# Homotopy Type Theory H-spaceoid > history (Rev #2)

## Idea

The oidification of an H-space

## Definition

An H-spaceoid $A$ 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 $a:A$, a morphism $1_a:hom_A(a,a)$, called the identity morphism.

• 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 $gf$.

• For each $a,b:A$ and $f:hom_A(a,b)$, we have $f=1_b \circ f$ and $f=f\circ 1_a$.

## See also

Revision on April 25, 2022 at 23:07:42 by Anonymous?. See the history of this page for a list of all contributions to it.