Ontology of partial magmoids

The idea of a category is often best understood in terms of partial algebra, and this perspective is available in many places in the literature. The composition operation $\circ$ of a category is a partial binary operation on morphisms. In order to make sense of the idea of a partial binary operation, I have come to consider the horizontal categorification of a partial magma. Towards that end, here is an ontology of the classes of partial magmoids: All of these objects including categories themselves are presheaves in the topos of compositional quivers $CQ$. Investigations of the objects of this topos have lead to considerations of partial magmoids, and their utility as a foundation for partial algebra is a further justification. In this context, a partial magma is simply a partial magmoid with a single object.

A thin partial magmoid is a quiver $Q$ with a specially defined set of composition triples $(a,b)(b,c)(a,c)$ of the underlying relation of its set of morphisms that forms its partial composition domain. Unlike for thin categories or thin semigroupoids, thin partial magmoids don't need to be transitive. Instead, every quiver has a thin partial magmoid with an empty composition domain so a thin partial magmoid can be installed on any quiver.

So these are just two of the most basic types of partial magmoids, but another direction you can go in is to consider magmoids themselves which get you closer to the familiar concepts of total algebra. In that context, for a given magmoid $M$ every endomorphism algebra is a magma. All these different types of partial magmoids are related by the inclusion relationships in our ontology.

References:
Horizontal categorification