Magma allows tensors and tensor spaces to change categories. Unless a user specifies otherwise, all tensors are assigned a category that is natural to the method by which it was created. For example a tensor created from an algebra will be assigned an algebra category, whereas a tensor created by structure constants will be assigned the Albert homotopism category [Alb42]. Tensor categories influence the behavior of commands such as kernels and images as well as the algebraic invariants such as derivation algebras of a tensor.
Our conventions follow [Wil13]. In particular, given a tensor T framed by [Uν, ..., U0] then a tensor category for T will specify a function A:[ν]to {-1, 0, 1} along with a partition (P) of [ν] such that the following rules apply to the tensors and morphisms in the category.
We use the phrase tensor category exclusively for categories that describe tensors and tensor spaces. In other words, the data structure of a tensor category is a function A:[ν] -> {-1, 0, 1} and a partition (P) of [ν]. It is useful to distinguish from tensors and cotensors at the categorical level, so a tensor category is either covariant or contravariant as well (in the latter case, referred to as a cotensor category).
Sets up a covariant tensor space category with specified direction of arrows A, and a partition (P) indicating variables to be treated as equivalent. The fiber A - 1(1) denotes the covariant variables, A - 1(0) identifies the constant variables, and A - 1( - 1) marks the contra-variant variables.
Sets up a contra-variant tensor space category with specified direction of arrows A, and a partition (P) indicating variables to be treated as equivalent. The fiber A - 1(1) denotes the covariant variables, A - 1(0) identifies the constant variables, and A - 1( - 1) marks the contra-variant variables.
We demonstrate the basic tensor category constructor. The difference between TensorCategory and CotensorCategory is only that the former is covariant and the latter is contravariant.
> C := TensorCategory([1,0,-1], {{0},{1},{2}});
> C;
Tensor category of valence 3 (->,==,<-) ({ 1 },{ 2 },{ 0 })
> IsCovariant(C);
true
>
> arrows := map< {1..5} -> {1} | x :-> 1 >;
> C := CotensorCategory(arrows, {{1..5}});
> C;
Cotensor category of valence 6 (->,->,->,->,->,==) ({ 0 },{ 1 .. 5 })
> IsContravariant(C);
true
Contravariant: BoolElt Default: false
Returns Albert's homotopism category -- all modules categories are covariant and no duplicates considered. Set the optional parameter Contravariant to true to make it a cotensor category.
Returns the cohomotopism category -- all domain modules categories are covariant, the codomain is contravariant, and no duplicates considered.
Returns the tensor category where all modules are constant except in position s and t. Both s and t are in [v]. Position s is covariant, position t is contravariant.
Now we look at a few special tensor category constructors. The default tensor category is the homotopism category, so we construct the homotopism category using TensorCategory and verify they are equivalent.
> C := TensorCategory([1,1,1,1], {{i} : i in [0..3]});
> C;
Tensor category of valence 4 (->,->,->,->) ({ 1 },{ 2 },{ 0 },{ 3 })
> HomotopismCategory(4) eq C;
true
The other special tensor categories can be constructed using TensorCategory as well, but we just construct a few to show their properties.
> CohomotopismCategory(3);
Tensor category of valence 3 (->,->,<-) ({ 1 },{ 2 },{ 0 })
>
> AdjointCategory(5, 4, 1);
Tensor category of valence 5 (<-,==,==,->,==) ({ 1 },{ 0, 2, 3 },{ 4 })
In this section the basic operations for tensor categories are described.
Returns true if the tensor categories are the same.
Returns the valence of the tensor category.
Returns the sequence of arrows of the tensor category. A -1 signifies a contravariant index, a 0 signifies a constant index, and a 1 signifies a covariant index.
Returns the repeat partition for the tensor category.
Returns true if the tensor category is covariant or contravariant.
We obtain basic properties of tensor categories.
