Bicat is the tricategory of bicategories, homomorphisms, pseudonatural transformations, and modifications. Gray is the subtricategory of 2-categories, 2functors, pseudonatural transformations, and modifications. We show that these two tricategories are not triequivalent. 1. Background. Weakening the notion of 2-category by replacing all equations between 1-cells by suitably coherent isomorphism...

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras” or “Kleisli monoids” relative to a “monad” on a bicategory. However, the meanings of these wor...

Nondeterministic concurrent strategies—those strategies compatible with copy-cat behaving as identity w.r.t. composition—have been characterised as certain maps of event structures. This leads to a bicategory of general concurrent games in which the maps are nondeterministic concurrent strategies. This paper explores the consequences of extending concurrent games with (1) winning, losing and, i...

Abstract We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied Ahrens, Kapulkin, and Shulman, we define study bicategories. To construct examples bicategories a modular fashion, displayed , an analog 1-categories introduced Ahrens Lumsdaine. demonstrate applicability this prove that several interest are univalent. Among these catego...

Abstract We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence bicategories with finite bilimits to show that every bicategory bicategorical structure is biequivalent 2-category 2-categorical structure. Then how extend this result Mac Lane-style “all pasting diagrams commute” theorem: precisely, in the free on graph, there at most one 2...

We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to be equivalent to the usual associativity, unit and braiding axioms. We also discuss the next dimensional version, that is, b-structures on bicategories. As an...