It has been known for some time that the cohomology theories of many classical algebraic objects – monoids, groups, associative algebras and Lie algebras for instance – have a common framework in terms of cohomology of internal monoids in a symmetric monoidal category [see, e.g., V. A. Pachuashvili, J. Pure Appl. Algebra 72, No. 2, 109-147 (1991); translation from Tr. Tbilis. Mat. Inst. Razmadze 77, 86-106 (1985; Zbl 0615.55015)]. But there are also important examples of algebraic structures which occur as monoids in non-symmetric monoidal categories, such as operads, monads, theories, categories, and square rings as described below. In this article we show that these structures are still susceptible to cohomological investigation, by developing the theory in the absence of the symmetry condition. Later we shall assume that the monoidal structure is left distributive over coproducts and the category is an abelian category; this is the case for operads, our original motivating example.
For the entire collection see [Zbl 0855.00018].