Re: partial categories

[selinger@mathstat.dal.ca (Peter Selinger)]

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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