Semigroupoid

Let $X$ be a set. Define $\forall a, b \in X, \text{Mor}(a, b)$ the set of morphisms from $a$ to $b$. Additionally, if $f \in \text{Mor}(a, b)$ then we write, $f : a \to b$.

A set $X$ with $\text{Mor}$ and a binary operator $compose : \text{Mor}(a, b) \times \text{Mor}(b, c) \to \text{Mor}(a, c)$ is a semigroupoid if $\forall f \in \text{Mor}(a, b), g \in \text{Mor}(b, c), \exists h \in \text{Mor}(a, c) : g \circ f = h.$

Remarks

This generalizes the notion of a 4b4b and a 4a1a. Additionally, this is also called a naked category.

• A semigroupoid is also a partial magma with associativity.