Projective Naturality in Heegaard Floer Homology

Let ${\text{Man}}_{\ast }$ denote the category of closed, connected, oriented and based $3$-manifolds, with basepoint preserving diffeomorphisms between them. Juh'asz, Thurston and Zemke showed that the Heegaard Floer invariants are natural with respect to diffeomorphisms, in the sense that there are functors $H{F}^{\circ }:{\text{Man}}_{\ast }\to {\mathbb{F}}_2[U]\text{-Mod}$ whose values agree with the invariants defined by Ozsv'ath and Szab'o. The invariant associated to a based $3$-manifold comes from a transitive system in ${\mathbb{F}}_2[U]\text{-Mod}$ associated to a graph of embedded Heegaard diagrams representing the $3$-manifold. We show that the Heegaard Floer invariants yield functors $HF^{\circ}: $\text{Man}$_{*} \rightarrow $\text{Trans}$(P($\mathbb{Z}$[U]$\text{-}$$\text{Mod}$))$ to the category of transitive systems in a projectivized category of $\mathbb{Z}[U]$-modules. In doing so, we will see that the transitive system of modules associated to a $3$-manifold actually comes from an underlying transitive system in the projectivized homotopy category of chain complexes over $\mathbb{Z}[U]\text{-Mod}$. We discuss an application to involutive Heegaard Floer homology, and potential generalizations of our results.