# SPDX-License-Identifier: GPL-2.0-or-later # CAP: Categories, Algorithms, Programming # # Implementations # #! @Chapter Universal Objects #################################### ## ## Coproduct and Pushout ## #################################### #################################### ## ## Coproduct ## #################################### #################################### ## Convenience methods #################################### ## function ( object_1, object_2 )end );## function ( object_1, object_2, object_3 )end );#################################### ## ## Direct Product and FiberProduct ## #################################### #################################### ## Convenience methods #################################### #################################### ## ## Direct Product ## #################################### #################################### ## Convenience methods #################################### ## # compatibility with GAP's DirectProduct function function ( diagram, object )end );## # compatibility with GAP's DirectProduct function function ( category_and_diagram, category )2 ];end );#################################### ## ## Direct sum ## #################################### #################################### ## Technical methods #################################### #################################### ## Convenience methods #################################### ## # compatibility with GAP's DirectSum function function ( diagram, object )end );## # compatibility with GAP's DirectSum function function ( category_and_diagram, category )2 ];if not ( IsBound( category!.supports_empty_limits ) and category!.supports_empty_limits = true ) and diagram = [ ] then fi ;end );# usually the type signatures should be part of the gd file, but `CapJitAddTypeSignature` is not available there "DirectSumFunctorial" , [ IsCapCategory, IsList ], function ( input_types )1 ].category );end );#################################### ## Add methods #################################### #################################### ## Categorical methods #################################### # convenience ## function ( morphism_matrix )if nr_rows = 0 then "The given matrix must not be empty" );fi ;1 ] );if nr_cols = 0 then "The given matrix must not be empty" );fi ;1 ][1 ] ),1 ] ) ),1 ], col -> Range( col ) )end );## function ( cat , morphism_matrix )if nr_rows = 0 then "The given matrix must not be empty" );fi ;1 ] );if nr_cols = 0 then "The given matrix must not be empty" );fi ;cat ,1 ] ) ),1 ], col -> Range( col ) )end );#################################### ## ## Equalizer ## #################################### #################################### ## Convenience methods #################################### ## function ( arg )if IsCapCategory( arg[1 ] ) then fi ;if Length( arg ) = 1 and1 ] ) and1 ], IsCapCategoryMorphism ) then 1 ][1 ] ), arg[1 ] );fi ;if Length( arg ) = 2 and1 ] ) and2 ] ) and2 ], IsCapCategoryMorphism ) then 1 ] ), arg[1 ], arg[2 ] );fi ;1 ] ), arg );end );## function ( list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms );end );function ( cat , list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Source( list_of_morphisms[1 ] ), list_of_morphisms );end );## function ( list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms );end );function ( cat , list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Source( list_of_morphisms[1 ] ), list_of_morphisms );end );## function ( list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms, E );end );function ( cat , list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Source( list_of_morphisms[1 ] ), list_of_morphisms, E );end );## function ( list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms );end );function ( cat , list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Source( list_of_morphisms[1 ] ), list_of_morphisms );end );## function ( list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;s[1 ] ), list_of_morphisms, E );end );function ( cat , list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Source( list_of_morphisms[1 ] ), list_of_morphisms, E );end );## function ( list_of_morphisms, tau )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms, tau );end );function ( cat , list_of_morphisms, tau )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Source( list_of_morphisms[1 ] ), list_of_morphisms, tau );end );## function ( list_of_morphisms, tau, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;isms[1 ] ), list_of_morphisms, tau, E );end );function ( cat , list_of_morphisms, tau, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Source( list_of_morphisms[1 ] ), list_of_morphisms, tau, E );end );#################################### ## ## FiberProduct ## #################################### #################################### ## Convenience methods #################################### ## function ( arg )if IsCapCategory( arg[1 ] ) then fi ;if Length( arg ) = 1 and1 ] ) and1 ], IsCapCategoryMorphism ) then 1 ][1 ] ), arg[1 ] );fi ;1 ] ), arg );end );## function ( diagram )1 ] ), diagram );end );## function ( cat , diagram )1 .. Length( diagram ) ], i -> ProjectionInFactorOfFiberProduct( cat , diagram, i ) );cat , sources_of_diagram, FiberProduct( cat , diagram ), test_source );end );## function ( diagram )1 ] ), diagram );end );## function ( cat , diagram )1 .. Length( diagram ) ], i -> ProjectionInFactorOfFiberProduct( cat , diagram, i ) );cat , sources_of_diagram, FiberProduct( cat , diagram ), test_source );end );"ProjectionInFactorOfFiberProduct" , 2 ],"UniversalMorphismIntoDirectSum" , 1 ],"FiberProduct" , 1 ] ],#################################### ## ## Coequalizer ## #################################### #################################### ## Convenience methods #################################### ## function ( arg )if IsCapCategory( arg[1 ] ) then fi ;if Length( arg ) = 1 and1 ] ) and1 ], IsCapCategoryMorphism ) then 1 ][1 ] ), arg[1 ] );fi ;if Length( arg ) = 2 and1 ] ) and2 ] ) and2 ], IsCapCategoryMorphism ) then 1 ] ), arg[1 ], arg[2 ] );fi ;1 ] ), arg );end );## function ( list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms );end );function ( cat , list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Range( list_of_morphisms[1 ] ), list_of_morphisms );end );## function ( list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms );end );function ( cat , list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Range( list_of_morphisms[1 ] ), list_of_morphisms );end );## function ( list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;s[1 ] ), list_of_morphisms, E );end );function ( cat , list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Range( list_of_morphisms[1 end );## function ( list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms );end );function ( cat , list_of_morphisms )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Range( list_of_morphisms[1 ] ), list_of_morphisms );end );## function ( list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;rphisms[1 ] ), list_of_morphisms, E );end );function ( cat , list_of_morphisms, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Range( list_of_morphisms[1 ] ), list_of_morphisms, E );end );## function ( list_of_morphisms, tau )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;1 ] ), list_of_morphisms, tau );end );function ( cat , list_of_morphisms, tau )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Range( list_of_morphisms[1 ] ), list_of_morphisms, tau );end );## function ( list_of_morphisms, tau, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;orphisms[1 ] ), list_of_morphisms, tau, E );end );function ( cat , list_of_morphisms, tau, E )if IsEmpty( list_of_morphisms ) then "the list of morphisms must not be empty" );fi ;cat , Range( list_of_morphisms[1 ] ), list_of_morphisms, tau, E );end );#################################### ## ## Pushout ## #################################### #################################### ## Convenience methods #################################### ## function ( mor1, mor2 )end );## function ( diagram )1 ] ), diagram );end );## function ( cat , diagram )1 .. Length( diagram ) ], i -> InjectionOfCofactorOfPushout( cat , diagram, i ) );cat , ranges_of_diagram, Pushout( cat , diagram ), test_sink );end );## function ( diagram )1 ] ), diagram );end );## function ( cat , diagram )1 .. Length( diagram ) ], i -> InjectionOfCofactorOfPushout( cat , diagram, i ) );cat , ranges_of_diagram, Pushout( cat , diagram ), test_sink );end );"InjectionOfCofactorOfPushout" , 2 ],"UniversalMorphismFromDirectSum" , 1 ],"Pushout" , 1 ] ],#################################### ## ## Homology object ## #################################### #################################### ## Convenience methods #################################### ## Morphism, IsCapCategoryMorphism ],function ( alpha, beta, epsilon, gamma, delta )end );#################################### ## ## Scheme for Universal Object ## #################################### #################################### ## Add Operations #################################### #################################### ## Attributes #################################### #################################### ## Implied Operations ####################################
Messung V0.5 in Prozent C=85 H=100 G=92
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