A colleague of mine is wondering if anyone has studied "partial categories," by which she means directed graphs with identities but with only some compositions (including all identity compositions) defined. A partial category can be thought of as a category enriched in pointed sets (with smash product as tensor and S^0 as unit). The slogan is that the basepoint in each hom-set stands in for "does not exist". But enriched functors don't give the right notion of maps; these should preserve identities and all specified compositions. Enriched functors behave appropriately with regards to the identites but may "forget" extant arrows and in particular need not preserve composites. So perhaps this perspective is not useful. I'll happily pass along any suggestions. Thanks, Emily Riehl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]