> C := CotensorCategory([1,0,-1,1],{{4,3},{1},{2}});
> C;
Cotensor category of valence 5 (->,==,<-,->,==) ({ 1 },{ 2 },{ 0 },{ 3, 4 })
>
> Valence(C);
5
> Arrows(C);
[ 1, 0, -1, 1 ]
> IsContravariant(C);
true
> RepeatPartition(C);
{
{ 1 },
{ 2 },
{ 3, 4 }
}
In this section, we define subtensors, local ideals, ideals, and quotients of tensors. For the following definitions fix a tensor t∈T, with frame oslasha∈[ν]Ua---that is Uν x ... x U1 ↣ U0.
A tensor s:Vν x ... x V1 ↣ V0 is a subtensor of t if for all a, Va≤Ua.
For A⊆[ν] - 0, a tensor s:Vν x ... x V1 ↣ V0 is an A-local ideal of t if s is a subtensor of t and for each a∈A, < s | Vν, ..., Va + 1, Ua, Va - 1, ..., V1 > ≤V0.
A tensor s is an ideal of t if it is a {1, ..., ν}-local ideal of t.
The A-local quotient of a tensor t by an A-local ideal s is the tensor q:Uν/Vν x ... x U1/V1 ↣ U0/V0 where for all | /line(ν) >, < q | /line(ν) > ≡ < q | ν > (mod) V0.
The quotient of a tensor t by an ideal s is the {1, ..., ν}-local quotient of t by s.
We include functions defined for the category of tensors. Most functions are currently defined only for the homotopism category.
Returns the smallest submap of T containing S.
Returns the smallest submap of T containing D in the domain and C in the codomain.
Decides whether S is a subtensor of T.
We construct the tensor t given by octonion multiplication. The quaternions H are a subalgebra of A= O generated by the first four basis elements. However, H cannot be coerced into A because of how Magma organizes algebras.
> A := OctonionAlgebra(Rationals(), -1, -1, -1); > t := Tensor(A); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 8 over Rational Field U1 : Full Vector space of degree 8 over Rational Field U0 : Full Vector space of degree 8 over Rational Field > H := sub< A | A.1, A.2, A.3, A.4 >; > H; Algebra of dimension 4 with base ring Rational Field
There are multiple ways to get the subtensor of multiplication from H. We will create H x H ↣ H as a subtensor of A x A ↣ A.
> H_gens := [A.i : i in [1..4]]; > s := Subtensor(t, [*H_gens, H_gens, A!0*]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 8, dimension 4 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) U1 : Vector space of degree 8, dimension 4 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) U0 : Vector space of degree 8, dimension 4 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0)Note that because H cannot be coerced into A (i.e. A!H produces an error), a subtensor of AlgGen cannot be done by Subtensor(t, [H, H, H]). Now we will construct the tensor straight from H. There is a subtle difference between the subtensor from A and the tensor from H---namely, the frame is different.
> s2 := Tensor(H); > s2; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 4 over Rational Field U1 : Full Vector space of degree 4 over Rational Field U0 : Full Vector space of degree 4 over Rational Field > s eq s2; false > Eltseq(s) eq Eltseq(s2); true
Returns the local ideal of T at I containing S.
Returns the local ideal of T at I containing D in the domain and C in the codomain.
Returns the local ideal of T at I containing S as a submap.
Decides if S is a local ideal of T at I.
We use the same tensor t as the previous example, multiplication in A= O , and we construct the subtensor t2 given by multiplication in H. We construct a subtensor s of t as the submap containing < A2 > x < A1, A4 > ↣ < 0 >, which is equal to < A2 > x < A1, A4 > ↣ < A2A1, A2A4 >.
> A := OctonionAlgebra(Rationals(), -1, -1, -1); > t := Tensor(A); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 8 over Rational Field U1 : Full Vector space of degree 8 over Rational Field U0 : Full Vector space of degree 8 over Rational Field > H_gens := [A.i : i in [1..4]]; > t2 := Subtensor(t, [*H_gens, H_gens, H_gens*]); > s := Subtensor(t, [* A.2, [A.1, A.4], A!0 *]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 8, dimension 1 over Rational Field Generators: (0 1 0 0 0 0 0 0) Echelonized basis: (0 1 0 0 0 0 0 0) U1 : Vector space of degree 8, dimension 2 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 0 0 1 0 0 0 0) U0 : Vector space of degree 8, dimension 2 over Rational Field Generators: (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) Echelonized basis: (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0)
The quaternions are a subalgebra of A generated by {A1, A2, A3, A4}. Therefore, the {2}-local ideal of s in t must contain A in the codomain. However, the {2}-local ideal of s in t2 must only contain < A1, A2, A3, A4} in the codomain.
