GAP (XModAlg) - Chapter 4: Conversion between cat1-algebras and crossed modules

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4 Conversion between cat1-algebras and crossed modules
4.1 Equivalent Categories
4.1-1 Cat1AlgebraOfXModAlgebra
4.1-2 XModAlgebraOfCat1Algebra

### 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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