# Homotopy Type Theory H-space (Rev #7, changes)

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# Contents

## Idea

H-spaces Sometimes are we simply can types equip equipped with the structure of a magma (from classical Algebra). They are useful classically in constructing fibrations.type with a certain structure, called a H-space, allowing us to derive some nice properties about the type or even construct fibrations

## Definition

A H-Space consists ofH-Space consists of

• A type $A$,
• A basepoint $e:A$
• A binary operation $\mu : A \to A \to A$
• for every $a:A$, equalities $\mu(e,a)=a$ and $\mu(a,e)=a$

## Properties

Let $A$ be a connected H-space. Then for every$a:A$connected? , the H-space. maps Then for every \mu(a,-),\mu(-,a):A a:A \to A , are the equivalences.maps? $\mu(a,-),\mu(-,a):A \to A$ are equivalences?.

## Examples

• There is a H-space structure on the circle. See Lemma 8.5.8 of the HoTT book. (TODO: Write out construction).

• Every loop space is naturally a H-space with path concatenation as the operation. In fact every loop space is a group?.

• The type of maps? $A \to A$ has the structure of a H-space, with basepoint $id_A$, operation function composition.