‣ IsCommutative( C ) | ( property ) |
Returns: true or false
Check whether the finitely presented category C is commutative.
‣ IsCounitary( B ) | ( property ) |
Returns: true or false
Check whether B is counitary.
‣ IsCoassociative( B ) | ( property ) |
Returns: true or false
Check whether B is coassociative.
‣ UnderlyingQuiver( C ) | ( attribute ) |
Returns: a QPA quiver
The quiver underlying the finitely presented category C.
‣ UnderlyingQuiverAlgebra( C ) | ( attribute ) |
Returns: a QPA path algebra
The quiver algebra (=path algebra with relations) underlying the finitely presented category C.
‣ Size( C ) | ( attribute ) |
Returns: a nonnegative integer
The number of morphisms in the finitely presented category C.
‣ BasisPathsByVertexIndex( C ) | ( attribute ) |
Returns: a matrix of basis paths of a QPA path algebra
The matrix of basis paths of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C, indexed by the vertex indices of source and target of the path.
‣ BasisMorphismsByVertexIndex( A ) | ( attribute ) |
Returns: a matrix of basis morphisms
The matrix of basis morphisms of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C, indexed by the vertex indices of source and target of the morphism.
‣ HomStructureOnBasisPaths( C ) | ( attribute ) |
Returns: a six-dimensional matrix of matrices
The hom structure on basis paths of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C: HomStructureOnBasisPaths( A )[ v_index ][ w_index ][ v'_index ][ w'_index ][ basis_path_1_index ][ basis_path_2_index ] = [ Hom(v,w) -> Hom(v',w'): x -> basis_path_1 * x * basis_path_2 ] for basis_path_1: v' -> v and basis_path_2: w -> w'.
‣ AssignSetOfObjects( C, label ) | ( operation ) |
Returns: nothing
Assigns the objects of the finitely presented category C to global variables. Names of the variables are the concatenation of label with the names of the defining vertices.
‣ SetOfGeneratingMorphisms( C, obj_1, obj_2 ) | ( operation ) |
Returns: a list
The subset of the generating morphisms that start at obj_1 and ends at obj_2.
‣ SetOfGeneratingMorphisms( obj_1, obj_2 ) | ( operation ) |
Returns: a list
The subset of the generating morphisms that start at obj_1 and ends at obj_2.
‣ SetOfGeneratingMorphisms( C, i, j ) | ( operation ) |
Returns: a list
Delegates to SetOfGeneratingMorphisms( C, SetOfObjects(C)[i], SetOfObjects(C)[j] ).
‣ AssignSetOfGeneratingMorphisms( C, label ) | ( operation ) |
Returns: nothing
Assigns the generating morphisms of the finitely presented category C to global variables. Names of the variables are the concatenation of label with the names of the defining arrows.
‣ RelationsOfFpCategory( C ) | ( attribute ) |
Returns: a QPA path algebra
The relations of the finitely presented category C corresponding to RelationsOfAlgebra( UnderlyingQuiverAlgebra( C ) ).
‣ OppositeFpCategory( C ) | ( attribute ) |
Returns: a CAP category
The finitely presented category defined by the opposite of the underlying quiver with relations.
‣ Antipode( B ) | ( attribute ) |
Returns: a CAP functor
The antipode of the Hopf finitely presented category B.
‣ UnderlyingVertex( obj ) | ( attribute ) |
Returns: a vertex in a QPA quiver
The vertex of the quiver underlying the object obj in a finitely presented category.
‣ UnderlyingQuiverAlgebraElement( mor ) | ( attribute ) |
Returns: an element in a QPA path algebra
The quiver algebra element underlying the morphism mor in a finitely presented category.
‣ UnderlyingAlgebra( C ) | ( attribute ) |
Returns: a ring
The underlying algebra of the finitely presented category C.
‣ Parity( C ) | ( attribute ) |
Returns: a string ("left" or "right")
The parity of the finitely presented category C.
‣ POW( C, n ) | ( operation ) |
Returns: a CAP category
The n-th power of the finitely presented category C. Admissible values for n are \(0,1,2\).
