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→ 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.
‣ 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,
‣ 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) ]
‣ 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:
Cat1AlgebraSelect( gf )
returns the list of possible size of Galois field or the list of possible size of groups with given Galois field.
Cat1AlgebraSelect( gf, gpsize )
returns the list of possible sizes of a group with given Galois field, or the list of possible number of groups with given Galois field and size of group.
Cat1AlgebraSelect( gf, gpsize, gpnum )
returns the list of possible number of group with given Galois field and size of group or the list of possible cat^1-structures with given Galois field and group.
Cat1AlgebraSelect( gf, gpsize, gpnum, num )
(or just Cat1Algebra( gf, gpsize, gpnum, num )
) returns the chosen cat^1-algebra.
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) ]
‣ 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) ]
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.
‣ 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
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.
‣ 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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