Holmstrom Monoidal category

Want to define rings and the like in the 2-category of cats and functors, and also in the 2-category of cats and adjunction. The algebraic notions of ring, commutative ring, modules over a ring, algebras over a ring, and central commutative algebras over a commutative ring, correspond to: monoidal category, symmetric monoidal category, modules over a monoidal category, algebras over a monoidal category, and central symmetric algebra over a symmetric monoidal category. If we consider adjunctions instead of functors, we add the word “closed” to each of these terms.

Def: A monoidal structure on a category $C$ is a “tensor product” bifunctor $C \times C \to C$, a unit object $S \in C$, a natural associativity isomorphism, and two natural (left and right) unit IMs, satisfying three coherence diagrams. A category with such a structure is called a monoidal category. Def of monoidal functor (essentially has to preserve unit and product up to a natural isomorphism), monoidal natural transformation, and of the 2-category of monoidal cats.

Examples: Sets, with Cartesian product. Topological spaces, with product. Simplicial sets, with product. Modules over a commutative ring, with tensor product. (All these are actually symmetric monoidal.)

Examples of monoidal functors: The free R-module functor from Sets to R-mods. The geometric realization functor.

Whenever $C$ is monoidal, this is also the case for $Map \ C$. (See Hovey p. 109)

Selinger: survey of graphical language

nLab page on Monoidal category