Cartesian Linear Bicategories

Abstract

In this thesis, we extend the theory of cartesian bicategories [14, 13] to linear bicategories [15] and introduce the concept of cartesian linear bicategories for locally ordered linear bicategories. We demonstrate that the linear bicategory 𝗥𝗲𝗹 of sets and relations, along with two other examples, fits within this framework. In our initial structure, called Cyclic cartesian linear bicategories, we believed that by taking the original definition of cartesian bicategories, adding a corresponding cartesian structure for the second horizontal composition, and replacing the adjunctions in bicategories with cyclic linear adjoints, we would achieve a proper cartesian structure on a locally ordered linear bicategory (𝓑,⊗,⊤,⊕,⊥). This approach was expected to make the tensor product of the cyclic cartesian structure a linear bicategorical product when restricted to the linear sub-bicategory of cyclic linear adjoints. However, we were surprised to discover that, in our main example, although the linear bicategory 𝗥𝗲𝗹 is cyclic cartesian, the linear bicategorical product of the linear sub-bicategory 𝗖𝗠𝗮𝗽(𝗥𝗲𝗹) does not coincide with the monoidal product.

Consequently, we refined our approach to accommodate the dual structures of tensor and par, which are linked in linear settings. By extending the theory of locally ordered cartesian linear bicategories, we introduce a characterization theorem for these structures, which ultimately leads us to a more general definition of cartesian linear bicategories that can be applied beyond the locally ordered case. Additionally, we explore the linear bicategory 𝗠𝗮𝘁(𝕏), where 𝕏 is a ★-autonomous linearly distributive category with linear products and coproducts [18], as an example of cartesian linear bicategories in the non-locally ordered case.

After studying the theory of cartesian linear bicategories, we introduce knowledge representation in linear bicategories of relations, inspired by Patterson’s work in [46]. This concept bridges categorical frameworks and logical systems, providing some applications of our work in databases and machine learning.