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4 Conversion between cat1-algebras and crossed modules
 4.1 Equivalent Categories

4 Conversion between cat1-algebras and crossed modules

4.1 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 (∂ : 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).

4.1-1 Cat1AlgebraOfXModAlgebra
‣ 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 ] ]

4.1-2 XModAlgebraOfCat1Algebra
‣ 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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