> s1 := LocalIdeal(t, s, {2});
> Codomain(s1);
Full Vector space of degree 8 over Rational Field
Generators:
(1 0 0 0 0 0 0 0)
(0 1 0 0 0 0 0 0)
(0 0 1 0 0 0 0 0)
(0 0 0 1 0 0 0 0)
(0 0 0 0 1 0 0 0)
(0 0 0 0 0 1 0 0)
(0 0 0 0 0 0 1 0)
(0 0 0 0 0 0 0 1)
> s2 := LocalIdeal(t2, s, {2});
> Codomain(s2);
Vector space of degree 8, dimension 4 over Rational Field
Generators:
(1 0 0 0 0 0 0 0)
(0 1 0 0 0 0 0 0)
(0 0 1 0 0 0 0 0)
(0 0 0 1 0 0 0 0)
Echelonized basis:
(1 0 0 0 0 0 0 0)
(0 1 0 0 0 0 0 0)
(0 0 1 0 0 0 0 0)
(0 0 0 1 0 0 0 0)
Returns the ideal of T containing S.
Returns the ideal of T containing D in the domain and C in the codomain.
Returns the ideal of T containing S as a submap.
Decides if S is an ideal of T.
First we will construct the tensor from the Q-algebra, Q5.
> T := KTensorSpace(Rationals(), [5,5,5]);
> A := VectorSpace(Rationals(), 5);
> t := T!0;
> for i in [1..5] do
> Assign(~t, [i,i,i], 1);
> end for;
> t;
Tensor of valence 3, U2 x U1 >-> U0
U2 : Full Vector space of degree 5 over Rational Field
U1 : Full Vector space of degree 5 over Rational Field
U0 : Full Vector space of degree 5 over Rational Field
> SystemOfForms(t);
[
[1 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0],
[0 0 0 0 0]
[0 1 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0],
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 1 0 0]
[0 0 0 0 0]
[0 0 0 0 0],
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 1 0]
[0 0 0 0 0],
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 1]
]
Now we will construct the ideal tensor from the subtensor containing < A1 > x < A2 > ↣ < A3 >. Note that the {2}-local ideal must include < A2, A3 > in the codomain, and the {1}-local ideal must contain < A1, A3 > in the codomain.
> s := Ideal(t, [A.1, A.2, A.3]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 5, dimension 1 over Rational Field Generators: (1 0 0 0 0) Echelonized basis: (1 0 0 0 0) U1 : Vector space of degree 5, dimension 1 over Rational Field Generators: (0 1 0 0 0) Echelonized basis: (0 1 0 0 0) U0 : Vector space of degree 5, dimension 3 over Rational Field Generators: (1 0 0 0 0) (0 1 0 0 0) (0 0 1 0 0) Echelonized basis: (1 0 0 0 0) (0 1 0 0 0) (0 0 1 0 0)
Finally, we verify that the subtensor containing < A1 > x < A2 > ↣ < A2, A3 > is not an ideal.
> r := Subtensor(t, [A.1, A.2], [A.2, A.3]); > IsIdeal(t, r); false
Check: BoolElt Default: true
Returns the local quotient of T by S at I⊆[ν] - 0. If you know S is a local ideal of T at I, set Check to false to skip the verification. A homotopism is also returned, mapping from T to T/S.
Check: BoolElt Default: true
Returns the quotient of T by S. If you know S is an ideal of T, set Check to false to skip the verification. A homotopism is also returned, mapping from T to T/S.
We will demonstrate one of the most common uses for quotienting tensors: constructing the associated fully nondegenerate tensor. We first construct a tensor with a nontrivial radical given by matrix multiplication: (Mat)3 x 2(Q) x Q3 ↣ Q3, where we take a projection of Q3 onto Q2 in the 1 coordinate.