‣ NerveTruncatedInDegree2AsFunctor( C ) | ( attribute ) |
Returns: a CAP functor
The input is a finitely presented category C equipped with a homomorphism structure with values in the skeletal category SkeletalFinSets of finite sets. The output is the nerve of B truncated in degree \(2\), as a presheaf on SimplicialCategoryTruncatedInDegree(\(2\)) with values in SkeletalFinSets.
We compute the nerve of the full subcategory of the simplicial category \(\Delta\) on the objects \([0], [1], [2]\).
gap> Delta2 := SimplicialCategoryTruncatedInDegree( 2 ); FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations gap> RelationsOfFpCategory( Delta2 ); [ [ (s*id), (C0) ], [ (t*id), (C0) ], [ (ps*is), (C1) ], [ (pt*it), (C1) ], [ (is*id), (it*id) ], [ (pt*is), (id*t) ], [ (ps*it), (id*s) ], [ (s*pt), (t*ps) ], [ (s*mu), (s*ps) ], [ (t*mu), (t*pt) ], [ (mu*is), (C1) ], [ (mu*it), (C1) ] ] gap> Size( Delta2 ); 31 gap> N := NerveTruncatedInDegree2AsFunctor( Delta2 ); Functor from FreeCategory( RightQuiver( "Delta_op(C0,C1,C2)[id:C0->C1,s:C1->C0,t:C1->C0, is:C1->C2,it:C1->C2, ps:C2->C1,pt:C2->C1,mu:C2->C1]" ) ) / relations -> SkeletalFinSets gap> Delta2op := SourceOfFunctor( N ); FreeCategory( RightQuiver( "Delta_op(C0,C1,C2)[id:C0->C1,s:C1->C0,t:C1->C0, is:C1->C2,it:C1->C2, ps:C2->C1,pt:C2->C1,mu:C2->C1]" ) ) / relations gap> N( Delta2op.C0 ); |3| gap> Display( N( Delta2op.C0 ) ); { 0, 1, 2 } gap> N( Delta2op.C1 ); |31| gap> Display( N( Delta2op.C1 ) ); { 0,..., 30 } gap> N( Delta2op.C2 ); |393| gap> Display( N( Delta2op.C2 ) ); { 0,..., 392 } gap> N( Delta2op.id ); |3| → |31| gap> Display( N( Delta2op.id ) ); { 0, 1, 2 } ⱶ[ 0, 5, 21 ]→ { 0,..., 30 }
‣ YonedaFibrationAsNaturalTransformation( B ) | ( attribute ) |
Returns: a CAP natural transformation
The input is a finitely presented category B. The output is a natural morphism. Its source is the functor \(B \to H, c \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,c), \psi \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,\psi)\). Its targe is the constant functor of \(0\)-cells \(B \to H, c \mapsto B_0, \psi \mapsto \mathrm{id}_{B_0}\).
‣ YonedaProjectionAsNaturalEpimorphism( B ) | ( attribute ) |
Returns: a CAP natural transformation
The input is a finitely presented category B. The output is a natural epimorphism. Its source is the functor \(B \to H, c \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(a,b) \times \mathrm{Hom}(b,c), \psi \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(1_a,1_b) \times \mathrm{Hom}(b,\psi)\). Its target is the functor \(B \to H, c \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,c), \psi \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,\psi)\).
‣ YonedaCompositionAsNaturalEpimorphism( B ) | ( attribute ) |
Returns: a CAP natural transformation
The input is a finitely presented category B. The output is a natural epimorphism. Its source is the functor \(B \to H, c \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(a,b) \times \mathrm{Hom}(b,c), \psi \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(1_a,1_b) \times \mathrm{Hom}(b,\psi)\). Its target is the functor \(B \to H, c \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,c), \psi \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,\psi)\).
We compute the Yoneda composition natural epimorphism of the full subcategory of the simplicial category \(\Delta\) on the objects \([0], [1], [2]\).
