Blow-up in manifolds with generalized corners

TL;DR

This paper develops a functorial framework connecting manifolds with generalized corners to complexes of toric monoids, enabling a systematic construction of blow-ups corresponding to refinements of these complexes.

Contribution

It introduces a functor from manifolds with generalized corners to complexes of toric monoids and establishes a universal blow-up construction for each refinement.

Findings

Constructed a functor from manifolds with generalized corners to complexes of toric monoids.

Proved the existence and uniqueness of blow-ups corresponding to refinements of complexes.

Provided a universal property characterizing these blow-ups.

Abstract

We construct a functor from the category of manifolds with generalized corners to the category of complexes of toric monoids, and for every `refinement' of the complex associated to a manifold, we show there is a unique `blow-up', i.e., a new manifold mapping to the original one, which satisfies a universal property and whose complex realizes the refinement. This was inspired in part by the work of Gillam and Molcho, though we work with manifolds with generalized corners, as developed by Joyce, which have embedded boundary faces, for which the appropriate objects (i.e., complexes of monoids) are simpler than they would be otherwise (i.e., monoidal spaces in the sense of Kato).