General representation theory in relatively closed monoidal categories

TL;DR

This paper generalizes the concept of closed monoidal categories using relative adjoint functors, introduces representations and their tensor products, and explores their behavior under lax monoidal functors, with applications to classical and quantum cases.

Contribution

It extends the theory of closed monoidal categories via relative adjoint functors and develops a framework for tensor products of representations in this setting.

Findings

Defined representations in generalized categories.

Introduced tensor product of bimonoid representations.

Analyzed the action of lax monoidal functors on these tensor products.

Abstract

We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax monoidal functors is investigated. We introduce tensor product of representations of bimonoids as a functorial binary operation and show how symmetric lax monoidal functors act on this product. Finally we apply the general theory to classical and quantum representations.