For a $C^{*}$-category with strict $G$-action we construct examples of equivariant coarse homology theories. To this end first introduce versions Roe categories objects in $C^{*}$-categories which are controlled over bornological spaces, and then apply homological functor. These theories employed to verify that certain functors on the orbit category CP-functors. This fact has consequences for i...

We construct a 2-equivalence $\mathfrak{CohTheory}^\text{op} \simeq \mathfrak{TypeSpaceFunc}$. Here $\mathfrak{CohTheory}$ is the 2-category of positive theories and $\mathfrak{TypeSpaceFunc}$ type space functors. give precise definition interpretations for logic, which will be 1-cells in $\mathfrak{CohTheory}$. The 2-cells are definable homomorphisms. restricts to duality categories, making ph...