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2 Cat1-algebras
 2.1 Definitions and examples
 2.2 Cat^1-algebra morphisms
 2.3 Equivalent Categories

2 Cat1-algebras

2.1 Definitions and examples

Algebraic structures which are equivalent to crossed modules of algebras include :

In this section we describe an implementation of cat^1-algebras and their morphisms.

The notion of cat^1-groups was defined as an algebraic model of 2-types by Loday in [Lod82]. Then Ellis defined the cat^1-algebras in [Ell88].

Let A and R be k-algebras, let t,h:A→ R be surjections, and let e:R→ A be an inclusion.

\xymatrix@R=50pt@C=50pt{ A \ar@{->}@<-1.5pt>[d]_{t} \ar@{->}@<1.5pt>[d]^{h} \\ R \ar@/^1.5pc/[u]^{e} }

If the conditions,

\mathbf{Cat1Alg1:} \quad te = id_{R} = he, \qquad \mathbf{Cat1Alg2:} \quad (\ker t)(\ker h) = \{0_{A}\}

are satisfied, then the algebraic system mathcalC := (e;t,h : A → R) is called a cat^1-algebra. A system which satisfies the condition mathbfCat1Alg1 is called a precat^1-algebra. The homomorphisms t,h and e are called the tail map, head map and range embedding homomorphisms, respectively.

2.1-1 Cat1Algebra
‣ Cat1Algebra( args )( function )
‣ PreCat1Obj( t, h, e )( operation )
‣ PreCat1AlgebraByEndomorphisms( t, h )( operation )
‣ PreCat1AlgebraObj( C )( operation )
‣ PreCat1Algebra( C )( operation )
‣ IsIdentityCat1Algebra( C )( property )
‣ IsCat1Algebra( C )( property )
‣ IsPreCat1Algebra( C )( property )

The operations listed above are used for construction of precat^1 and cat^1-algebra structures. The function Cat1Algebra selects the operation from the above implementations up to user's input. The operation PreCat1AlgebraObj is used for preserving the implementations,

2.1-2 Source
‣ Source( C )( attribute )
‣ Range( C )( attribute )
‣ TailMap( C )( attribute )
‣ HeadMap( C )( attribute )
‣ RangeEmbedding( C )( attribute )
‣ Kernel( C )( method )
‣ Boundary( C )( attribute )

These are the seven main attributes of a pre-cat^1-algebra.


gap> Ac6 := GroupRing( GF(2), Group( (1,2,3)(4,5) ) );
<algebra-with-one over GF(2), with 1 generators>
gap> Rc3 := GroupRing( GF(2), Group( (1,2,3) ) );
<algebra-with-one over GF(2), with 1 generators>
gap> homAR := AllHomsOfAlgebras( Ac6, Rc3 );;
gap> mgiAR := List( homAR, h -> MappingGeneratorsImages(h) );;
gap> Print( mgiAR, "\n" );
[ [ [ (Z(2)^0)*(1,3,2)(4,5) ], [ <zero> of ... ] ], 
  [ [ (Z(2)^0)*(1,3,2)(4,5) ], [ (Z(2)^0)*() ] ], 
  [ [ (Z(2)^0)*(1,3,2)(4,5) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3) ] ], 
  [ [ (Z(2)^0)*(1,3,2)(4,5) ], 
      [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ] ], 
  [ [ (Z(2)^0)*(1,3,2)(4,5) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,3,2) ] ], 
  [ [ (Z(2)^0)*(1,3,2)(4,5) ], [ (Z(2)^0)*(1,2,3) ] ], 
  [ [ (Z(2)^0)*(1,3,2)(4,5) ], [ (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ] ], 
  [ [ (Z(2)^0)*(1,3,2)(4,5) ], [ (Z(2)^0)*(1,3,2) ] ] ]
gap> homRA := AllHomsOfAlgebras( Rc3, Ac6 );;
gap> mgiRA := List( homRA, h -> MappingGeneratorsImages(h) );;
gap> Print( mgiRA, "\n" );
[ [ [ (Z(2)^0)*(1,2,3) ], [ <zero> of ... ] ], 
  [ [ (Z(2)^0)*(1,2,3) ], [ (Z(2)^0)*() ] ], 
  [ [ (Z(2)^0)*(1,2,3) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3) ] ], 
  [ [ (Z(2)^0)*(1,2,3) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ] ],
  [ [ (Z(2)^0)*(1,2,3) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,3,2) ] ], 
  [ [ (Z(2)^0)*(1,2,3) ], [ (Z(2)^0)*(1,2,3) ] ], 
  [ [ (Z(2)^0)*(1,2,3) ], [ (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2) ] ], 
  [ [ (Z(2)^0)*(1,2,3) ], [ (Z(2)^0)*(1,3,2) ] ] ]
gap> C4 := PreCat1Obj( homAR[6], homAR[6], homRA[8] );
[AlgebraWithOne( GF(2), [ (Z(2)^0)*(1,2,3)(4,5) ] ) -> AlgebraWithOne( GF(2), 
[ (Z(2)^0)*(1,2,3) ] )]
gap> IsCat1Algebra( C4 );
true
gap> Size( C4 );
[ 64, 8 ]
gap> Display( C4 );