> K := Rationals(); > F := [*KMatrixSpace(K, 3, 2), VectorSpace(K, 3), VectorSpace(K, 3)*]; > mult := function(x) > return Transpose(x[1]*Matrix(2, 1, Eltseq(x[2])[2..3])); > end function; > t := Tensor(F, mult); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 6 over Rational Field U1 : Full Vector space of degree 3 over Rational Field U0 : Full Vector space of degree 3 over Rational Field > s := Subtensor(t, [*[F[1].1, F[1].4], F[2], F[3]*]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 6, dimension 2 over Rational Field Generators: (1 0 0 0 0 0) (0 0 0 1 0 0) Echelonized basis: (1 0 0 0 0 0) (0 0 0 1 0 0) U1 : Full Vector space of degree 3 over Rational Field Generators: (1 0 0) (0 1 0) (0 0 1) U0 : Full Vector space of degree 3 over Rational Field Generators: (1 0 0) (0 1 0) (0 0 1) > IsFullyNondegenerate(s); false
Now we will construct the ideal r that evaluates to 0 and the largest subspace of Q3 not contained in the image.
> r := Ideal(t, [*F[1]!0, F[2].1, F[3].3*]); > r; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 6, dimension 0 over Rational Field Generators: U1 : Vector space of degree 3, dimension 1 over Rational Field Generators: (1 0 0) Echelonized basis: (1 0 0) U0 : Vector space of degree 3, dimension 1 over Rational Field Generators: (0 0 1) Echelonized basis: (0 0 1) > IsIdeal(t, r); true
Finally, we quotient s by r to obtain a fully nondegenerate tensor.
> q := s/r; > q; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 2 over Rational Field > IsFullyNondegenerate(q); true
We have categorical notions for tensor spaces as well, and these are inherited from the module structure on tensor spaces.
Returns the subtensor space of T generated by the tensors in the sequence L.
Decides if the tensor space S is a subtensor space of T.
We will construct the subspace S of symmetric forms from the tensor space T with frame Q2 x Q2 ↣ Q.
> K := Rationals(); > T := KTensorSpace(K, [2,2,1]); > T; Tensor space of dimension 4 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > S := sub< T | T.1, T.2+T.3, T.4 >; > S; Tensor space of dimension 3 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > IsSymmetric(S); true
Now we will construct the subspace A of alternating forms from T.
> A := sub< T | T.2-T.3 >; > A; Tensor space of dimension 1 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > IsAlternating(A); true
Now we verify that A is not a subtensor space of S, and thus, we have constructed a direct decomposition of T into its symmetric and alternating space.
> IsSubtensorSpace(S, A); false
Returns the quotient tensor space of T by S.
We pick up with the same tensor spaces as the previous example: T has frame Q2 x Q2 ↣ Q, S is the symmetric subspace, and A is the alternating subspace.
> K := Rationals(); > T := KTensorSpace(K, [2,2,1]); > T; Tensor space of dimension 4 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > S := sub< T | T.1, T.2+T.3, T.4 >; > A := sub< T | T.2-T.3 >;
Now we construct the quotient of T by A. The result is not a symmetric tensor space. Note that Q2 is equivalent to a symmetric tensor modulo A, but this choice is arbitrary.
> Q := T/A;
> Q;
Tensor space of dimension 3 over Rational Field with valence 3
U2 : Full Vector space of degree 2 over Rational Field
U1 : Full Vector space of degree 2 over Rational Field
U0 : Full Vector space of degree 1 over Rational Field
> SystemOfForms(Q.1);
[
[1 0]
[0 0]
]
> SystemOfForms(Q.2);
[
[0 0]
[1 0]
]
> SystemOfForms(Q.3);
[
[0 0]
[0 1]
]
Magma provides functions for homotopisms, i.e. morphisms of tensors. Homotopisms are also equipped with a tensor category. Because homotopisms can contain multiple maps between modules, it is not really clear what is meant by the domain or codomain of a given homotopism. In this context, the domain of a homotopism H will refer to the tensor, t:Uν x ... x U1 ↣ U0, where an arrow equal to 1 at coordinate a will mean that the corresponding map at coordinate a has domain equal to Ua. And in the same vain, the tensor s:Vν x ... x V1 ↣ V0 is the codomain of H if an arrow equal to 1 at coordinate a implies that the corresponding map at coordinate a has codomain equal to Va.
