Homotopy Type Theory abelian group > history (Rev #15)

Definition

As a twice-delooping of a pointed simply connected 2-groupoid

A pointed simply connected 2-groupoid consists of

  • A type GG
  • A basepoint e:Ge:G
  • A 1-connector
    κ 1: f:G𝟙 a:𝟙isContr([fiber(f,a)] 1)\kappa_1:\prod_{f:G \to \mathbb{1}} \prod_{a:\mathbb{1}} \mathrm{isContr}(\left[\mathrm{fiber}(f, a)\right]_{1})
  • A 2-truncator:
    τ 2:isTwoGroupoid(G)\tau_2:\mathrm{isTwoGroupoid}(G)

An abelian group is the type Aut(Aut(G))\mathrm{Aut}(\mathrm{Aut}(G)) of automorphisms of automorphisms in GG.

As a group

An abelian group or consists of

  • A type GG,
  • A basepoint e:Ge:G
  • A binary operation μ:GGG\mu : G \to G \to G
  • A unary operation ι:GG\iota: G \to G
  • A contractible left unit identity
    c λ: (a:G)isContr(μ(e,a)=a)c_\lambda:\prod_{(a:G)} isContr(\mu(e,a)=a)
  • A contractible right unit identity
    c ρ: (a:G)isContr(μ(a,e)=a)c_\rho:\prod_{(a:G)} isContr(\mu(a,e)=a)
  • A contractible associative identity
    c α: (a:G) (b:G) (c:G)isContr(μ(μ(a,b),c)=μ(a,μ(b,c)))c_\alpha:\prod_{(a:G)} \prod_{(b:G)} \prod_{(c:G)} isContr(\mu(\mu(a, b),c)=\mu(a,\mu(b,c)))
  • A contractible left inverse identity
    c l: (a:G)isContr(μ(ι(a),a)=e)c_l:\prod_{(a:G)} isContr(\mu(\iota(a), a)=e)
  • A contractible right inverse identity
    c r: (a:G)isContr(μ(a,ι(a))=e)c_r:\prod_{(a:G)} isContr(\mu(a,\iota(a))=e)
  • A contractible commutative identity
    c κ: (a:A) (b:A)isContr(μ(a,b)=μ(b,a))c_\kappa:\prod_{(a:A)} \prod_{(b:A)} isContr(\mu(a, b)=\mu(b, a))
  • A 0-truncator
    τ 0: (a:G) (b:G)isProp(a=b)\tau_0: \prod_{(a:G)} \prod_{(b:G)} isProp(a=b)

Examples

See also

References

Revision on June 15, 2022 at 18:27:33 by Anonymous?. See the history of this page for a list of all contributions to it.