Achiral and chiral involutive categories
Robin Cockett
University of Calgary
Abstract:
A (contravariant) involutive category is a category equipped with an
equivalence to its dual: it is achiral in case the two adjoints are
exactly the same and, furthermore, the counit and unit are the same.
It is a chiral involutive category when the two adjoints are distinct.
From such an achiral category one can construct a "unitary" involutive
category. These can be alternately viewed as involutive categories
with an inner product and (using the inner product structure) they can
be shown to be canonically equivalent to dagger categories.
We shall discuss the question of what the appropriate functors are
between general chiral involutive categories. Following Paul-Andre
Mellies, we shall argue that his symmetry breaking chiral functors
fulfill this role. Given time, we shall discuss how these functors
give rise to dagger linearly distributive categories ($\dagger$-LDC).
These were used by Priyaa Srinivasan in her PhD. to support an
interpretation of infinite dimensional quantum mechanics. The talk,
therefore, provides an alternative mathematical provenance to explain
mathematically how $\dagger$-LDCs arise.
This is joint work with Priyaa and Durgesh.