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5 Conversion between cat1-algebras and crossed modules

5.1 Equivalent Categories

The categories \(\mathbf{Cat1Alg}\) (cat\(^{1}\)-algebras) and \(\mathbf{XModAlg}\) (crossed modules) are naturally equivalent [Ell88]. This equivalence is outlined in what follows. For a given crossed module \((\partial : S \rightarrow R)\) we can construct the semidirect product \(R \ltimes S\) thanks to the action of \(R\) on \(S\). If we define \(t,h : R \ltimes S \rightarrow R\) and \(e : R \rightarrow R \ltimes S\) by

\[ t(r,s) = r, \qquad h(r,s) = r + \partial(s), \qquad e(r) = (r,0), \]

respectively, then \(\mathcal{C} = (e;t,h : R \ltimes S \rightarrow R)\) 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 \(\mathcal{C}=(e;t,h : A \rightarrow R)\), the map \(\partial : \ker t \rightarrow R\) is a crossed module, where the action is multiplication action by \(eR\), and \(\partial\) 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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