Читать книгу Algebra and Applications 2 - Группа авторов - Страница 36

Any associative algebra A is some degenerate form of operad: indeed, defining by and for n ≠ 1, the collection is obviously an operad. An algebra over is the same as an A-module.

This point of view leads to a more conceptual definition of operads: an operad is nothing but an associative unital algebra in the category of “S-objects”, that is, collections of vector spaces with a right action of Sn on . There is a suitable “tensor product” ⌧ on S-objects, however not symmetric, such that the global composition γ and the unit (defined by u(1) = e) make the following diagrams commute:

These two diagrams commute if and only if e verifies the unit axiom and the partial compositions verify the two associativity axioms and the equivariance axiom (Loday and Vallette 2012).