Symmetric monoidal categories for operads

The notion of a symmetric monoidal category is used to give a general background for the theory of operads. Standard examples of symmetric monoidal categories include categories of modules over a commutative ring, categories of differential graded modules, various categories of coalgebras, the category of sets together with the cartesian product, the category of simplicial sets, and the category of topological spaces. Other possible examples include the modern categories of spectra used to model stable homotopy, but in applications operads come often from the category of topological spaces or simplicial sets and categories of spectra are not used as the base category in our sense (see next).

The first purpose of this chapter is to survey definitions of symmetric monoidal categories.