An introduction to quantaloid-enriched categories

Abstract

This survey paper, specifically targeted at a readership of fuzzy logicians and fuzzy set theorists, aims to provide a gentle introduction to the basic notions of quantaloid-enriched category theory. We discuss at length the definitions of quantaloid, quantaloid-enriched category, distributor and functor, always giving several examples that – hopefully – appeal to the intended readership. To indicate the strength of this general theory, we explain in considerable detail how (co)limits are dealt with, and particularly how the Yoneda embedding of a quantaloid-enriched category in its free (co)completion comes to be. Our insistence on quantaloid-enrichment (rather than quantale-enrichment) is duly explained by examples requiring a notion of “partial elements” (sheaves, partial metric spaces). A final section glosses over some further topics, providing ample references for the interested reader.