# Abstract bivariant Cuntz semigroups

## TL;DR

This paper develops a new categorical framework for Cuntz semigroups, introducing a bivariant theory that generalizes morphisms and connects to C$^*$-algebra invariants, with computable cases and applications to order-zero maps.

## Contribution

It introduces a closed symmetric monoidal category structure on abstract Cuntz semigroups and defines a bivariant semigroup $[[S,T]]$ for morphisms, extending the theory.

## Key findings

- The category of abstract Cuntz semigroups is closed and symmetric monoidal.
- The bivariant semigroup $[[S,T]]$ generalizes morphisms between Cuntz semigroups.
- Order-zero maps induce elements in the bivariant Cuntz semigroup.

## Abstract

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[[S,T]]$ playing the role of morphisms from $S$ to $T$. Applied to C$^*$-algebras $A$ and $B$, the semigroup $[[\mathrm{Cu}(A),\mathrm{Cu}(B)]]$ should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between C$^*$-algebras naturally define elements in the respective bivariant Cuntz semigroup.