the category of relational precubical sets. This is the combinatorial analog of eulidean locally ordered spaces and the blowup construction from directed topology. Secondly, we show how to give an abstract proof of Kleene's theorem from automata theory by defining "manifold automata" that behave well with respect to concatenation.