Picard Groupoids and Spectra

The goal of this lecture is to explain in detail the statement that a Picard groupoid is “the same thing as” an Ω-spectrum E with πi(E) = 0 for i 6= 0, 1. Along the way we introduce the notion of a (very special) Γ-space, which provides one of the possible ways of formalising the concept of an “abelian group structure defined up to all the higher homotopies”, and we present an approach to K-theory based on the notion of a permutative category.