Algebraic structures which are equivalent to crossed modules of algebras include :
cat\(^{1}\)-algebras, (Ellis, [Ell88]);
simplicial algebras with Moore complex of length 1, (Z. Arvasi and T.Porter, [AP96]);
algebra-algebroids, (Gaffar Musa's Ph.D. thesis, [Mos86]).
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 \(t,h:A\rightarrow R\) be surjections, and let \(e:R\rightarrow A\) 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 \(\mathcal{C} := (e;t,h : A \rightarrow R)\) is called a cat\(^{1}\)-algebra. A system which satisfies the condition \(\mathbf{Cat1Alg1}\) is called a precat\(^{1}\)-algebra. The homomorphisms \(t,h\) and \(e\) are called the tail map, head map and range embedding homomorphisms, respectively.
‣ Cat1Algebra( args ) | ( function ) |
‣ PreCat1AlgebraByEndomorphisms( t, h ) | ( operation ) |
‣ PreCat1AlgebraByTailHeadEmbedding( t, h, e ) | ( operation ) |
‣ PreCat1Algebra( args ) | ( 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 operations PreCat1AlgebraByEndomorphisms and PreCat1AlgebraByTailHeadEmbedding are used with particular choices of algebra homomorphisms.
‣ Source( C ) | ( attribute ) |
‣ Range( C ) | ( attribute ) |
‣ TailMap( C ) | ( attribute ) |
‣ HeadMap( C ) | ( attribute ) |
‣ RangeEmbedding( C ) | ( attribute ) |
‣ Kernel( C ) | ( method ) |
‣ Boundary( C ) | ( attribute ) |
‣ Size2d( C ) | ( attribute ) |
These are the eight main attributes of a pre-cat\(^{1}\)-algebra.
In the example we use homomorphisms between A2c6 and I2c6 constructed in section 2.3.
gap> t4 := homAR[8]; [ (Z(2)^0)*(1,6,5,4,3,2) ] -> [ (Z(2)^0)*(7,9,8) ] gap> e4 := homRA[8]; [ (Z(2)^0)*(7,8,9) ] -> [ (Z(2)^0)*(1,5,3)(2,6,4) ] gap> C4 := PreCat1AlgebraByTailHeadEmbedding( t4, t4, e4 ); [AlgebraWithOne( GF(2), [ (Z(2)^0)*(1,2,3,4,5,6) ] ) -> AlgebraWithOne( GF(2), [ (Z(2)^0)*(7,8,9) ] )] gap> IsCat1Algebra( C4 ); true gap> Size2d( C4 ); [ 64, 8 ] gap> Display( C4 ); Cat1-algebra [..=>..] :- : 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)*(7,8,9) ] : tail homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*(7,8,9) ] : head homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*(7,8,9) ] : range embedding maps range generators to: [ (Z(2)^0)*(), (Z(2)^0)*(1,5,3)(2,6,4) ] : kernel has generators: [ (Z(2)^0)*()+(Z(2)^0)*(1,4)(2,5)(3,6), (Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)* (1,5,3)(2,6,4), (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,6,5,4,3,2) ] : 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)*(1,4)(2,5)(3,6), (Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)* (1,5,3)(2,6,4), (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,6,5,4,3,2) ]
‣ Cat1AlgebraSelect( n, gpsize, gpnum, num ) | ( operation ) |
The Cat1Algebra (3.1-1) 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:
Cat1AlgebraSelect( n ) returns the list of possible sizes of groups for group algebras with Galois field \(GF(n)\).
Cat1AlgebraSelect( n, m ) returns the list of allowable group numbers with given Galois field \(GF(n)\) and groups of size \(m\).
Cat1AlgebraSelect( n, m, k ) returns the list of possible cat\(^{1}\)-algebra structures with given Galois field \(GF(n)\) and group number \(k\) of size \(m\).
Cat1AlgebraSelect( n, m, k, j ) (or simply Cat1Algebra( n, m, k, j )) returns the \(j\)-th cat\(^{1}\)-algebra structure with these other parameters.
Now, we give examples of the use of this function.
gap> C := Cat1AlgebraSelect( 11 ); |--------------------------------------------------------| | 11 is invalid number for Galois Field (GFnum) | | Possible numbers for GFnum in the Data : | |--------------------------------------------------------| [ 2, 3, 4, 5, 7 ] Usage: Cat1Algebra( GFnum, 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( GFnum, 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( GFnum, 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( GFnum, 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> Size2d( 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) ]
‣ SubCat1Algebra( arg ) | ( operation ) |
‣ SubPreCat1Algebra( arg ) | ( operation ) |
‣ IsSubCat1Algebra( arg ) | ( property ) |
‣ IsSubPreCat1Algebra( arg ) | ( property ) |
Let \(\mathcal{C} = (e;t,h:A\rightarrow R)\) be a cat\(^{1}\)-algebra, and let \(A^{\prime}\), \(R^{\prime}\) 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 \(\mathbf{Cat1Alg1}\) and \(\mathbf{Cat1Alg2}\) conditions, then the system \(\mathcal{C}^{\prime } = (e^{\prime};t^{\prime},h^{\prime} : A^{\prime} \rightarrow R^{\prime})\) is called a subcat\(^{1}\)-algebra of \(\mathcal{C} = (e;t,h:A\rightarrow R)\).
If the morphisms satisfy only the \(\mathbf{Cat1Alg1}\) condition then \(\mathcal{C}^{\prime }\) is called a sub-precat\(^{1}\)-algebra of \(\mathcal{C}\).
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) ]
Let \(\mathcal{C} = (e;t,h:A\rightarrow R)\), \(\mathcal{C}^{\prime } = (e^{\prime}; t^{\prime }, h^{\prime } : A^{\prime} \rightarrow R^{\prime})\) be cat\(^{1}\)-algebras, and let \(\phi : A\rightarrow A^{\prime}\) and \(\varphi : R \rightarrow R^{\prime}\) 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. \(t^{\prime} \circ \phi = \varphi \circ t\), \(h^{\prime} \circ \phi = \varphi \circ h\) and \(e^{\prime } \circ \varphi = \phi \circ e\)), then the pair \((\phi ,\varphi )\) is called a cat\(^{1}\)-algebra morphism.
‣ 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].
‣ 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
‣ ImagesSource2DimensionalMapping( m ) | ( operation ) |
When \((\theta,\varphi)\) is a homomorphism of cat1-algebras (or crossed modules) this operation returns the image crossed module.
gap> imm := ImagesSource2DimensionalMapping( m );; gap> Display( imm ); Cat1-algebra [..=>..] :- : source algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*() ] : range algebra has generators: [ (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)*() ] : the kernel is trivial.
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