Check: BoolElt Default: true
Check: BoolElt Default: true
Returns the homotopism from T to S given by the list of maps M and the category C. The default tensor category is the same as tensor categories for T and S. If the maps M will produce a homotopism, then set Check to false to skip the verification.
Returns the homotopism given by the maps in M with tensor category C.
Decides if the list of maps M induces a homotopism from T to S in the tensor category C. The default tensor category is the homotopism category. If it does induce a homotopism, it is also returned.
We will construct two symmetric tensors t, s: GF(3)3 x GF(3)3 ↣ GF(3)3 and apply permutations to the bases.
> T := KTensorSpace(GF(3), [3,3,3]);
> t := T.1+T.14+T.27;
> SystemOfForms(t);
[
[1 0 0]
[0 0 0]
[0 0 0],
[0 0 0]
[0 1 0]
[0 0 0],
[0 0 0]
[0 0 0]
[0 0 1]
]
> s := (T.4+T.10)+(T.8+T.20)+(T.18+T.24);
> SystemOfForms(s);
[
[0 1 0]
[1 0 0]
[0 0 0],
[0 0 1]
[0 0 0]
[1 0 0],
[0 0 0]
[0 0 1]
[0 1 0]
]
Now we construct a homotopism from t to t given by apply a permutation matrix in every coordinate.
> P := PermutationMatrix(GF(3), [2,1,3]); > H := Homotopism(t, t, [*P, P, P*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: [0 1 0] [1 0 0] [0 0 1] U1 -> V1: [0 1 0] [1 0 0] [0 0 1] U0 -> V0: [0 1 0] [1 0 0] [0 0 1]
Note that this permutation matrix does not induce a homotopism of s.
> IsHomotopism(s, s, [*P, P, P*]); false
> V := VectorSpace(GF(3), 3); > T := TensorSpace([V, V, V]); > t := T.1+T.14+T.27; > P := PermutationMatrix(GF(3), [2,1,3]); > f := hom< V -> V | [<V.1, V.2>, <V.2, V.1>, <V.3, V.3>] >; > H := Homotopism(t, t, [*f, f, f*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3) U1 -> V1: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3) U0 -> V0: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3)
Furthermore, we can input lists with types Mtrx and Map included.
> H2 := Homotopism(t, t, [*P, f, P*]); > H2; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: [0 1 0] [1 0 0] [0 0 1] U1 -> V1: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3) U0 -> V0: [0 1 0] [1 0 0] [0 0 1]
We provide some operations for homotopisms.
Returns the composition of the homotopisms H1 and H2.
Returns the map on the ath coordinate.
We construct a nondegenerate alternating form t on V=Q6. The group of isometries are isomorphic to (Sp)(6, Q), the group generated by all transvections. We construct a transvection L and a corresponding matrix.
> V := VectorSpace(Rationals(), 6); > T := KTensorSpace(Rationals(), [6, 6, 1]); > t := T.2-T.7+T.16-T.21+T.30-T.35; > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 6 over Rational Field U1 : Full Vector space of degree 6 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > > u := V.2+2*V.3-V.5; > L := map< V -> V | x :-> x + (x*t*u)[1]*u >; > L; Mapping from: ModTupFld: V to ModTupFld: V given by a rule [no inverse] > M := Matrix(6, 6, [V.i @ L : i in [1..6]]); > M; [ 1 1 2 0 -1 0] [ 0 1 0 0 0 0] [ 0 0 1 0 0 0] [ 0 -2 -4 1 2 0] [ 0 0 0 0 1 0] [ 0 1 2 0 -1 1]
We construct a homotopism from the transvection.
