From 8d021fa3f07aabf575166c87ccf6f94de7addfc1 Mon Sep 17 00:00:00 2001 From: Mohamed Barakat Date: Mon, 2 May 2022 22:18:55 +0200 Subject: [PATCH] comply with FunctorCategories v2022.04-04 --- PackageInfo.g | 4 +- examples/CategoryOfRepresentations.g | 130 +++++++------------ examples/DecomposeOnceByRandomEndomorphism.g | 42 +++--- examples/RepresentingC4C4.g | 17 +-- 4 files changed, 73 insertions(+), 120 deletions(-) diff --git a/PackageInfo.g b/PackageInfo.g index 350da99..5b6ae3f 100644 --- a/PackageInfo.g +++ b/PackageInfo.g @@ -10,7 +10,7 @@ SetPackageInfo( rec( PackageName := "CatReps", Subtitle := "Representations and cohomology of finite categories", -Version := "2022.04-05", +Version := "2022.04-06", Date := ~.Version{[ 1 .. 10 ]}, Date := Concatenation( "01/", ~.Version{[ 6, 7 ]}, "/", ~.Version{[ 1 .. 4 ]} ), @@ -109,7 +109,7 @@ Dependencies := rec( [ "MatricesForHomalg", ">= 2020.02.02" ], [ "Toposes", ">= 2022.04-29" ], [ "Algebroids", ">= 2022.01-02" ], - [ "FunctorCategories", ">= 2022.02-01" ], + [ "FunctorCategories", ">= 2022.04-04" ], ], SuggestedOtherPackages := [ ], ExternalConditions := [ ], diff --git a/examples/CategoryOfRepresentations.g b/examples/CategoryOfRepresentations.g index e0e5128..cb4e5c6 100644 --- a/examples/CategoryOfRepresentations.g +++ b/examples/CategoryOfRepresentations.g @@ -42,11 +42,6 @@ CommutativeRingOfLinearCategory( CatReps ); zero := ZeroObject( CatReps ); #! <(1)->0, (2)->0; (a)->0x0, (b)->0x0, (c)->0x0> Display( zero ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 0 #! @@ -58,25 +53,22 @@ Display( zero ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! (an empty 0 x 0 matrix) #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! (an empty 0 x 0 matrix) #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data unit := TensorUnit( CatReps ); #! <(1)->1, (2)->1; (a)->1x1, (b)->1x1, (c)->1x1> Display( unit ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 1 #! @@ -88,28 +80,25 @@ Display( unit ); #! #! An identity morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! 1 #! #! An identity morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 #! #! An identity morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data d := [[1,1,0,0,0],[0,1,1,0,0],[0,0,1,0,0],[0,0,0,1,1],[0,0,0,0,1]];; e := [[0,1,0,0],[0,0,1,0],[0,0,0,0],[0,1,0,1],[0,0,1,0]];; f := [[1,1,0,0],[0,1,1,0],[0,0,1,0],[0,0,0,1]];; nine := AsObjectInFunctorCategory( A, [ 5, 4 ], [ d, e, f ] ); #! <(1)->5, (2)->4; (a)->5x5, (b)->5x4, (c)->4x4> Display( nine ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 5 #! @@ -125,7 +114,6 @@ Display( nine ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! . 1 . . #! . . 1 . @@ -135,7 +123,6 @@ Display( nine ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 1 . . #! . 1 1 . @@ -143,6 +130,10 @@ Display( nine ); #! . . . 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data nine( A.1 ); #! nine( A.2 ); @@ -176,11 +167,6 @@ fortyone( A.b ) = TensorProductOnMorphisms( nine( A.b ), nine( A.b ) ); fortyone( A.c ) = TensorProductOnMorphisms( nine( A.c ), nine( A.c ) ); #! true Display( fortyone ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 25 #! @@ -216,7 +202,6 @@ Display( fortyone ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! . . . . . 1 . . . . . . . . . . #! . . . . . . 1 . . . . . . . . . @@ -246,7 +231,6 @@ Display( fortyone ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 1 . . 1 1 . . . . . . . . . . #! . 1 1 . . 1 1 . . . . . . . . . @@ -266,6 +250,10 @@ Display( fortyone ); #! . . . . . . . . . . . . . . . 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data etas := WeakDirectSumDecomposition( fortyone : random := false );; dec := List( etas, eta -> List( SetOfObjects( A ), o -> Dimension( Source( eta( o ) ) ) ) ); @@ -276,11 +264,6 @@ iso := UniversalMorphismFromDirectSum( etas ); IsIsomorphism( iso ); #! true Display( Source( iso ) ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 25 #! @@ -316,7 +299,6 @@ Display( Source( iso ) ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! . . . . . . . . . . . . . . . . #! . . . . . . . . . . . . . . . . @@ -346,7 +328,6 @@ Display( Source( iso ) ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 . . . . . . . . . . . . . . . #! . 