# H-spaces

Content created by Egbert Rijke, Fredrik Bakke and Jonathan Prieto-Cubides.

Created on 2022-06-03.

module structured-types.h-spaces where

Imports
open import foundation.action-on-identifications-binary-functions
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.evaluation-functions
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.functoriality-dependent-pair-types
open import foundation.identity-types
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.unital-binary-operations
open import foundation.universe-levels
open import foundation.whiskering-identifications-concatenation

open import foundation-core.endomorphisms

open import structured-types.magmas
open import structured-types.noncoherent-h-spaces
open import structured-types.pointed-homotopies
open import structured-types.pointed-maps
open import structured-types.pointed-sections
open import structured-types.pointed-types


## Idea

A (coherent) H-space is a "wild unital magma", i.e., it is a pointed type equipped with a binary operation for which the base point is a unit, with a coherence law between the left and the right unit laws.

## Definitions

### Unital binary operations on pointed types

coherent-unit-laws-mul-Pointed-Type :
{l : Level} (A : Pointed-Type l)
(μ : (x y : type-Pointed-Type A) → type-Pointed-Type A) → UU l
coherent-unit-laws-mul-Pointed-Type A μ =
coherent-unit-laws μ (point-Pointed-Type A)

coherent-unital-mul-Pointed-Type :
{l : Level} → Pointed-Type l → UU l
coherent-unital-mul-Pointed-Type A =
Σ ( type-Pointed-Type A → type-Pointed-Type A → type-Pointed-Type A)
( coherent-unit-laws-mul-Pointed-Type A)


### H-spaces

An H-space consists of a pointed type X and a coherent unital multiplication on X. The entry make-H-Space is provided to break up the construction of an H-space into two components: the construction of its underlying pointed type and the construction of the coherently unital multiplication on this pointed type. Furthermore, this definition suggests that any construction of an H-space should be refactored by first defining its underlying pointed type, then defining its coherently unital multiplication, and finally combining those two constructions using make-H-Space.

H-Space : (l : Level) → UU (lsuc l)
H-Space l =
Σ ( Pointed-Type l) coherent-unital-mul-Pointed-Type

make-H-Space :
{l : Level} →
(X : Pointed-Type l) → coherent-unital-mul-Pointed-Type X → H-Space l
make-H-Space X μ = (X , μ)

{-# INLINE make-H-Space #-}

module _
{l : Level} (M : H-Space l)
where

pointed-type-H-Space : Pointed-Type l
pointed-type-H-Space = pr1 M

type-H-Space : UU l
type-H-Space = type-Pointed-Type pointed-type-H-Space

unit-H-Space : type-H-Space
unit-H-Space = point-Pointed-Type pointed-type-H-Space

coherent-unital-mul-H-Space :
coherent-unital-mul-Pointed-Type pointed-type-H-Space
coherent-unital-mul-H-Space = pr2 M

mul-H-Space :
type-H-Space → type-H-Space → type-H-Space
mul-H-Space = pr1 coherent-unital-mul-H-Space

mul-H-Space' :
type-H-Space → type-H-Space → type-H-Space
mul-H-Space' x y = mul-H-Space y x

ap-mul-H-Space :
{a b c d : type-H-Space} → Id a b → Id c d →
Id (mul-H-Space a c) (mul-H-Space b d)
ap-mul-H-Space p q = ap-binary mul-H-Space p q

magma-H-Space : Magma l
pr1 magma-H-Space = type-H-Space
pr2 magma-H-Space = mul-H-Space

coherent-unit-laws-mul-H-Space :
coherent-unit-laws mul-H-Space unit-H-Space
coherent-unit-laws-mul-H-Space =
pr2 coherent-unital-mul-H-Space

left-unit-law-mul-H-Space :
(x : type-H-Space) →
Id (mul-H-Space unit-H-Space x) x
left-unit-law-mul-H-Space =
pr1 coherent-unit-laws-mul-H-Space

right-unit-law-mul-H-Space :
(x : type-H-Space) →
Id (mul-H-Space x unit-H-Space) x
right-unit-law-mul-H-Space =
pr1 (pr2 coherent-unit-laws-mul-H-Space)

coh-unit-laws-mul-H-Space :
Id
( left-unit-law-mul-H-Space unit-H-Space)
( right-unit-law-mul-H-Space unit-H-Space)
coh-unit-laws-mul-H-Space =
pr2 (pr2 coherent-unit-laws-mul-H-Space)

unit-laws-mul-H-Space :
unit-laws mul-H-Space unit-H-Space
pr1 unit-laws-mul-H-Space = left-unit-law-mul-H-Space
pr2 unit-laws-mul-H-Space = right-unit-law-mul-H-Space

is-unital-mul-H-Space : is-unital mul-H-Space
pr1 is-unital-mul-H-Space = unit-H-Space
pr2 is-unital-mul-H-Space = unit-laws-mul-H-Space

is-coherently-unital-mul-H-Space :
is-coherently-unital mul-H-Space
pr1 is-coherently-unital-mul-H-Space = unit-H-Space
pr2 is-coherently-unital-mul-H-Space =
coherent-unit-laws-mul-H-Space


## Properties

### Every noncoherent H-space can be upgraded to a coherent H-space

h-space-Noncoherent-H-Space :
{l : Level} → Noncoherent-H-Space l → H-Space l
pr1 (h-space-Noncoherent-H-Space A) = pointed-type-Noncoherent-H-Space A
pr1 (pr2 (h-space-Noncoherent-H-Space A)) = mul-Noncoherent-H-Space A
pr2 (pr2 (h-space-Noncoherent-H-Space A)) =
coherent-unit-laws-unit-laws
( mul-Noncoherent-H-Space A)
( unit-laws-mul-Noncoherent-H-Space A)


### The type of H-space structures on A is equivalent to the type of sections of ev-point : (A → A) →∗ A

module _
{l : Level} (A : Pointed-Type l)
where

ev-endo-Pointed-Type : endo-Pointed-Type (type-Pointed-Type A) →∗ A
pr1 ev-endo-Pointed-Type = ev-point-Pointed-Type A
pr2 ev-endo-Pointed-Type = refl

pointed-section-ev-point-Pointed-Type : UU l
pointed-section-ev-point-Pointed-Type =
pointed-section ev-endo-Pointed-Type

compute-pointed-section-ev-point-Pointed-Type :
pointed-section-ev-point-Pointed-Type ≃ coherent-unital-mul-Pointed-Type A
compute-pointed-section-ev-point-Pointed-Type =
( equiv-tot
( λ _ →
equiv-Σ _
( equiv-funext)
( λ _ →
equiv-tot (λ _ → inv-equiv (equiv-right-whisker-concat refl))))) ∘e
( associative-Σ _ _ _)