> H := Homotopism(t, t, [*L, L, IdentityMatrix(Rationals(), 1)*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U1 -> V1: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U0 -> V0: [1]
Since H is an isometry of t, H2 is also an isometry of t. We verify that the 2-coordinate map of H2 is exactly M2.
> H2 := H*H; > H2; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field Composition of Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] and Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U1 -> V1: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field Composition of Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] and Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U0 -> V0: [1] > M2 := Matrix(6, 6, [V.i @ H2.2 : i in [1..6]]); > M2; [ 1 2 4 0 -2 0] [ 0 1 0 0 0 0] [ 0 0 1 0 0 0] [ 0 -4 -8 1 4 0] [ 0 0 0 0 1 0] [ 0 2 4 0 -2 1] > M^2 eq M2; true
If a>0, then the tensor returned is the tensor that has been pre-composed by the map f or matrix M.
If H is a cohomotopism (a homotopism in the cohomotopism category), then either the domain or codomain of H is returned, depending on the orientation of the arrows of H.
Returns the domain tensor of H.
Returns the codomain tensor of H.
Returns the list of maps for the various modules in the domain and codomain tensors.
Returns the tensor category of H.
Changes the tensor category of H to the given category.
Returns the valence of the underlying tensor category of the homotopism H.
Returns the kernel of H as an ideal of its domain tensor.
Returns the image of H as a submap of the codomain tensor.
We demonstrate how to access properties of a homotopism. We construct tensors t:Q4 x Q4 ↣ Q and s:Q6 x Q6 ↣ Q given by the dot product.
> t := Tensor(IdentityMatrix(Rationals(), 4), 2, 1); > s := Tensor(IdentityMatrix(Rationals(), 6), 2, 1); > Z := ZeroMatrix(Rationals(), 4, 6); > M := InsertBlock(Z, IdentityMatrix(Rationals(), 4), 1, 1); > H := Homotopism(t, s, [*M, M, IdentityMatrix(Rationals(), 1)*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] U1 -> V1: [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] U0 -> V0: [1]
Like with maps, we can obtain standard properties of homotopisms.
> Domain(H);
Tensor of valence 3, U2 x U1 >-> U0
U2 : Full Vector space of degree 4 over Rational Field
U1 : Full Vector space of degree 4 over Rational Field
U0 : Full Vector space of degree 1 over Rational Field
> Codomain(H);
Tensor of valence 3, U2 x U1 >-> U0
U2 : Full Vector space of degree 6 over Rational Field
U1 : Full Vector space of degree 6 over Rational Field
U0 : Full Vector space of degree 1 over Rational Field
> Maps(H);
[*
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0],
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0],
[1]
*]
> TensorCategory(H);
Tensor category of valence 3 (->,->,->) ({ 1 },{ 2 },{ 0 })
When the image and kernel can be computed for the each of the maps in the homotopism, then the image and kernel can be computed for the homotopism.
> Im := Image(H); > Im; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 6, dimension 4 over Rational Field Echelonized basis: (1 0 0 0 0 0) (0 1 0 0 0 0) (0 0 1 0 0 0) (0 0 0 1 0 0) U1 : Vector space of degree 6, dimension 4 over Rational Field Echelonized basis: (1 0 0 0 0 0) (0 1 0 0 0 0) (0 0 1 0 0 0) (0 0 0 1 0 0) U0 : Full Vector space of degree 1 over Rational Field > Ker := Kernel(H); > Ker; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 4, dimension 0 over Rational Field U1 : Vector space of degree 4, dimension 0 over Rational Field U0 : Vector space of degree 1, dimension 0 over Rational Field
Just like the shuffle for tensors, this returns the shuffle of the homotopism H. This is a functor from one tensor category to another and changes the order of the maps to {Hνg, ..., H1g, H0g}. In order to be defined, g∈Sym({0, ..., ν }). If 0g≠0, then both the image and pre-image of 0 under g will be replaced by their K-dual space. For cotensors, g∈Sym({1, ..., ν}). Sequences [a1, ..., aν + 1] will be interpreted as a permutation in one-line notation.