Monotone-light factorizations in coarse geometry

Abstract

We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar way to classically monotone maps, and prove that coarsely monotone maps are stable under those pullbacks which exist in the coarse category. For the case of maps between proper metric spaces, we exhibit some connections between the coarse and classical notions of monotone and light using the Higson corona. Finally, we look at some coarse properties which are preserved by coarsely light maps such as finite asymptotic dimension and exactness, and make some remarks on the situation for groups and group homomorphisms.

Coarse quotients by group actions and the maximal Roe algebra

“…Note that by Proposition 3.8 in this paper and Corollary 9.4 in [7], X has Property A if X G does, for any countable group G. The following corollary was already proved in [6].…”

“…We recall the definition of coarsely light map from [7]. For A a subset of a large scale space X and U a cover of X, let A U be the set of equivalence classes of A under the equivalence relation:…”

“…is uniformly bounded in X. Equivalently, f is coarsely light if for every uniformly bounded family V in Y , the family of subsets f −1 (V) has asymptotic dimension zero uniformly (see [7] for details). Proposition 3.8.…”

Higson compactification and dimension raising

“…They also contain the class of quotient maps Z → G\Z by the finite group action and as a corollary we deduce the main result of [20]. For another application of the Higson corona in coarse geometry see [16].…”