nLab grouplike A3-space

Contents

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.

See also

• higher algebra
• A3-space
• group