INVOLUTIVE CATEGORIES, COLORED *-OPERADS AND QUANTUM FIELD THEORY

Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field) theory, we develop the involutive analogs of colored operads and their algebras, named colored (*)-operads and (*)-algebras. Central to the definition of colored (*)-operads is the involutive monoidal category of symmetric sequences, which we obtain from a general product-exponential 2-adjunction whose right adjoint forms involutive functor categories. For (*)-algebras over (*)-operads we obtain involutive analogs of the usual change of color and operad adjunctions. As an application, we turn the colored operads for algebraic quantum field theory into colored (*)-operads. The simplest instance is the associative (*)-operad, whose (*)-algebras are unital and associative (*)-algebras.