Cat1-algebra [..=>..] :- 
: source algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3)(4,5) ]
:  range algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3) ]
: tail homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,3,2) ]
: head homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,3,2) ]
: range embedding maps range generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,3,2) ]
: kernel 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) ]
: boundary homomorphism maps generators of kernel to:
  [ <zero> of ..., <zero> of ..., <zero> of ... ]
: kernel embedding maps generators of kernel to:
  [ (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) ]

2.1-3 Cat1AlgebraSelect
‣ Cat1AlgebraSelect( gf, gpsize, gpnum, num )( operation )

The Cat1Algebra (2.1-1) function may also be used to select a cat^1-algebra from a data file. All cat^1-structures on commutative algebras are stored in a list in file cat1algdata.g. The data is read into the list CAT1ALG_LIST only when this function is called.

The function Cat1AlgebraSelect may be used in four ways:

Now, we will give an example for the usage of this function.


gap> C := Cat1AlgebraSelect( 11 );
|--------------------------------------------------------|
| 11 is invalid number for Galois Field (gf)             |
| Possible numbers for the gf in the Data :              |
|--------------------------------------------------------|
 [ 2, 3, 4, 5, 7 ]
Usage: Cat1Algebra( gf, gpsize, gpnum, num );
fail
gap> C := Cat1AlgebraSelect( 4, 12 );
|--------------------------------------------------------|
| 12 is invalid number for size of group (gpsize)        |
| Possible numbers for the gpsize for GF(4) in the Data: |
|--------------------------------------------------------|
 [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]
Usage: Cat1Algebra( gf, gpsize, gpnum, num );
fail
gap> C := Cat1AlgebraSelect( 2, 6, 3 );
|--------------------------------------------------------|
| 3 is invalid number for group of order 6               |
| Possible numbers for the gpnum in the Data :           |
|--------------------------------------------------------|
 [ 1, 2 ]
Usage: Cat1Algebra( gf, gpsize, gpnum, num );
fail
gap> C := Cat1AlgebraSelect( 2, 6, 2 );
There are 4 cat1-structures for the algebra GF(2)_c6.
 Range Alg      Tail                    Head
|--------------------------------------------------------|
| GF(2)_c6      identity map            identity map     |
| -----         [ 2, 10 ]               [ 2, 10 ]        |
| -----         [ 2, 14 ]               [ 2, 14 ]        |
| -----         [ 2, 50 ]               [ 2, 50 ]        |
|--------------------------------------------------------|
Usage: Cat1Algebra( gf, gpsize, gpnum, num );
Algebra has generators [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3)(4,5) ]
4
gap> C0 := Cat1AlgebraSelect( 4, 6, 2, 2 );
[GF(2^2)_c6 -> Algebra( GF(2^2), 
[ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)(3,6)+(
    Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] )]
gap> Size( C0 ); 
[ 4096, 1024 ]
gap> Display( C0 ); 

Cat1-algebra [GF(2^2)_c6=>..] :- 
: source algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3,4,5,6) ]
:  range algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)
    (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ]
: tail homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)
    (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ]
: head homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)
    (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ]
: range embedding maps range generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)
    (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ]
: kernel has generators:
  [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)
    (2,5)(3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ]
: boundary homomorphism maps generators of kernel to:
  [ <zero> of ... ]
: kernel embedding maps generators of kernel to:
  [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)
    (2,5)(3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ]

2.1-4 SubCat1Algebra
‣ SubCat1Algebra( arg )( operation )
‣ SubPreCat1Algebra( arg )( operation )
‣ IsSubCat1Algebra( arg )( property )
‣ IsSubPreCat1Algebra( arg )( property )

