# Positive enumerable functors

We investigate different effectivizations of functors using Turing operators and enumeration operators and compare these notions. The most promising of these notions are positive enumerable functors. A positive enumerable functors from $$\mathcal A$$ to $$\mathcal B$$ consists of two enumeration operators $$\Psi$$ and $$\Psi_*$$ such that when given an enumeration of the positive diagram $$P(\hat{\mathcal A})$$ of a structure isomorphic to $$\hat{\mathcal A}\cong \mathcal A$$, $$\Phi$$ enumerates the positive diagram of $$F(\hat{\mathcal A})$$. The operator $$\Phi_*$$ will take care of the isomorphisms by enumerating the graph of $$F(f:\hat{\mathcal A}\to\tilde{\mathcal A})$$ given $$P(\hat{\mathcal A})\oplus Graph(f)\oplus P(\tilde{\mathcal A})$$.
We show that if $$\mathcal A$$ is positive enumerable bi-transformable to $$\mathcal B$$ (if there are positve enumerable functors $$F:\mathcal A\to\mathcal B$$ and $$G:\mathcal B\to \mathcal A$$ such that $$F$$ and $$G$$ are “enumerable pseudo-inverses”) then $$\mathcal A$$ and $$\mathcal B$$ have the same enumeration degree spectrum and that positive enumerable functors are independent from the already established enumerable and computable functors.