nLab A3-space

Contents

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.

See also

• higher algebra
• H-space
• monoid
• ring
• H-category
• An-space
• grouplike A3-space