Let mathcalC = (e;t,h:A→ R) be a cat^1-algebra, and let A^', R^' be subalgebras of A and R respectively. If the restriction morphisms

t^{\prime} = t|_{A^{\prime}} : A^{\prime}\rightarrow R^{\prime}, \qquad h^{\prime} = h|_{A^{\prime}} : A^{\prime}\rightarrow R^{\prime}, \qquad e^{\prime} = e|_{R^{\prime}} : R^{\prime}\rightarrow A^{\prime}

satisfy the mathbfCat1Alg1 and mathbfCat1Alg2 conditions, then the system mathcalC^' = (e^';t^',h^' : A^' → R^') is called a subcat^1-algebra of mathcalC = (e;t,h:A→ R).

If the morphisms satisfy only the mathbfCat1Alg1 condition then mathcalC^' is called a sub-precat^1-algebra of mathcalC.

The operations in this subsection are used for constructing subcat^1-algebras of a given cat^1-algebra.


gap> C3 := Cat1AlgebraSelect( 2, 6, 2, 4 );; 
gap> A3 := Source( C3 );
GF(2)_c6
gap> B3 := Range( C3 ); 
GF(2)_c3
gap> eA3 := Elements( A3 );;
gap> eB3 := Elements( B3 );;
gap> AA3 := Subalgebra( A3, [ eA3[1], eA3[2], eA3[3] ] );
<algebra over GF(2), with 3 generators>
gap> [ Size(A3), Size(AA3) ]; 
[ 64, 4 ]
gap> BB3 := Subalgebra( B3, [ eB3[1], eB3[2] ] ); 
<algebra over GF(2), with 2 generators>
gap> [ Size(B3), Size(BB3) ]; 
[ 8, 2 ]
gap> CC3 := SubCat1Algebra( C3, AA3, BB3 );
[Algebra( GF(2), [ <zero> of ..., (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(4,5) 
 ] ) -> Algebra( GF(2), [ <zero> of ..., (Z(2)^0)*() ] )]
gap> Display( CC3 );

Cat1-algebra [..=>..] :-
: source algebra has generators:
  [ <zero> of ..., (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(4,5) ]
:  range algebra has generators:
  [ <zero> of ..., (Z(2)^0)*() ]
: tail homomorphism maps source generators to:
  [ <zero> of ..., (Z(2)^0)*(), <zero> of ... ]
: head homomorphism maps source generators to:
  [ <zero> of ..., (Z(2)^0)*(), <zero> of ... ]
: range embedding maps range generators to:
  [ <zero> of ..., (Z(2)^0)*() ]
: kernel has generators:
  [ <zero> of ..., (Z(2)^0)*()+(Z(2)^0)*(4,5) ]
: boundary homomorphism maps generators of kernel to:
  [ <zero> of ..., <zero> of ... ]
: kernel embedding maps generators of kernel to:
  [ <zero> of ..., (Z(2)^0)*()+(Z(2)^0)*(4,5) ]

2.2 Cat^1-algebra morphisms

Let mathcalC = (e;t,h:A→ R), mathcalC^' = (e^'; t^' , h^' : A^' → R^') be cat^1-algebras, and let ϕ : A→ A^' and φ : R → R^' be algebra homomorphisms. If the diagram

\xymatrix@R=50pt@C=50pt{ A \ar@{->}@<-1.5pt>[d]_{t} \ar@{->}@<1.5pt>[d]^{h} \ar@{->}[r]^{\phi} & A' \ar@{->}@<-1.5pt>[d]_{t'} \ar@{->}@<1.5pt>[d]^{h'} \\ R \ar@/^1.5pc/[u]^{e} \ar@{->}[r]_{\varphi} & R' \ar@/_1.5pc/[u]_{e'} }

commutes, (i.e. t^' ∘ ϕ = φ ∘ t, h^' ∘ ϕ = φ ∘ h and e^' ∘ φ = ϕ ∘ e), then the pair (ϕ ,φ ) is called a cat^1-algebra morphism.

2.2-1 Cat1AlgebraMorphism
‣ Cat1AlgebraMorphism( arg )( operation )
‣ IdentityMapping( C )( method )
‣ PreCat1AlgebraMorphismByHoms( f, g )( operation )
‣ Cat1AlgebraMorphismByHoms( f, g )( operation )
‣ IsPreCat1AlgebraMorphism( C )( property )
‣ IsCat1AlgebraMorphism( arg )( property )

These operations are used for constructing cat^1-algebra morphisms. Details of the implementations can be found in [Oda09].

