This monograph is a detailed study of the Grothendieck construction over a bipermutative category in the context of categorically enriched multicategories, with new and important applications to inverse -theory and pseudo symmetric -algebras. After carefully recalling preliminaries in enriched categories, bipermutative categories, and enriched multicategories, we show that the Grothendieck construction over a small tight bipermutative category is a pseudo symmetric Cat-multifunctor and generally not a Cat-multifunctor in the symmetric sense. Pseudo symmetry of Cat-multifunctors is a new concept we introduce in this work. The following features make it accessible as a graduate text or reference for experts: This book is suitable for graduate students and researchers with an interest in category theory, algebraic -theory, homotopy theory, and related fields. The presentation is thorough and self-contained, with complete details and background material for non-expert readers.