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.