Homotopy Type Theory group > history (Rev #3, changes)

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Definition

A group 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 left unitor contractor
    λc λ: (a:G)isContr(μ(e,a)=a) \lambda:\prod_{(a:G)} c_\lambda:\prod_{(a:G)} \mu(e,a)=a isContr(\mu(e,a)=a)
  • A right unitor contractor
    ρc ρ: (a:G)isContr(μ(a,e)=a) \rho:\prod_{(a:G)} c_\rho:\prod_{(a:G)} \mu(a,e)=a isContr(\mu(a,e)=a)
  • An asssociator contractor
    αc α: (a:G) (b:G) (c:G)isContr(μ(μ(a,b),c)=μ(a,μ(b,c))) \alpha:\prod_{(a:G)} c_\alpha:\prod_{(a:G)} \prod_{(b:G)} \prod_{(c:G)} \mu(\mu(a, isContr(\mu(\mu(a, b),c)=\mu(a,\mu(b,c)) b),c)=\mu(a,\mu(b,c)))
  • A left invertor contractor
    lc l: (a:G)isContr(μ(ι(a),a)=e) l:\prod_{(a:G)} c_l:\prod_{(a:G)} \mu(\iota(a), isContr(\mu(\iota(a), a)=e a)=e)
  • A right invertor contractor
    rc r: (a:G)isContr(μ(a,ι(a))=e) r:\prod_{(a:G)} c_r:\prod_{(a:G)} \mu(a,\iota(a))=e isContr(\mu(a,\iota(a))=e)
  • A 0-truncator
    τ 0: (a:G) (b:G) (c:a=b)isProp (d:a=b)( (x:c=d)a (y:c=d)x= y b) \tau_0: \prod_{(a:G)} \prod_{(b:G)} \prod_{(c:a=b)} isProp(a=b) \prod_{(d:a=b)} \sum_{(x:c=d)} \prod_{(y:c=d)} x=y

Examples

  • The integers are a group.

  • Given a set AA, the type of automorphisms Aut(A)Aut(A) has the structure of a group, with basepoint id Aid_A, binary operation function composition, and unary operation inverse automorphism () 1{(-)}^{-1}.

See also

References

Revision on February 13, 2022 at 22:56:02 by Anonymous?. See the history of this page for a list of all contributions to it.