🤖 AI Summary
This work proposes a systematic approach to constructing a category of combinatorial manifolds satisfying prescribed local properties and applies it to directed topology and automata theory. By forming a coreflective subcategory within the category of relational presheaves and endowing it with a model structure via a unique factorization system, the cofibrant objects are precisely the combinatorial manifolds. This framework introduces manifold-theoretic ideas into automata theory for the first time, yielding a categorical abstraction and proof of Kleene’s theorem. Furthermore, it establishes the category of Euclidean precubical sets as a coreflective subcategory of relational precubical sets and reveals its combinatorial correspondence with the blowup construction in directed topology.
📝 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.