Constructing the big relative Fukaya category, and its open--closed maps

This paper continues previous work of the author with Perutz, in which the `small' version of Seidel's relative Fukaya category of a smooth complex projective variety relative to a normal crossings divisor was constructed, under a semipositivity assumption. In the present work, we generalize this to construct the `big' relative Fukaya category in the same setting, as well as its closed--open and open--closed maps, and prove Abouzaid's split-generation criterion in this context. We also establish a general framework for constructing chain-level Floer-theoretic operations in this context, and dealing efficiently with signs and bounding cochains, which will be used in follow-up work with Ganatra to construct the cyclic open--closed map and establish its properties.