  The Grothendieck construction is a fundamental link between indexed
categories and opfibrations. This work is a detailed study of the Grothendieck
construction over a small tight bipermutative category in the context of
Cat-enriched multicategories, with applications to inverse K-theory and pseudo
symmetric E-infinity-algebras.
  The ordinary Grothendieck construction over a small category C is a
2-equivalence that sends a C-indexed category to an opfibration over C. We show
that the Grothendieck construction over a small tight bipermutative category D
is a pseudo symmetric Cat-multifunctor that is generally not a Cat-multifunctor
in the symmetric sense. When the projection to D is taken into account, we
prove that the Grothendieck construction over D lifts to a non-symmetric
Cat-multiequivalence whose codomain is a non-symmetric Cat-multicategory with
small permutative opfibrations over D as objects.
  As applications we show that inverse K-theory, from Gamma-categories to small
permutative categories, is a pseudo symmetric Cat-multifunctor but not a
Cat-multifunctor in the symmetric sense. It follows that inverse K-theory
preserves algebraic structures parametrized by non-symmetric and pseudo
symmetric Cat-multifunctors but not Cat-multifunctors in general. As a special
case, we observe that inverse K-theory sends pseudo symmetric
E-infinity-algebras in Gamma-categories to pseudo symmetric E-infinity-algebras
in small permutative categories.
