Further 2-Dimensional Categorical Structures

In this chapter, further 2-dimensional categorical structures are presented and discussed. These include monoidal bicategories, as one-object tricategories, along with braided monoidal bicategories, sylleptic monoidal bicategories, and symmetric monoidal bicategories. The rest of the chapter discusses the Gray tensor product on 2-categories, Gray monoids, double categories, and monoidal double categories.

2-Dimensional Categories

2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

Minimal realization in bicategories of automata

The context of this article is the programme to develop monoidal bicategories with a feedback operation as an algebra of processes, with applications to concurrency theory. The objective here is to study reachability, minimization and minimal realization in these bicategories. In this setting the automata are 1-cells, in contrast with previous studies where they appeared as objects. As a consequence, we are able to study the relation of minimization and minimal realization to serial composition of automata using (co)lax (co)monads. We are led to define suitable behaviour categories and prove minimal realization theorems that extend classical results.