On quantaloids and quantal categories
type: Item
instance of: scholarly article
title: On quantaloids and quantal categories
zbMATH Open document ID: 0866.18005
DOI: 10.1007/BF02350358
published in: Lithuanian Mathematical Journal
publication date: 1997-07-17T00:00:00Z
review text: In ``Quantal sets and sheaves over quantales'' [Lith. Math. J. 34, No. 1, 8-29 (1994) and Liet. Mat. Rink. 34, No. 1, 9-31 (1994; Zbl 0837.18003)], the author presented an approach to quantal sets and sheaves on quantales somewhat more general than those that had been considered earlier. The restriction of idempotence on the quantale was replaced by rather technical, but less restrictive axioms. This approach to sheaves on quantales was a modification of some ideas of Höhle on \(M\)-valued sets and integral commutative \(cl\)-monoids. The article under review extends this earlier work of the author on quantales to the setting of quantaloids. Quantaloids are a natural and useful categorical generalization of quantales (which are the one object quantaloids). The recent book by the reviewer [``The theory of quantaloids'', Pitman Res. Notes Math. 348 (1996; Zbl 0845.18003)] contains a detailed look at quantaloids and some of their applications. The author looks at a certain class of quantaloids, (not necessarily unital) and proceeds to define a quantal category for a quantaloid \(Q\) as a set \(X\) ``enriched'' in \(Q\) with an existence predicate. This notion differs from that of \(Q\)-enriched category considered by the reviewer based on the standard notion of category enriched in a base bicategory. The author does provide a useful comparison between the two concepts. He then proceeds to define a category of quantal categories on which is defined a monad assigning to every \(Q\)-category its ``singletons''. The algebras for this monad then lead to the notion of sheaf on \(Q\). No concrete examples are considered and the next step would be to apply these ideas in a particular setting.
Mathematics Subject Classification ID: 18B35
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 939528
zbMATH Keywords: quantaloids
zbMATH Keywords: categorical generalization of quantales
zbMATH Keywords: quantal category
zbMATH Keywords: monad
zbMATH Keywords: sheaf
author: Remigijus Petras Gylys
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
cites work: A theory of quantal sets
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cites work: A categorical look at tree automata and context-free languages
cites work: Quantales and (noncommutative) linear logic
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description: scientific article; zbMATH DE number 939528