Crthuky

Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $fin Hom(A,B)$ and $g in Hom(B,C)$ then $gcirc f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. 'Category-like' structures derived from directed graphs with at most one edge in each direction, where the vertices are the objects and the edges are the morphisms, provide plentiful examples, as do (equivalently) not-necessarily-transitive relations on a set $X$. Could anyone provide references which discuss this from a categorical perspective? Thanks in advance!

As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equivalent to the category of pointed sets with its monoidal smash product, where the basepoint in a pointed set is a formal way to represent "not defined", it is equivalent to talk about categories enriched over the latter.

A different notion of "category with partially defined composition" is called a paracategory. This has $n$-ary partial composition functions for all $n$, which are associative insofar as defined in an "unbiased" way. It was apparently defined by Peter Freyd in unpublished work, and studied further by Hermida and Mateus; see the references at the link.

Jørgen Ellegaard Andersen calls this a "categroid". I'm not particularly fond of that term.