# nLab A3-space

### Context

#### Higher algebra

higher algebra

universal algebra

## Idea

Sometimes we can equip a type with a certain structure, called an $A_3$-algebra structure, allowing us to derive some nice properties about the type and 0-truncate it to form monoids.

## Definition

### In classical mathematics

An $A_3$-space is a homotopy associative H-space (but no coherence is required of the associator).

An $H$-monoid is a monoid 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

Both notions coincide in homotopy type theory. An $A_3$-space or H-monoid consists of

• A type $A$,
• A basepoint $e:A$
• A binary operation $\mu : A \to A \to A$
• A left unitor
$\lambda:\prod_{(a:A)} \mu(e,a)=a$
• A right unitor
$\rho:\prod_{(a:A)} \mu(a,e)=a$
• An asssociator
$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$

### Homomorphisms of $A_3$-spaces

A homomorphism of $A_3$-spaces between two $A_3$-spaces $A$ and $B$ consists of

• A function $\phi:A \to B$ such that

• The basepoint is preserved
$\phi(e_A) = e_B$
• The binary operation is preserved
$\prod_{(a:A)} \prod_{(b:A)} \phi(\mu_A(a, b)) = \mu_B(\phi(a),\phi(b))$
• A function

$\phi_\lambda:\left(\prod_{(a:A)} \mu(e_A,a)=a\right) \to \left(\prod_{(b:B)} \mu(e_B,b)=b\right)$

such that the left unitor is preserved:

$\phi_\lambda(\lambda_A) = \lambda_B$
• A function
$\phi_\rho:\left(\prod_{(a:A)} \mu(a, e_A)=a\right) \to \left(\prod_{(b:B)} \mu(b, e_B)=b\right)$

such that the right unitor is preserved:

$\phi_\rho(\rho_A) = \rho_B$
• A function
$\phi_\alpha:\left(\prod_{(a_1:A)} \prod_{(a_2:A)} \prod_{(a_3:A)} \mu(\mu(a_1, a_2),a_3)=\mu(a_1,\mu(a_2,a_3))\right) \to \left(\prod_{(b_1:B)} \prod_{(b_2:B)} \prod_{(b_3:B)} \mu(\mu(b_1, b_2),b_3)=\mu(b_1,\mu(b_2,b_3))\right)$

such that the associator is preserved:

$\phi_\alpha(\alpha_A) = \alpha_B$

## Examples

• The integers are an $A_3$-space.

• Every loop space is naturally an $A_3$-space with path concatenation as the operation. In fact every loop space is a $\infty$-group.

• The type of endofunctions $A \to A$ has the structure of an $A_3$-space, with basepoint $id_A$, operation function composition.

• A monoid is a 0-truncated $A_3$-space.