The categories mathbfCat1Alg (cat^1-algebras) and mathbfXModAlg (crossed modules) are naturally equivalent [Ell88]. This equivalence is outlined in what follows. For a given crossed module (∂ : A → R) we can construct the semidirect product R⋉ A thanks to the action of R on A. If we define t,h : R⋉ A → R and e : R → R ⋉ A by
t(r,a) = r, \qquad h(r,a) = r+\partial(a), \qquad e(r) = (r,0),
respectively, then mathcalC = (e;t,h : R ⋉ A → R) is a cat^1-algebra.
Conversely, for a given cat^1-algebra mathcalC=(e;t,h : A → R), the map ∂ : ker t → R is a crossed module, where the action is multiplication action and ∂ is the restriction of h to ker t.
Since all of these operations are linked to the functions Cat1Algebra
(2.1-1) and XModAlgebra
(3.1-1), they can be permormed by calling these two functions. We may also use the function Cat1Algebra
(2.1-1) instead of the operation Cat1AlgebraSelect
(2.1-3).
‣ Cat1AlgebraOfXModAlgebra ( X0 ) | ( operation ) |
‣ PreCat1AlgebraOfPreXModAlgebra ( X0 ) | ( operation ) |
These operations are used for constructing a cat^1-algebra from a given crossed module of algebras.
gap> CXM := Cat1AlgebraOfXModAlgebra( XM ); [GF(2^2)[k4] IX <e5> -> GF(2^2)[k4]] gap> Display( CXM ); Cat1-algebra [..=>GF(2^2)[k4]] :- : range algebra has generators: [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1, (Z(2)^0)*f2 ] : tail homomorphism maps source generators to: : range embedding maps range generators to: [ [ (Z(2)^0)*<identity> of ..., <zero> of ... ], [ (Z(2)^0)*f1, <zero> of ... ], [ (Z(2)^0)*f2, <zero> of ... ] ] : kernel has generators: [ [ <zero> of ..., <zero> of ... ], [ <zero> of ..., (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^ 0)*f1*f2 ], [ <zero> of ..., (Z(2^2))*<identity> of ...+(Z(2^2))*f1+(Z(2^2))*f2+( Z(2^2))*f1*f2 ], [ <zero> of ..., (Z(2^2)^2)*<identity> of ...+(Z(2^2)^2)*f1+(Z(2^2)^2)*f2+( Z(2^2)^2)*f1*f2 ] ]
‣ XModAlgebraOfCat1Algebra ( C ) | ( operation ) |
‣ PreXModAlgebraOfPreCat1Algebra ( C ) | ( operation ) |
These operations are used for constructing a crossed module of algebras from a given cat^1-algebra.
gap> X3 := XModAlgebraOfCat1Algebra( C3 ); [ <algebra of dimension 3 over GF(2)> -> <algebra of dimension 3 over GF(2)> ] gap> Display( X3 ); Crossed module [..->..] :- : Source algebra has generators: [ (Z(2)^0)*()+(Z(2)^0)*(4,5), (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,2,3)(4,5), (Z(2)^0)*(1,3,2)+(Z(2)^0)*(1,3,2)(4,5) ] : Range algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3), (Z(2)^0)*(1,3,2) ] : Boundary homomorphism maps source generators to: [ <zero> of ..., <zero> of ..., <zero> of ... ]
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