1 1 . . . . . . . . . . . . . @@ -366,12 +347,11 @@ Display( Source( iso ) ); #! . . . . . . . . . . . . . . . 1 #! #! A morphism in Category of matrices over GF(3) -Display( iso ); -#! A morphism in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! #! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data +Display( iso ); #! Image of <(1)>: #! . . 1 . . . 2 1 . 1 1 . . 2 . . . 1 . . . . . . . #! . . . . . . . . . . . . . . . . . . . . . . 1 . . @@ -401,7 +381,6 @@ Display( iso ); #! #! An isomorphism in Category of matrices over GF(3) #! -#! #! Image of <(2)>: #! . . 2 . . 1 . . 2 2 1 . . . . . #! 2 1 . . . . . . . . . . . 1 . . @@ -421,6 +400,10 @@ Display( iso ); #! . . . . . . . . . . . . . . . 1 #! #! An isomorphism in Category of matrices over GF(3) +#! +#! A morphism in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data eta := etas[9]; #! <(1)->3x25, (2)->3x16> TensorProductOnMorphisms( eta, eta ); @@ -428,11 +411,6 @@ TensorProductOnMorphisms( eta, eta ); six := Source( eta ); #! <(1)->3, (2)->3; (a)->3x3, (b)->3x3, (c)->3x3> Display( six ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 3 #! @@ -446,7 +424,6 @@ Display( six ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! . . 1 #! . . . @@ -454,39 +431,35 @@ Display( six ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 1 . #! . 1 1 #! . . 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data emb := EmbeddingOfSumOfImagesOfAllMorphisms( unit, six ); #! <(1)->1x3, (2)->0x3> Display( emb ); -#! A morphism in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! . 1 . #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of <(2)>: #! (an empty 0 x 3 matrix) #! #! A morphism in Category of matrices over GF(3) +#! +#! A morphism in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data s1 := Source( emb ); #! <(1)->1, (2)->0; (a)->1x1, (b)->1x0, (c)->0x0> Display( s1 ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 1 #! @@ -498,17 +471,19 @@ Display( s1 ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! (an empty 1 x 0 matrix) #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! (an empty 0 x 0 matrix) #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data Aop := OppositeAlgebroid( A ); #! Algebroid generated by the right quiver q_op(2)[a:1->1,b:2->1,c:2->2] Yop := YonedaEmbedding( Aop ); @@ -526,11 +501,6 @@ proj1 := Yop( Aop.1 ); IsProjective( proj1 ); #! true Display( proj1 ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 3 #! @@ -544,7 +514,6 @@ Display( proj1 ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! 1 . . #! . 1 . @@ -552,13 +521,16 @@ Display( proj1 ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! . 1 . #! . . 1 #! 1 . . #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data e1 := EmbeddingOfSumOfImagesOfAllMorphisms( unit, proj1 ); #! <(1)->1x3, (2)->1x3> Source( e1 ); @@ -568,11 +540,6 @@ IsEpimorphism( EmbeddingOfSumOfImagesOfAllMorphisms( proj1, six ) ); five := CokernelObject( emb ); #! <(1)->2, (2)->3; (a)->2x2, (b)->2x3, (c)->3x3> Display( five ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 2 #! @@ -585,20 +552,22 @@ Display( five ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! . . 1 #! . 2 . #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 1 . #! . 1 1 #! . . 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data #! @EndExample #! The next calculation shows that the $3$-dimensional representation of $C_3$ #! associated to object $1$ is a single copy of the regular representation of $C_3$. @@ -627,11 +596,6 @@ proj2 := Yop( Aop.2 ); IsProjective( proj2 ); #! true Display( proj2 ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 0 #! @@ -643,17 +607,19 @@ Display( proj2 ); #! #! A zero morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! (an empty 0 x 3 matrix) #! #! A zero morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! . 1 . #! . . 1 #! 