# Involutive categories, colored $\ast$-operads and quantum field theory

February 26, 2018

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 $\ast$-operads and $\ast$-algebras. Central
to the definition of colored $\ast$-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. Using a
novel criterion for trivializability of involutive structures, we show that the
involutive monoidal category of symmetric sequences admits a very simple
description. For $\ast$-algebras over $\ast$-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
$\ast$-operads. The simplest instance is the associative $\ast$-operad, whose
$\ast$-algebras are unital and associative $\ast$-algebras.

Keywords:

involutive categories, involutive monoidal categories, *-monoids, colored operads, *-algebras, algebraic quantum field theory