ModulePresentationNaturalTransformations.gd

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##                                       ModulePresentationsForCAP package
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##  Copyright 2014, Sebastian Gutsche, TU Kaiserslautern
##                  Sebastian Posur,   RWTH Aachen
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#! @Chapter Module Presentations
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#! @Section Natural Transformations
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#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural isomorphism from the identity functor
#! to the left standard module functor.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{StandardModuleLeft}$
#! @Arguments R
DeclareAttribute( "NaturalIsomorphismFromIdentityToStandardModuleLeft",
                  IsHomalgRing );

#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural isomorphism from the identity functor
#! to the right standard module functor.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{StandardModuleRight}$
#! @Arguments R
DeclareAttribute( "NaturalIsomorphismFromIdentityToStandardModuleRight",
                  IsHomalgRing );

#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural isomorphism from the identity functor
#! to the functor that gets rid of zero generators of left modules.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsLeft}$
#! @Arguments R
DeclareAttribute( "NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsLeft",
                  IsHomalgRing );

#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural isomorphism from the identity functor
#! to the functor that gets rid of zero generators of right modules.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{GetRidOfZeroGeneratorsRight}$
#! @Arguments R
DeclareAttribute( "NaturalIsomorphismFromIdentityToGetRidOfZeroGeneratorsRight",
                  IsHomalgRing );

#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural morphism from the identity functor
#! to the left less generators functor.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{LessGeneratorsLeft}$
#! @Arguments R
DeclareAttribute( "NaturalIsomorphismFromIdentityToLessGeneratorsLeft",
                  IsHomalgRing );

#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural morphism from the identity functor
#! to the right less generator functor.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{LessGeneratorsRight}$
#! @Arguments R
DeclareAttribute( "NaturalIsomorphismFromIdentityToLessGeneratorsRight",
                  IsHomalgRing );

#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural morphism from the identity functor
#! to the double dual functor in left Presentations category.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualLeft}$
#! @Arguments R
DeclareAttribute( "NaturalTransformationFromIdentityToDoubleDualLeft",
                  IsHomalgRing );

#! @Description
#! The argument is a homalg ring $R$.
#! The output is the natural morphism from the identity functor
#! to the double dual functor in right Presentations category.
#! @Returns a natural transformation $\mathrm{Id} \rightarrow \mathrm{FunctorDoubleDualRight}$
#! @Arguments R
DeclareAttribute( "NaturalTransformationFromIdentityToDoubleDualRight",
                  IsHomalgRing );