Homotopy Type Theory
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Definition
A group consists of
A type G G ,
A basepoint e : G e:G
A binary operation μ : G → G → G \mu : G \to G \to G
A unary operation ι : G → G \iota: G \to G
A left unitor contractor λ c 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 ρ c 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 α c 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 l c 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 r c 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-truncatorτ 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 A A , the type of automorphisms Aut ( A ) Aut(A) has the structure of a group, with basepoint id A id_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
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