gap> Delta2 := SimplicialCategoryTruncatedInDegree( 2 ); FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations gap> RelationsOfFpCategory( Delta2 ); [ [ (s*id), (C0) ], [ (t*id), (C0) ], [ (ps*is), (C1) ], [ (pt*it), (C1) ], [ (is*id), (it*id) ], [ (pt*is), (id*t) ], [ (ps*it), (id*s) ], [ (s*pt), (t*ps) ], [ (s*mu), (s*ps) ], [ (t*mu), (t*pt) ], [ (mu*is), (C1) ], [ (mu*it), (C1) ] ] gap> Size( Delta2 ); 31 gap> Ymu := YonedaCompositionAsNaturalEpimorphism( Delta2 ); Natural transformation from Functor from FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations -> SkeletalFinSets -> Functor from FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations -> SkeletalFinSets gap> Ymu := YonedaProjectionAsNaturalEpimorphism( Delta2 ); Natural transformation from Functor from FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations -> SkeletalFinSets -> Functor from FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations -> SkeletalFinSets gap> Ys := YonedaFibrationAsNaturalTransformation( Delta2 ); Natural transformation from Functor from FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations -> SkeletalFinSets -> Functor from FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations -> SkeletalFinSets
‣ TruthMorphismOfTrueToSieveFunctor( B ) | ( attribute ) |
Returns: a CAP functor
Return the truth morphism of true from terminal functor to the functor of sieves from OppositeFpCategory( B ) to RangeCategoryOfHomomorphismStructure( B ).
gap> q := RightQuiver( "q(1)[a:1->1]" ); q(1)[a:1->1] gap> Fq := FreeCategory( q ); FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) gap> M0 := Fq / [ [ Fq.a^3, Fq.1 ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = 1 ] gap> S0 := SieveFunctor( M0 ); Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ] -> SkeletalFinSets gap> M0op := AsCapCategory( Source( S0 ) ); FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ] gap> S0( M0op.1 ); |2| gap> Display( S0( M0op.1 ) ); { 0, 1 } gap> S0( M0op.a ); |2| → |2| gap> Display( S0( M0op.a ) ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 } gap> M1 := Fq / [ [ Fq.a^3, Fq.a ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a ] gap> S1 := SieveFunctor( M1 ); Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ] -> SkeletalFinSets gap> M1op := AsCapCategory( Source( S1 ) ); FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ] gap> S1( M1op.1 ); |3| gap> Display( S1( M1op.1 ) ); { 0, 1, 2 } gap> S1( M1op.a ); |3| → |3| gap> Display( S1( M1op.a ) ); { 0, 1, 2 } ⱶ[ 0, 2, 2 ]→ { 0, 1, 2 } gap> M2 := Fq / [ [ Fq.a^3, Fq.a^2 ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a ] gap> S2 := SieveFunctor( M2 ); Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets gap> M2op := AsCapCategory( Source( S2 ) ); FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] gap> S2( M2op.1 ); |4| gap> Display( S2( M2op.1 ) ); { 0 ,..., 3 } gap> S2( M2op.a ); |4| → |4| gap> Display( S2( M2op.a ) ); { 0,..., 3 } ⱶ[ 0, 2, 3, 3 ]→ { 0,..., 3 } gap> eta2 := TruthMorphismOfTrueToSieveFunctor( M2 ); Natural transformation from Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets -> Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets gap> eta2( M2op.1 ); |1| → |4| gap> Display( eta2( M2op.1 ) ); { 0 } ⱶ[ 3 ]→ { 0,..., 3 } gap> iota2 := EmbeddingOfSieveFunctor( M2 ); Natural transformation from Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets -> Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets gap> iota2( M2op.1 ); |4| → |8| gap> Display( iota2( M2op.1 ) ); { 0,..., 3 } ⱶ[ 0, 4, 6, 7 ]→ { 0,..., 7 }
‣ SieveFunctor( B ) | ( attribute ) |
Returns: a CAP functor
Return the functor of sieves from OppositeFpCategory( B ) to RangeCategoryOfHomomorphismStructure( B ).
