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5 Conversion between cat1-algebras and crossed modules

5.1 Equivalent Categories

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).

5.1-1 Cat1AlgebraOfXModAlgebra

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 XModAlgebraByIdeal (4.1-2) (The output from Display needs to be improved.)

gap> CAB := Cat1AlgebraOfXModAlgebra( XAB );
[Algebra( GF(5), [ v.1, v.2, v.3, v.4, v.5 ] ) -> A(l,m)]
gap> Display( CAB );

Cat1-algebra [..=>A(l,m)] :- 
:  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:
: range embedding maps range generators to:
  [ v.1, v.2 ]
: kernel has generators:
  Algebra( GF(5), [ v.4, v.5 ] )

5.1-2 XModAlgebraOfCat1Algebra

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) ]
: Boundary homomorphism maps source generators to:
  [ <zero> of ..., <zero> of ..., <zero> of ... ]

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