Paracategories and Saturated Partial Algebras
Freyd´s notion of paracategory embodies a system of morphisms subject to partial compositions. We give an abstract axiomatisation of this notion internally in a regular category admitting free monoids. This
leads us to consider the more general notion of partial algebras relative to a monad. We introduce for these the crucial notion of
saturation (which is characteristic of paracategories) in order to characterise their representability. We explore also the
2-dimensional aspects of the theory of paracategories, most notably the notion of adjunction, in order to capture Freyd´s proposed example of the cartesian closed paracategory of dinatural transformations.
Paracategories I: Internal Paracategories and Saturated Partial Algebras by C. Hermida and P. Mateus (Theoretical Computer Science, 309, 125-156 2003).
Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory by C. Hermida and P. Mateus (Theoretical Computer Science, 311, 71-103 2004).
Areas of interest
Claudio Hermida (Instituto Superior Técnico, Lisboa)
November 12, 2004