Add natural transformation for graded modules by Freyd category

FreydCategoriesForCAP/gap/GradedModulePresentationsByFreyd/GradedModulePresentationsByFreydFunctors.gd

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+#############################################################################
+##
+## GradedModulePresentationsbyFreyd
+##
+## Copyright 2019,  Martin Bies,       ULB Brussels
+##
+#! @Chapter Functors in the CAP category of graded module presentations  by use of Freyd categories
+##
+#############################################################################
+
+
+# an attribute for homalg_matrices, that allows to represent that matrix 'smaller'
+DeclareAttribute( "LessGradedGeneratorsTransformationTripleLeft",
+                  IsHomalgMatrix );
+DeclareAttribute( "LessGradedGeneratorsTransformationTripleRight",
+                  IsHomalgMatrix );
+
+
+#######################################################
+##
+#! @Section Functor less_generators for FpGradedModules
+##
+#######################################################
+
+#! @Description
+#! The argument is a graded left or right module presentation $M$. We then return a presentation of this
+#! module which uses less generators.
+#! @Returns a graded left or right module presentation
+#! @Arguments M
+DeclareOperation( "LessGradedGenerators",
+                  [ IsFpGradedLeftOrRightModulesObject ] );
+
+#! @Description
+#! The argument is a graded left or right module presentation morphism $a$. We then return a presentation of this
+#! morphism which uses less generators.
+#! @Returns a graded left or right module presentation morphism
+#! @Arguments a
+DeclareOperation( "LessGradedGenerators",
+                  [ IsFpGradedLeftOrRightModulesMorphism ] );
+
+# a function that computes the functor 'LessGenerators' for both left and right presentations
+DeclareGlobalFunction( "FunctorLessGradedGenerators" );
+
+#! @Description
+#! The argument is a homalg graded ring $R$. The output is functor which takes a left 
+#! presentation in FpGradedLeftModules as input and computes a presentation having less generators.
+#! @Returns a functor
+#! @Arguments R
+DeclareAttribute( "FunctorLessGradedGeneratorsLeft",
+                  IsHomalgGradedRing );
+
+#! @Description
+#! The argument is a homalg graded ring $R$. The output is functor which takes a right 
+#! presentation in FpGradedRightModules as input and computes a presentation having less generators.
+#! @Returns a functor
+#! @Arguments R
+DeclareAttribute( "FunctorLessGradedGeneratorsRight",
+                  IsHomalgGradedRing );
+
+
+
+
+######################################################
+##
+#! @Section Functor StandardModule for FpGradedModules
+##
+######################################################
+
+#! @Description
+#! The argument is a graded left or right module presentation $M$. We then try to reduce the relations and
+#! thereby return a new presentation - the Standard module.
+#! @Returns a graded left or right module presentation
+#! @Arguments M
+DeclareOperation( "GradedStandardModule",
+                  [ IsFpGradedLeftOrRightModulesObject ] );
+
+#! @Description
+#! The argument is a graded left or right module presentation morphism $a$. We then try to reduce the relations
+#! and thereby return a new presentation, which we term the Standard module morphism.
+#! @Returns a graded left or right module presentation morphism
+#! @Arguments a
+DeclareOperation( "GradedStandardModule",
+                  [ IsFpGradedLeftOrRightModulesMorphism ] );
+
+# a function that computes the functor 'StandardModule' for both left and right presentations
+DeclareGlobalFunction( "FunctorGradedStandardModule" );
+
+#! @Description
+#! The argument is a homalg graded ring $R$. The output is functor which takes a left
+#! presentation in FpGradedLeftModules as input and computes its standard presentation.
+#! @Returns a functor
+#! @Arguments R
+DeclareAttribute( "FunctorGradedStandardModuleLeft",
+                  IsHomalgGradedRing );
+
+#! @Description
+#! The argument is a homalg graded ring $R$. The output is functor which takes a right
+#! presentation in FpGradedRightModules as input and computes its standard presentation.
+#! @Returns a functor
+#! @Arguments R
+DeclareAttribute( "FunctorGradedStandardModuleRight",
+                  IsHomalgGradedRing );
+
+
+
+##############################################################
+##
+#! @Section Functor ByASmallerPresentation for FpGradedModules
+##
+##############################################################
+
+#! @Description
+#! The argument is a graded left or right module presentation $M$. We then return a smaller presentation of this
+#! module. This is obtained by first applying 'LessGenerators' and then 'StandardModule'.
+#! @Returns a graded left or right module presentation
+#! @Arguments M
+DeclareOperation( "ByASmallerPresentation",
+                  [ IsFpGradedLeftOrRightModulesObject ] );
+
+#! @Description
+#! The argument is a graded left or right module presentation morphism $a$. We then return a smaller presentation of this
+#! morphism. This is obtained by first applying 'LessGenerators' and then 'StandardModule'.
+#! @Returns a graded left or right module presentation morphism
+#! @Arguments a
+DeclareOperation( "ByASmallerPresentation",
+                  [ IsFpGradedLeftOrRightModulesMorphism ] );
+
+# a function that computes the functor 'LessGenerators' for both left and right presentations
+DeclareGlobalFunction( "FunctorByASmallerPresentation" );
+
+#! @Description
+#! The argument is a homalg graded ring $R$. The output is functor which takes a left
+#! presentation in FpGradedLeftModules as input and computes a smaller presentation. The latter is achieved by
+#! first applying 'LessGenerators' and then acting with 'StandardModule'.
+#! @Returns a functor
+#! @Arguments R
+DeclareAttribute( "FunctorByASmallerPresentationLeft",
+                  IsHomalgGradedRing );
+
+#! @Description
+#! The argument is a homalg graded ring $R$. The output is functor which takes a right
+#! presentation in FpGradedRightModules as input and computes a smaller presentation. The latter is achieved by
+#! first applying 'LessGenerators' and then acting with 'StandardModule'.
+#! @Returns a functor
+#! @Arguments R
+DeclareAttribute( "FunctorByASmallerPresentationRight",
+                  IsHomalgGradedRing );