1 . . #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data #! @EndExample diff --git a/examples/DecomposeOnceByRandomEndomorphism.g b/examples/DecomposeOnceByRandomEndomorphism.g index 74907f5..2f9e63e 100644 --- a/examples/DecomposeOnceByRandomEndomorphism.g +++ b/examples/DecomposeOnceByRandomEndomorphism.g @@ -42,11 +42,6 @@ iota := result[1]; kappa := result[2]; #! <(1)->22x25, (2)->15x16> Display( fortyone ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 25 #! @@ -82,7 +77,6 @@ Display( fortyone ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! . . . . . 1 . . . . . . . . . . #! . . . . . . 1 . . . . . . . . . @@ -112,7 +106,6 @@ Display( fortyone ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 1 . . 1 1 . . . . . . . . . . #! . 1 1 . . 1 1 . . . . . . . . . @@ -132,14 +125,13 @@ Display( fortyone ); #! . . . . . . . . . . . . . . . 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data S := DirectSum( [ Source( iota ), Source( kappa ) ] ); #! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16> Display( S ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 25 #! @@ -175,7 +167,6 @@ Display( S ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! 2 . . . . . . . . . . . . . . . #! 1 . . . . . . . . . . . . . . . @@ -205,7 +196,6 @@ Display( S ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 . . . . . . . . . . . . . . . #! . 1 1 . . 1 1 . . . . . . . . . @@ -225,6 +215,10 @@ Display( S ); #! . . . . . . . . . . . . . . . 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data #! @EndExample #! Comparing the matrices of fortyone with those of @@ -236,11 +230,6 @@ Display( S ); #! @BeginExample Display( iota ); -#! A morphism in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! 2 2 . 1 1 . . . . . . . . . . 1 2 1 2 1 . . . . . #! 1 2 1 2 1 1 2 1 2 1 . . . . . 2 . . 1 . 2 . . 1 . @@ -248,16 +237,15 @@ Display( iota ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of <(2)>: #! . . . . . . . . . . . . . . . 1 #! #! A morphism in Category of matrices over GF(3) -Display( Source( iota) ); -#! An object in FunctorCategory( Algebroid generated by the +#! +#! A morphism in FunctorCategory( Algebroid generated by the #! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! +#! over GF(3) ) given by the above data +Display( Source( iota ) ); #! Image of <(1)>: #! A vector space object over GF(3) of dimension 3 #! @@ -271,7 +259,6 @@ Display( Source( iota) ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (1)-[{ Z(3)^0*(b) }]->(2): #! 2 #! 1 @@ -279,11 +266,14 @@ Display( Source( iota) ); #! #! A morphism in Category of matrices over GF(3) #! -#! #! Image of (2)-[{ Z(3)^0*(c) }]->(2): #! 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data #! @EndExample #! We can then look at the other embedding of the direct sum diff --git a/examples/RepresentingC4C4.g b/examples/RepresentingC4C4.g index 020880c..6029994 100644 --- a/examples/RepresentingC4C4.g +++ b/examples/RepresentingC4C4.g @@ -120,11 +120,6 @@ eleven := AsObjectInFunctorCategory( A, [ 6, 5 ], [ amat, bmat, cmat ] ); IsWellDefined( eleven ); #! true Display( eleven ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 6 #! @@ -161,6 +156,9 @@ Display( eleven ); #! . . . . 1 #! #! A morphism in Category of matrices over GF(3) +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data gammas := WeakDirectSumDecomposition( eleven ); #! [ <(1)->1x6, (2)->0x5>, <(1)->1x6, (2)->1x5>, <(1)->1x6, (2)->1x5>, #! <(1)->0x6, (2)->2x5>, <(1)->2x6, (2)->0x5>, <(1)->1x6, (2)->1x5> ] @@ -171,11 +169,6 @@ gammas := WeakDirectSumDecomposition( eleven ); #! @Example Display( Source( UniversalMorphismFromDirectSum( gammas ) ) ); -#! An object in FunctorCategory( Algebroid generated by the -#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices -#! over GF(3) ) defined by the following data: -#! -#! #! Image of <(1)>: #! A vector space object over GF(3) of dimension 6 #! @@ -212,5 +205,9 @@ Display( Source( UniversalMorphismFromDirectSum( gammas ) ) ); #! . . . . 1 #! #! A morphism in Category of matrices over GF(3) +#! +#! An object in FunctorCategory( Algebroid generated by the +#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices +#! over GF(3) ) given by the above data #! @EndExample #! @EndChunk