ABSTRACT

Cohomology operations are absolutely essential in making cohomology an effective tool for studying spaces. In ordinary algebra, commutativity is an extremely useful property possessed by certain monoids and algebras. A multicategory encodes the structure of a category where functions have multiple input objects. They serve as a useful way to encode many multilinear structures in stable homotopy theory: multiplications, module structures, graded rings, and coherent structures on categories. The chapter considers compatibility relations between operations on different homotopy degrees. Every Adem relation between Dyer–Lashof operations produces a secondary Dyer–Lashof operations. Secondary operations are part of the homotopy theory of C, and there is typically no method to determine secondary operations purely in terms of the homotopy category. Topological Andre-Quillen homology and cohomology are invariants of ring spectra developed by I. Kriz and Maria Basterra.