Specified mathematical mirror datum

A specified mathematical mirror datum is a typed record. It includes (i) declared categories ${\mathcal{C}}_A,{\mathcal{C}}_B$ in which the two family arrows $p_A:X_A\to B_A$ and $p_B:X_B\to B_B$ live, together with the chosen complex, symplectic, or other structures on their objects and arrows and the axioms those structures satisfy; (ii) a declared category $\mathcal{U}$ containing the selected parameter objects $U_A,U_B$, specified inclusions $j_A:U_A\hookrightarrow B_A$ and $j_B:U_B\hookrightarrow B_B$ in the respective base categories, and a specified morphism $\phi :U_A\to U_B$ in $\mathcal{U}$; (iii) coefficient rings $R_A,R_B$, specified completions or filtrations on them, a comparison ring $R$, and ring homomorphisms $R_A\to R$ and $R_B\to R$ (or a specified ring isomorphism when the rings are identified); (iv) for every category whose objects are designated as branes in the instance, its objects and morphisms, grading, and local-system conventions; and (v) either a functor, specified by its object and morphism assignments, between the declared categories or invariant functors from them to a common declared target category, together with the objects and comparison morphism to be compared. Every symbol in the record therefore has an ambient type, and every additional structure is accompanied by its defining axioms. The selected categories, bases, and structures may vary by instance; the tuple does not select them implicitly.

Here a functor $F:\mathcal{C}\to \mathcal{D}$ consists of an assignment $x\mapsto F(x)$ on objects and maps $F_{x,y}:{\mathrm{Hom}}_{\mathcal{C}}(x,y)\to {\mathrm{Hom}}_{\mathcal{D}}(F(x),F(y))$ satisfying $F({\mathrm{id}}_x)={\mathrm{id}}_{F(x)}$ and $F(g\circ f)=F(g)\circ F(f)$ for every composable pair $f,g$. These requirements apply to each functor in (v), including invariant functors.

The datum makes a later comparison proposition well-formed; it does not assert that $\phi$ is an isomorphism or that the functor or invariant is an equivalence. An A-side is required to be symplectic only when that is part of the selected datum. Compact Calabi–Yau hypotheses are instance-specific and are not built into this generic definition. If a symplectic form is selected, use the convention in Symplectic form and symplectic manifold.

Symplectic form and symplectic manifold