# Homotopy Type Theory H-magmoid > history (Rev #1)

## Definition

An H-magmoid $A$ consists of the following.

• A type $A_0$, whose elements are called objects. Typically $A$ is coerced to $A_0$ in order to write $x:A$ for $x:A_0$.

• For each $a,b:A$, a type $hom_A(a,b)$, whose elements are called arrows or morphisms.

• For each $a,b,c:A$, a function

$hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)$

called composition, and denoted infix by $g \mapsto f \mapsto g \circ f$, or sometimes $gf$.