Paracategories I: internal parategories and saturated partial algebras

Details Paracategories I: internal parategories and saturated partial algebras Paracategories I: internal parategories and saturated partial algebras

2003-03-06

arxiv.org/abs/math/0303083v1

Mathematics

Category Theory

Scientific paper

Based on the monoid classifier, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras. We introduce for the latter the crucial notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via epimorphisms and (monic) Kleene morphisms and relate the latter to saturation.

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