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Range categories I: General theory

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J.R.B. Cockett, Xiuzhan Guo and Pieter Hofstra

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.

Published 2012-09-09.

http://www.tac.mta.ca/tac/volumes/26/17/26-17.dvi

http://www.tac.mta.ca/tac/volumes/26/17/26-17.ps

http://www.tac.mta.ca/tac/volumes/26/17/26-17.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/17/26-17.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/17/26-17.ps

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