Compact Closed Bicategories

TL;DR

This paper introduces the concept of compact closed bicategories, provides examples, and proves their properties, including a new direct definition and applications to spans and resistor networks.

Contribution

It offers a direct definition of compact closed bicategories, extends previous indirect definitions, and demonstrates their presence in spans and resistor network bicategories.

Findings

Bicategory of spans of sets is compact closed.

Certain bicategories of resistor networks are compact closed.

A new direct definition of compact closed bicategories is provided.

Abstract

A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual "zig-zag" identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day and Street defined compact closed bicategories indirectly via Gray monoids and then appealed to a coherence theorem to extend the concept to bicategories; we restate the definition directly. We prove that given a 2-category T with finite products and weak pullbacks, the bicategory of objects of C, spans, and isomorphism classes of maps of spans is compact closed. As corollaries, the bicategory of spans of sets and certain bicategories of "resistor networks" are compact closed.