Meromorphic connections in filtered $A_{\mathrm{\infty }}$ categories

In this note, introducing notions of CH module, CH morphism and CH connection, we define a meromorphic connection in the “$z$-direction” on periodic cyclic homology of an $A_{\mathrm{\infty }}$ category as a connection on cohomology of a CH module. Moreover, we study and clarify compatibility of our meromorphic connections under a CH module morphism preserving CH connections at chain level. Our motivation comes from symplectic geometry. The formulation given in this note designs to fit algebraic properties of open-closed maps in symplectic geometry.