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The Grothendieck construction is a fundamental link between indexedcategories and opfibrations. This work is a detailed study of the Grothendieckconstruction over a small tight bipermutative category in the context ofCat-enriched multicategories, with applications to inverse K-theory and pseudosymmetric E-infinity-algebras.  The ordinary Grothendieck construction over a small category C is a2-equivalence that sends a C-indexed category to an opfibration over C. We showthat the Grothendieck construction over a small tight bipermutative category Dis a pseudo symmetric Cat-multifunctor that is generally not a Cat-multifunctorin the symmetric sense. When the projection to D is taken into account, weprove that the Grothendieck construction over D lifts to a non-symmetricCat-multiequivalence whose codomain is a non-symmetric Cat-multicategory withsmall permutative opfibrations over D as objects.  As applications we show that inverse K-theory, from Gamma-categories to smallpermutative categories, is a pseudo symmetric Cat-multifunctor but not aCat-multifunctor in the symmetric sense. It follows that inverse K-theorypreserves algebraic structures parametrized by non-symmetric and pseudosymmetric Cat-multifunctors but not Cat-multifunctors in general. As a specialcase, we observe that inverse K-theory sends pseudo symmetricE-infinity-algebras in Gamma-categories to pseudo symmetric E-infinity-algebrasin small permutative categories.

The Grothendieck Construction of Bipermutative-Indexed Categories and Pseudo Symmetric Inverse K-Theory