A categorical study on the generalized type semigroup

TL;DR

This paper proves that the generalized type semigroup belongs to a specific category by verifying certain axioms, especially highlighting its properties in zero-dimensional cases and its analogy to Cuntz semigroups.

Contribution

It demonstrates that the generalized type semigroup satisfies key axioms of category W and establishes its properties in zero-dimensional spaces, linking it to Cuntz semigroups.

Findings

The semigroup satisfies axioms (W1)-(W4) and (W6).

In zero-dimensional spaces, it also satisfies (W5).

Supports analogy with the Cuntz semigroup for C*-algebras.

Abstract

In this short note, we show that the generalized type semigroup $\CW(X, \Gamma)$ introduced by the author in \cite{M3} belongs to the category \textnormal{W}. In particular, we demonstrate that $\CW(X, \Gamma)$ satisfies axioms (W1)-(W4) and (W6). When $X$ is zero-dimensional, we also establish (W5) for the semigroup. This supports the analogy between the generalized type semigroup and pre-completed Cuntz semigroup $W(\cdot)$ for $C^*$-algebras.