Supercoherence

Abstract

The basic technical point of this paper is that a pseudo-simplicial category can be produced from primitive data consisting of face functors and degeneracy functors, natural isomorphisms corresponding to the standard simplicial identities, and a small list of higher order commutativity conditions relating these isomorphisms. A similar machine exists for constructing contravariant pseudo-functors on Segal's category Γ. Thus, a monoidal category M gives rise canonically to a pseudo-simplicial category BM which enjoys many of the properties of a classifying space construction, while a symmetric monoidal category A determines a Γ o category Γ o A which then can be used to directly construct a Γ o -space Γ o ∗ A and a spectrum Spt( A). These constructions generalize the basic classical categorical coherence results, and they lead to several applications in homotopy theory and algebraic K-theory. The applications given here include a generalized Quillen S 1 S-construction, a pseudo-functional version of the group-completion theorem, an explicit construction to the K-theory and L-theory presheaves of spectra, and a presheaf level delooping of the Q = + theorem.