gap> q := RightQuiver( "q(1)[a:1->1]" ); q(1)[a:1->1] gap> Fq := FreeCategory( q ); FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) gap> M0 := Fq / [ [ Fq.a^3, Fq.1 ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = 1 ] gap> S0 := SieveFunctor( M0 ); Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ] -> SkeletalFinSets gap> M0op := AsCapCategory( Source( S0 ) ); FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ] gap> S0( M0op.1 ); |2| gap> Display( S0( M0op.1 ) ); { 0, 1 } gap> S0( M0op.a ); |2| → |2| gap> Display( S0( M0op.a ) ); { 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 } gap> M1 := Fq / [ [ Fq.a^3, Fq.a ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a ] gap> S1 := SieveFunctor( M1 ); Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ] -> SkeletalFinSets gap> M1op := AsCapCategory( Source( S1 ) ); FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ] gap> S1( M1op.1 ); |3| gap> Display( S1( M1op.1 ) ); { 0, 1, 2 } gap> S1( M1op.a ); |3| → |3| gap> Display( S1( M1op.a ) ); { 0, 1, 2 } ⱶ[ 0, 2, 2 ]→ { 0, 1, 2 } gap> M2 := Fq / [ [ Fq.a^3, Fq.a^2 ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a ] gap> S2 := SieveFunctor( M2 ); Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets gap> M2op := AsCapCategory( Source( S2 ) ); FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] gap> S2( M2op.1 ); |4| gap> Display( S2( M2op.1 ) ); { 0 ,..., 3 } gap> S2( M2op.a ); |4| → |4| gap> Display( S2( M2op.a ) ); { 0,..., 3 } ⱶ[ 0, 2, 3, 3 ]→ { 0,..., 3 } gap> eta2 := TruthMorphismOfTrueToSieveFunctor( M2 ); Natural transformation from Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets -> Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets gap> eta2( M2op.1 ); |1| → |4| gap> Display( eta2( M2op.1 ) ); { 0 } ⱶ[ 3 ]→ { 0,..., 3 } gap> iota2 := EmbeddingOfSieveFunctor( M2 ); Natural transformation from Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets -> Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ] -> SkeletalFinSets gap> iota2( M2op.1 ); |4| → |8| gap> Display( iota2( M2op.1 ) ); { 0,..., 3 } ⱶ[ 0, 4, 6, 7 ]→ { 0,..., 7 }
‣ TrivialCategory( str ) | ( operation ) |
2.3-2 \*‣ \*( C, D ) | ( operation ) |
‣ ElementaryTensor( a, b, T ) | ( operation ) |
Returns: a morphism in a CAP category
Given an object a in a finitely presented category A and an object b in a finitely presented category B and the tensor product T of A and B, return the tensor product of a and b in T.
‣ ElementaryTensor( a, g, T ) | ( operation ) |
Returns: a morphism in a CAP category
Given an object a in a finitely presented category A and a morphism g in a finitely presented category B and the tensor product T of A and B, return the tensor product of a and g in T.
‣ ElementaryTensor( f, b, T ) | ( operation ) |
Returns: a morphism in a CAP category
Given a morphism f in a finitely presented category A and an object b in a finitely presented category B and the tensor product T of A and B, return the tensor product of f and b in T.
‣ QuiverVertexAsIdentityPath( vertex ) | ( operation ) |
Returns: a path
Simply returns vertex, but with the semantics of being an identity path.
‣ FreeCategory( q ) | ( operation ) |
‣ Category( q, L ) | ( operation ) |
‣ QuotientCategory( C, L ) | ( operation ) |
‣ /( C, L ) | ( operation ) |
Returns: a CAP category
Construct the finitely presented category generated by the quiver q, possibly modulo the relations L.
‣ ObjectInFpCategory( A, V ) | ( operation ) |
Returns: an object in a CAP category
The constructor of objects in a finitely presented category C given a vertex V in the underlying quiver.
‣ MorphismInFpCategory( S, path, T ) | ( operation ) |
‣ MorphismInFpCategory( A, path ) | ( operation ) |
Returns: an object in a CAP category a morphism in a CAP category
Delegates to ObjectInFpCategory( C, V ). The constructor of morphisms in a finitely presented category C given the source S, the target T, and the underlying path path. If neither S nor T are provided they are read off from path.
2.4-4 \/‣ \/( path, A ) | ( operation ) |
Returns: a morphism in a CAP category
Delegates to MorphismInFpCategory( path ).
‣ SimplicialCategoryTruncatedInDegree( n ) | ( function ) |
Returns: a CAP category
The full subcategory of the simplicial category \(\Delta\) on the objects \([0], \ldots, [n]\).
‣ IsFpCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of finitely presented categories.
‣ IsMonoidAsCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of algebras.
‣ IsCellInFpCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of cells in a finitely presented category.
‣ IsObjectInFpCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of objects in a finitely presented category.
‣ IsMorphismInFpCategory( arg ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a finitely presented category.
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