Thank you. This seems very helpful. I've already passed your message along. Emily On Wed, 28 Sep 2011, Peter Selinger wrote:
Dear Emily,
one notion of "partial category" that has been studied is Freyd's notion of "paracategory". It differs from what you wrote below as follows: instead of taking compositions of two arrows as the primitive operation, one takes compositions of n arrows as primitive operations, for all n. So if f1, ..., fn are n arrows (so that the codomain of each is the domain of the next), then [f1, ..., fn] is their composition, which may be defined or undefined.
In the case of (total) categories, adding n-ary compositions makes no difference, since they are already definable in terms of identities and binary composition. But in the partial case, it does make a difference, as it is possible, for example, that [f,g,h] is defined, but [f,g] and [g,h] are undefined.
The axioms are:
(a) [] : A -> A is defined (the composition of the empty path from A to A) (b) [f] is defined and equal to f, for all arrows f, (c) if ff, gg, hh are are (composable) paths, and if [gg] is defined, then [ff,[gg],hh] is defined if and only if [ff,gg,hh] is defined, and in this case, they are both equal.
The main representation theorem is:
Every reflexive subgraph of a category is a paracategory; conversely, every paracategory can be faithfully completed to a category.
It is not the only possible notion of partial category, and may not always be the notion that one wants.
-- Peter
Emily Riehl 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
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