nLab H-magmoid

Contents

Idea

The oidification of an “H-magma”, which is to magmas what H-spaces are to unital magmas.

Definition

Classically

A H-magmoid is a 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, 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 $gf$.

See also

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