2.2-2 Source
‣ Source( m )( attribute )
‣ Range( m )( attribute )
‣ IsTotal( m )( method )
‣ IsSingleValued( m )( method )
‣ Name( m )( method )
‣ Boundary( m )( attribute )

These are the six main attributes of a cat^1-algebra morphism.


gap> C1 := Cat1Algebra( 2, 1, 1, 1 );
[GF(2)_triv -> GF(2)_triv]
gap> Display( C1 );

Cat1-algebra [GF(2)_triv=>GF(2)_triv] :-
: source algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
:  range algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: tail homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: head homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: range embedding maps range generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: the kernel is trivial.

gap> C2 := Cat1Algebra( 2, 2, 1, 2 );
[GF(2)_c2 -> GF(2)_triv]
gap> Display( C2 );

Cat1-algebra [GF(2)_c2=>GF(2)_triv] :-
: source algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,2) ]
:  range algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: tail homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: head homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: range embedding maps range generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: kernel has generators:
  [ (Z(2)^0)*()+(Z(2)^0)*(1,2) ]
: boundary homomorphism maps generators of kernel to:
  [ <zero> of ... ]
: kernel embedding maps generators of kernel to:
  [ (Z(2)^0)*()+(Z(2)^0)*(1,2) ]

gap> C1 = C2;
false
gap> R1 := Source( C1 );;
gap> R2 := Source( C2 );;
gap> S1 := Range( C1 );;
gap> S2 := Range( C2 );;
gap> gR1 := GeneratorsOfAlgebra( R1 );
[ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> gR2 := GeneratorsOfAlgebra( R2 );
[ (Z(2)^0)*(), (Z(2)^0)*(1,2) ]
gap> gS1 := GeneratorsOfAlgebra( S1 );
[ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> gS2 := GeneratorsOfAlgebra( S2 );
[ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> im1 := [ gR2[1], gR2[1] ];
[ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> f1 := AlgebraHomomorphismByImages( R1, R2, gR1, im1 );
[ (Z(2)^0)*(), (Z(2)^0)*() ] -> [ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> im2 := [ gS2[1], gS2[1] ];
[ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> f2 := AlgebraHomomorphismByImages( S1, S2, gS1, im2 );
[ (Z(2)^0)*(), (Z(2)^0)*() ] -> [ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> m := Cat1AlgebraMorphism( C1, C2, f1, f2 );
[[GF(2)_triv=>GF(2)_triv] => [GF(2)_c2=>GF(2)_triv]]
gap> Display( m );
Morphism of cat1-algebras :-
: Source = [GF(2)_triv=>GF(2)_triv] with generating sets:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
:  Range = [GF(2)_c2=>GF(2)_triv] with generating sets:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,2) ]
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: Source Homomorphism maps source generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
: Range Homomorphism maps range generators to:
  [ (Z(2)^0)*(), (Z(2)^0)*() ]
gap> IsSurjective( m );
false
gap> IsInjective( m );
true
gap> IsBijective( m );
false


2.3 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.

2.3-1 PreCat1ByPreXMod
‣ PreCat1ByPreXMod( X0 )( operation )
‣ PreXModAlgebraByPreCat1Algebra( C )( operation )
‣ Cat1AlgebraByXModAlgebra( X0 )( operation )
‣ XModAlgebraByCat1Algebra( C )( operation )

These operations are used for constructing a cat^1-algebra from a given crossed module, and conversely.


gap> CXM := Cat1AlgebraByXModAlgebra( XM );
[GF(2^2)[k4] IX <e5> -> GF(2^2)[k4]]
gap> X3 := XModAlgebraByCat1Algebra( C3 ); 
[Algebra( GF(2), [ <zero> of ..., <zero> of ..., <zero> of ... 
 ] )->Algebra( GF(2), 
[ (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) ] )]
gap> Display( X3 ); 

Crossed module [..->..] :- 
: Source algebra has generators:
  [ <zero> of ..., <zero> of ..., <zero> of ... ]
: Range 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) ]
: Boundary homomorphism maps source generators to:
  [ <zero> of ..., <zero> of ..., <zero> of ... ]

Since all these operations are linked to the functions Cat1Algebra (2.1-1) and XModAlgebra (3.1-1), all of them can be done by using these two functions. We may also use the function Cat1Algebra (2.1-1) instead of the operation Cat1AlgebraSelect (2.1-3).

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