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Involutive categories, colored *-operads and quantum field theory

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Marco Benini, Alexander Schenkel and Lukas Woike

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.

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

2010 MSC:
18Dxx, 81Txx

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 2, pp 13-57.

Published 2019-02-11.

http://www.tac.mta.ca/tac/volumes/34/2/34-02.pdf

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