ABSTRACT
Part II is about the Grothendieck construction over a small tight bipermutative category. This chapter first discusses in detail the Cat-multicategory D-Cat of D-indexed categories for a small permutative category D. The rest of this chapter discusses a different Cat-multicategory DCat, which serves as the domain of the Grothendieck construction, for a small tight bipermutative category D. The objects of DCat are symmetric monoidal functors from D to Cat, instead of just D-indexed categories. Moreover, the multimorphism categories of DCat involve the full extent of the bipermutative structure of D and Laplaza's Coherence Theorem. Furthermore, while the Cat-multicategory structure on D-Cat is induced by its symmetric monoidal closed structure, the Cat-multicategory structure on DCat is not induced by a monoidal structure in general.
