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 (∂ : S → R) we can construct the semidirect product R ⋉ S thanks to the action of R on S. If we define t,h : R ⋉ S → R and e : R → R ⋉ S by
t(r,s) = r, \qquad h(r,s) = r + \partial(s), \qquad e(r) = (r,0),
respectively, then mathcalC = (e;t,h : R ⋉ S → R) is a cat^1-algebra.
Notice that h is an algebra homomorphism, since:
h(r_1r_2,~ r_1 \cdot s_2 + r_2 \cdot s_1 + s_1s_2) ~=~ r_1r_2 + r_1(\partial s_2) + r_2(\partial s_1) + (\partial s_1)(\partial s_2) ~=~ (r_1 + \partial s_1)(r_2 + \partial s_2).
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 by eR, and ∂ is the restriction of h to ker t.
Since all of these operations are linked to the functions Cat1Algebra (3.1-1) and XModAlgebra (4.1-1), they can be performed by calling these two functions. We may also use the function Cat1Algebra (3.1-1) instead of the operation Cat1AlgebraSelect (3.1-3).
‣ Cat1AlgebraOfXModAlgebra( X0 ) | ( operation ) |
‣ PreCat1AlgebraOfPreXModAlgebra( X0 ) | ( operation ) |
These operations are used for constructing a cat^1-algebra from a given crossed module of algebras. As an example we use the crossed module XAB constructed in section 4.1-2.
gap> Cn := Cat1AlgebraOfXModAlgebra( Xn ); [An |X Bn -> An] gap> Display( Cn ); Cat1-algebra [An |X Bn => An] :- : range algebra has generators: [ [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^0 ] ], [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ] : tail homomorphism maps source generators to: [ [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^0 ] ], [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], [ [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], [ [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ] : head homomorphism maps source generators to: [ [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^0 ] ], [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ], [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ] : range embedding maps range generators to: [ v.1, v.2 ] : kernel has generators: [ v.4, v.5 ]
As a second example, we convert the crossed module X4 constructed in section 4.1-8
gap> C3 := Cat1AlgebraOfXModAlgebra( X3 ); [A3 |X GR(c3) -> A3] gap> Display( C3 ); Cat1-algebra [A3 |X GR(c3) => A3] :- : range algebra has generators:[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ] : tail homomorphism maps source generators to: [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ] : head homomorphism maps source generators to: [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ] ] : range embedding maps range generators to: [ v.1 ] : kernel has generators: [ v.4, v.5, v.6 ]
‣ XModAlgebraOfCat1Algebra( C ) | ( operation ) |
‣ PreXModAlgebraOfPreCat1Algebra( C ) | ( operation ) |
These operations are used for constructing a crossed module of algebras from a given cat^1-algebra. The example uses the cat^1-algebra C3 constructed in section 3.1-4.
gap> X6 := XModAlgebraOfCat1Algebra( C6 ); [ <algebra of dimension 3 over GF(2)> -> <algebra of dimension 3 over GF(2)> ] gap> Display( X6 ); 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) ] : Boundary homomorphism maps source generators to: [ <zero> of ..., <zero> of ..., <zero> of ... ]
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