In this two-part paper, we undertake a systematic study of abstract partial map categories in which every map has both a restriction (domain of definition) and a range (image). In this first part, we explore connections with related structures such as inverse categories and allegories, and establish two representational results. The first of these explains how every range category can be fully and faithfully embedded into a category of partial maps equipped with a suitable factorization system. The second is a generalization of a result from semigroup theory by Boris Schein, and says that every small range category satisfying the additional condition that every map is an epimorphism onto its range can be faithfully embedded into the category of sets and partial functions with the usual notion of range. Finally, we give an explicit construction of the free range category on a partial map category in terms of certain types of labeled trees.
Keywords: Categories of partial maps, restriction category, factorization systems
2010 MSC: 18A15, 18A32, 18D20
Theory and Applications of Categories, Vol. 26, 2012, No. 17, pp 412-452.