Abstract

In this chapter, we develop the involutive versions of (symmetric) monoidal categories, (commutative) monoids, monads, and algebras over a monad. In an involutive symmetric monoidal category, in addition to involutive commutative monoids, there is also a notion of a reversing involutive monoid, which incorporates the symmetry isomorphism in a way that is different from a commutative monoid. Reversing involutive monoids are important because they include complex unital ∗-algebras and their differential graded analogues as examples.

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