nLab H-spaceoid

Contents

Idea

The oidification of an H-space.

Definition

Classically

A H-spaceoid is a unital magmoid internal to the classical homotopy category of topological spaces Ho(Top), or in the homotopy category $Ho(Top)_*$ of pointed topological spaces, which has a unit up to homotopy.

In homotopy type theory

In homotopy type theory, a H-spaceoid is a type $A$ with

• 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 functor

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

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

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

See also

• H-space
• unital magmoid
• H-magmoid
• H-category (category object internal to a homotopy category)
• homotopy category