On Wed, Sep 28, 2011 at 4:34 PM, Emily Riehl <eriehl@math.harvard.edu>wrote:

...

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

Dear Emily Riehl,

The notion you seem to refer to is that which has appeared in the literature as 'precategory' as used in

- Universal Aspects of Probabilistic Automata

L. Schröder and P. Mateus Mathematical Structures in Computer Science, Volume 12 Issue 4, August 2002 - Precategories for Combining Probabilistic Automata P. Mateus, A. Sernadas, C. Sernadas *Electronic Notes in Theoretical Computer Science*, 1999, Pages 169-186 CTCS '99, Conference on Category Theory and Computer Science Enriching over pointed sets you will end up with functors which must preserve the undefined composites, something a little awkward to require. The above treatment does not do that, but the concepts are developed ad hoc from scratch. A more sophisticated notion of 'partial category' is that of *paracategory*, introduced by P. Freyd. Here, the partial composites must satisfy a certain 'saturation' condition. The general context for such a theory is that of *partial algebras*, which admit an abstract notion of *saturation*. In this context, saturation is equivalent to a representation result which embeds any partial algebra in a total one (using the notion of Kleene morphism, which reflect definedness of composites, rather than preserving them) . This theory is given in * - *Paracategories I: Internal Paracategories and Saturated Partial Algebras C. Hermida, P. Mateus *Theoretical Computer Science*<http://www.sciencedirect.com/science/journal/03043975> Volume 309, Issues 1-3, 2 December 2003, Pages 125-156 Such a general theory can then be instantiated in various contexts: categories, multicategories and indexed categories are treated in - Paracategories II: adjunctions, fibrations and examples from probabilistic automata theory C. Hermida, P. Mateus *Theoretical Computer Science*<http://www.sciencedirect.com/science/journal/03043975> Volume 311, Issues 1-3, 23 January 2004, Pages 71-103 which develops the basic 2-category theory of paracategories, inlcuding adjunctions, limits, etc. The most relevant example (for which paracategories were introduced) is that of bivariant functors and dinatural transformations, which constitute a cartesian closed paracategory. Finally, I'd like to mention that paracategories have been used to study certain models of quatum compuation in the thesis of O. Malherbe: Categorical models of computation: Partially traced categories and presheaf models of quantum computation by *Malherbe, Octavio, * Ph.D., *UNIVERSITY OF OTTAWA , 2010, 215 pages; NR73903 * Sincerely, Claudio Hermida [For admin and other information see: http://www.mta.ca/~cat-dist/ ]