Compact Closed Bicategories

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.

Fixed typo, published to TAC

References in corpus (7)

• Constructing symmetric monoidal bicategories
• Infinite loop spaces, and coherence for symmetric monoidal bicategories
• Quasistrict symmetric monoidal 2-categories via wire diagrams
• Spans in 2-Categories: A monoidal tricategory
• The Categorified Heisenberg Algebra I: A Combinatorial Representation
• Linear Logic without Units
• Exotic Statistics for Strings in 4d BF Theory

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