# nLab grouplike A3-space

### Context

#### Higher algebra

higher algebra

universal algebra

## Idea

The invertible version of the A3-space up to homotopy, without any higher coherences for inverses.

## Definition

A grouplike $A_3$-space or grouplike $A_3$-algebra in homotopy types or H-group consists of

• A type $A$,
• A basepoint $e:A$
• A binary operation $\mu : A \to A \to A$
• A unary operation $\iota: A \to A$
• A left unitor
$\lambda_u:\prod_{(a:A)} \mu(e,a)=a$
• A right unitor
$\rho_u:\prod_{(a:A)} \mu(a,e)=a$
• An asssociator
$\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))$
• A left invertor
$l:\prod_{(a:A)} \mu(\iota(a), a)=e$
• A right invertor
$r:\prod_{(a:A)} \mu(a,\iota(a))=e$

One could also speak of grouplike $A_3$-spaces where the existence of left and right inverses are mere property rather than structure, which is a grouplike $A_3$-space as defined above with additional structure specifying that the types $\prod_{(a:A)} \mu(\iota(a), a)=e$ and $\prod_{(a:A)} \mu(a,\iota(a))=e$ are contractible:

$c_l: \sum_{l:\prod_{(a:A)} \mu(\iota(a), a)=e} \prod_{b:\prod_{(a:A)} \mu(\iota(a), a)=e} l = b$
$c_r: \sum_{r:\prod_{(a:A)} \mu(a,\iota(a))=e} \prod_{b:\prod_{(a:A)} \mu(a,\iota(a))=e} r = b$

or equivalently,

$c_l:\prod_{(a:A)} \sum_{(l:\mu(\iota(a), a)=e)} \prod_{(b:\mu(\iota(a), a)=e)} l = b$
$c_r:\prod_{(a:A)} \sum_{(r:\mu(a,\iota(a))=e)} \prod_{(r:\mu(a,\iota(a))=e)} r = b$

## Examples

• The integers are an grouplike $A_3$-space.

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

• A group is a 0-truncated grouplike $A_3$-space.