The theory of quantaloids.

*(English)*Zbl 0845.18003
Pitman Research Notes in Mathematics Series. 348. Harlow: Addison Wesley Longman. 147 p. (1996).

Quantales are complete lattices endowed with a sup-preserving associative binary relation, and the author’s former monograph [Quantales and their applications, Pitman Res. Notes Math. Ser. 234 (1990; Zbl 0703.06007)] serves as an indispensable handbook to active researchers related with the field and as a good introduction to novices in the field. Quantaloids, being a natural generalization of quantales, are locally small categories whose hom-sets are complete lattices with composition preserving sups in both variables, and the author’s present monograph, consisting of five chapters, gives an up-to-date perspective on their art.

After giving a definition of quantaloid with examples in chapter 1, the author discusses several methods of producing new quantaloids from old ones in chapter 2. Chapter 3 is devoted to free quantaloids \({\mathcal P} ({\mathcal A})\) on locally small categories \({\mathcal A}\). Chapter 4 deals with automata and tree automata from a standpoint of enriched category theory. It is stressed that the passage from automata to tree automata is essentially the passage from a one object base quantaloid to a more general one. The last chapter discusses the general theory of modules and bimodules over quantaloids as well as its relation to the theory of *-autonomous categories.

After giving a definition of quantaloid with examples in chapter 1, the author discusses several methods of producing new quantaloids from old ones in chapter 2. Chapter 3 is devoted to free quantaloids \({\mathcal P} ({\mathcal A})\) on locally small categories \({\mathcal A}\). Chapter 4 deals with automata and tree automata from a standpoint of enriched category theory. It is stressed that the passage from automata to tree automata is essentially the passage from a one object base quantaloid to a more general one. The last chapter discusses the general theory of modules and bimodules over quantaloids as well as its relation to the theory of *-autonomous categories.

Reviewer: Hirokazu Nishimura (Tsukuba)

##### MSC:

18B35 | Preorders, orders, domains and lattices (viewed as categories) |

06B23 | Complete lattices, completions |

18D20 | Enriched categories (over closed or monoidal categories) |

03G30 | Categorical logic, topoi |

68Q70 | Algebraic theory of languages and automata |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |