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1 Finite strict coproduct cocompletion
 1.1 Attributes
 1.2 Constructors
 1.3 GAP categories
 1.4 Examples

1 Finite strict coproduct cocompletion

1.1 Attributes

1.1-1 PairOfIntAndList
‣ PairOfIntAndList( arg )( attribute )

1.1-2 PairOfLists
‣ PairOfLists( arg )( attribute )

1.1-3 UnderlyingCategory
‣ UnderlyingCategory( UC )( attribute )

Return the category C underlying the finite coproduct cocompletion category UC := FiniteStrictCoproductCompletion( C ).

1.1-4 EmbeddingOfUnderlyingCategory
‣ EmbeddingOfUnderlyingCategory( UC )( attribute )

Returns: a CAP functor

The full embedding functor from the category C underlying the finite coproduct cocompletion UC into UC.

1.1-5 ExtendFunctorToFiniteStrictCoproductCompletion
‣ ExtendFunctorToFiniteStrictCoproductCompletion( UC )( attribute )

Returns: a CAP functor

The full embedding functor from the category C underlying the finite coproduct cocompletion UC into UC.

1.1-6 ExtendEmbeddingToFiniteStrictCoproductCompletion
‣ ExtendEmbeddingToFiniteStrictCoproductCompletion( C )( attribute )

Returns: a CAP functor

Extend (i.e., lift) the (full) embedding the category C into PreSheaves( C ) to the full embedding of FiniteStrictCoproductCompletion( C ) into PreSheaves( C ).

1.2 Constructors

1.2-1 FiniteStrictCoproductCompletion
‣ FiniteStrictCoproductCompletion( cat )( attribute )

Return the finite coproduct cocompletion of the category cat in which the cocartesian associators are given by identities.

gap> LoadPackage( "FiniteCocompletions" );
true
gap> T := FiniteStrictCoproductCompletion( InitialCategory( ) );
FiniteStrictCoproductCompletion( InitialCategory( ) )
gap> Display( T );
A CAP category with name FiniteStrictCoproductCompletion( InitialCategory( ) ):

110 primitive operations were used to derive 672 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsFiniteCategory
* IsEquippedWithHomomorphismStructure
* IsLinearCategoryOverCommutativeRing
* IsLeftClosedMonoidalCategory
* IsLeftCoclosedMonoidalCategory
* IsAbelianCategoryWithEnoughInjectives
* IsAbelianCategoryWithEnoughProjectives
* IsElementaryTopos
* IsSymmetricClosedMonoidalLattice
* IsSymmetricCoclosedMonoidalLattice
* IsBooleanAlgebra
* IsRigidSymmetricClosedMonoidalCategory
* IsRigidSymmetricCoclosedMonoidalCategory
and not yet algorithmically
* IsLinearCategoryOverCommutativeRingWithFinitelyGeneratedFreeExternalHoms
and furthermore mathematically
* IsDiscreteCategory
* IsFinitelyPresentedLinearCategory
* IsLinearClosureOfACategory
* IsLocallyOfFiniteInjectiveDimension
* IsLocallyOfFiniteProjectiveDimension
* IsStableProset
* IsTerminalCategory
* IsTotalOrderCategory
gap> i := InitialObject( T );
<A zero object in FiniteStrictCoproductCompletion( InitialCategory( ) )>
gap> t := TerminalObject( T );
<A zero object in FiniteStrictCoproductCompletion( InitialCategory( ) )>
gap> z := ZeroObject( T );
<A zero object in FiniteStrictCoproductCompletion( InitialCategory( ) )>
gap> Display( i );
[ 0, [  ] ]

An object in FiniteStrictCoproductCompletion( InitialCategory( ) )
given by the above data
gap> Display( t );
[ 0, [  ] ]

An object in FiniteStrictCoproductCompletion( InitialCategory( ) )
given by the above data
gap> Display( z );
[ 0, [  ] ]

An object in FiniteStrictCoproductCompletion( InitialCategory( ) )
given by the above data
gap> IsEqualForObjects( i, z );
true
gap> IsIdenticalObj( i, z );
false
gap> IsEqualForObjects( t, z );
true
gap> IsIdenticalObj( t, z );
false
gap> IsWellDefined( z );
true
gap> id_z := IdentityMorphism( z );
<A zero, identity morphism in
 FiniteStrictCoproductCompletion( InitialCategory( ) )>
gap> fn_z := ZeroObjectFunctorial( T );
<A zero, isomorphism in FiniteStrictCoproductCompletion( InitialCategory( ) )>
gap> IsWellDefined( fn_z );
true
gap> IsEqualForMorphisms( id_z, fn_z );
true
gap> IsCongruentForMorphisms( id_z, fn_z );
true
gap> LoadPackage( "FunctorCategories", ">= 2023.10-04" );
true
gap> FinBouquets;
FinBouquets
gap> Chat := ModelingCategory( FinBouquets );
FiniteCocompletion( FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )
gap> UC := FiniteStrictCoproductCompletion(
>                  UnderlyingCategory( FinBouquets ) );
FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )
gap> P := UC.P;
<An object in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )>
gap> Display( P );
[ 1, [ <(P)> ] ]

An object in FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data
gap> L := UC.L;
<An object in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )>
gap> Display( L );
[ 1, [ <(L)> ] ]

An object in FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data
gap> b := UC.b;
<A morphism in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )>
gap> Display( b );
{ 0 } ⱶ[ 0 ]→ { 0 }

[ (P)-[(b)]->(L) ]

A morphism in FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) ) given by the above data
gap> HomStructure( UC );
|1|
gap> HomStructure( P, L );
|1|
gap> homPL := MorphismsOfExternalHom( P, L );
[ <A morphism in FiniteStrictCoproductCompletion(
   FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )> ]
gap> List( homPL, m -> HomStructure( P, L, HomStructure( m ) ) = m );
[ true ]
gap> HomStructure( b, b );
|0| → |1|
gap> HomStructure( b, L );
|1| → |1|
gap> HomStructure( P, b );
|1| → |1|
gap> M := Coproduct( P, L, P );
<An object in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )>
gap> N := Coproduct( L, P, L );
<An object in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "q(P,L)[b:P->L]" ) ) )>
gap> HomStructure( M, N );
|18|
gap> HomStructure( P, L );
|1|
gap> homMN := MorphismsOfExternalHom( M, N );;
gap> List( homMN, m -> HomStructure( M, N, HomStructure( m ) ) = m );
[ true, true, true, true, true, true, true, true, true,
  true, true, true, true, true, true, true, true, true ]

1.3 GAP categories

1.3-1 IsFiniteStrictCoproductCompletion
‣ IsFiniteStrictCoproductCompletion( arg )( filter )

Returns: true or false

The GAP category of finite coproduct cocompletion categories.

1.3-2 IsCellInFiniteStrictCoproductCompletion
‣ IsCellInFiniteStrictCoproductCompletion( arg )( filter )

Returns: true or false

The GAP category of cells in a finite coproduct cocompletion category.

1.3-3 IsObjectInFiniteStrictCoproductCompletion
‣ IsObjectInFiniteStrictCoproductCompletion( arg )( filter )

Returns: true or false

The GAP category of objects in a finite coproduct cocompletion category.

1.3-4 IsMorphismInFiniteStrictCoproductCompletion
‣ IsMorphismInFiniteStrictCoproductCompletion( arg )( filter )

Returns: true or false

The GAP category of morphisms in a finite coproduct cocompletion category.

1.4 Examples

1.4-1 Cocartesian associator
gap> LoadPackage( "FiniteCocompletions" );
true
gap> LoadPackage( "Algebroids" );
true
gap> LoadPackage( "LazyCategories", ">= 2023.01-02" );
true
gap> Q := RightQuiver( "Q(a,b,c)[]" );
Q(a,b,c)[]
gap> C := FreeCategory( Q );
FreeCategory( RightQuiver( "Q(a,b,c)[]" ) )
gap> UC := FiniteStrictCoproductCompletion( C );
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )
gap> a := UC.a;
<An object in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )>
gap> b := UC.b;
<An object in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )>
gap> c := UC.c;
<An object in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )>
gap> ab_c := Coproduct( Coproduct( a, b ), c );
<An object in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )>
gap> a_bc := Coproduct( a, Coproduct( b, c ) );
<An object in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )>
gap> ab_c = a_bc;
true
gap> HomStructure( ab_c, a_bc );
|1|
gap> hom := MorphismsOfExternalHom( ab_c, a_bc );
[ <A morphism in FiniteStrictCoproductCompletion(
   FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )> ]
gap> alpha := hom[1];
<A morphism in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )>
gap> Source( alpha ) = ab_c;
true
gap> IsOne( alpha );
true
gap> IsWellDefined( alpha );
true
gap> alpha = CocartesianAssociatorLeftToRight( a, b, c );
true
gap> LUC := LazyCategory( UC );
LazyCategory(
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) ) )
gap> Emb := EmbeddingFunctorOfUnderlyingCategory( LUC );
Embedding functor into lazy category
gap> Display( Emb );
Embedding functor into lazy category:

FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )
  |
  V
LazyCategory(
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) ) )
gap> F := PreCompose( EmbeddingOfUnderlyingCategory( UC ), Emb );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category
gap> Display( F );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category:

FreeCategory( RightQuiver( "Q(a,b,c)[]" ) )
  |
  V
LazyCategory(
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) ) )
gap> G := ExtendFunctorToFiniteStrictCoproductCompletion( F );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) )
gap> Display( G );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) ):

FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) )
  |
  V
LazyCategory( FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) ) )
gap> Galpha := G( alpha );
<A morphism in LazyCategory( FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(a,b,c)[]" ) ) ) )>
gap> IsWellDefined( Galpha );
true

1.4-2 Cocartesian braiding
gap> LoadPackage( "FiniteCocompletions" );
true
gap> LoadPackage( "Algebroids" );
true
gap> LoadPackage( "LazyCategories", ">= 2023.01-02" );
true
gap> Q := RightQuiver( "Q(a,b)[]" );
Q(a,b)[]
gap> C := FreeCategory( Q );
FreeCategory( RightQuiver( "Q(a,b)[]" ) )
gap> SetName( C.a, "C.a" ); SetName( C.b, "C.b" );
gap> UC := FiniteStrictCoproductCompletion( C );
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )
gap> a := UC.a;
<An object in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )>
gap> b := UC.b;
<An object in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )>
gap> ab := Coproduct( a, b );
<An object in
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )>
gap> Display( ab );
[ 2, [ C.a, C.b ] ]

An object in
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )
given by the above data
gap> ba := Coproduct( b, a );
<An object in
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )>
gap> Display( ba );
[ 2, [ C.b, C.a ] ]

An object in
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )
given by the above data
gap> HomStructure( ab, ba );
|1|
gap> hom := MorphismsOfExternalHom( ab, ba );
[ <A morphism in FiniteStrictCoproductCompletion(
   FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )> ]
gap> gamma := hom[1];
<A morphism in
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )>
gap> Source( gamma ) = ab;
true
gap> Target( gamma ) = ba;
true
gap> IsWellDefined( gamma );
true
gap> Display( gamma );
{ 0, 1 } ⱶ[ 1, 0 ]→ { 0, 1 }

[ (a)-[(a)]->(a), (b)-[(b)]->(b) ]

A morphism in
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )
given by the above data
gap> LUC := LazyCategory( UC );
LazyCategory(
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) ) )
gap> Emb := EmbeddingFunctorOfUnderlyingCategory( LUC );
Embedding functor into lazy category
gap> Display( Emb );
Embedding functor into lazy category:

FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )
  |
  V
LazyCategory(
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) ) )
gap> F := PreCompose( EmbeddingOfUnderlyingCategory( UC ), Emb );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category
gap> Display( F );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category:

FreeCategory( RightQuiver( "Q(a,b)[]" ) )
  |
  V
LazyCategory(
FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) ) )
gap> G := ExtendFunctorToFiniteStrictCoproductCompletion( F );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) )
gap> Display( G );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) ):

FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) )
  |
  V
LazyCategory( FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(a,b)[]" ) ) ) )
gap> Ggamma := G( gamma );
<A morphism in LazyCategory(
 FiniteStrictCoproductCompletion( FreeCategory( RightQuiver( "Q(a,b)[]" ) ) ) )>
gap> IsWellDefined( Ggamma );
true

1.4-3 Cocartesian codiagonal
gap> LoadPackage( "FiniteCocompletions" );
true
gap> LoadPackage( "Algebroids" );
true
gap> LoadPackage( "LazyCategories", ">= 2023.01-02" );
true
gap> Q := FinQuiver( "Q(a)[]" );
FinQuiver( "Q(a)[]" )
gap> C := PathCategory( Q );
PathCategory( FinQuiver( "Q(a)[]" ) )
gap> SetName( C.a, "C.a" );
gap> UC := FiniteStrictCoproductCompletion( C );
FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )
gap> a := UC.a;
<An object in
 FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )>
gap> aa := Coproduct( a, a );
<An object in
 FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )>
gap> Display( aa );
[ 2, [ C.a, C.a ] ]

An object in
FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )
given by the above data
gap> a = aa;
false
gap> HomStructure( a, aa );
|2|
gap> hom_a_aa := MorphismsOfExternalHom( a, aa );
[ <A morphism in
   FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )> ]
gap> hom_a_aa[1] = InjectionOfCofactorOfCoproduct( [ a, a ], 1 );
true
gap> hom_a_aa[2] = InjectionOfCofactorOfCoproduct( [ a, a ], 2 );
true
gap> HomStructure( aa, a );
|1|
gap> hom_aa_a := MorphismsOfExternalHom( aa, a );
[ <A morphism in
   FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )> ]
gap> delta := hom_aa_a[1];
<A morphism in
 FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )>
gap> Source( delta ) = aa;
true
gap> Target( delta ) = a;
true
gap> IsOne( ComponentOfMorphismFromCoproduct( delta, [ a, a ], 1 ) );
true
gap> IsOne( ComponentOfMorphismFromCoproduct( delta, [ a, a ], 2 ) );
true
gap> IsWellDefined( delta );
true
gap> Display( delta );
{ 0, 1 } ⱶ[ 0, 0 ]→ { 0 }

[ id(a):C.a -≻ C.a, id(a):C.a -≻ C.a ]

A morphism in
FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )
given by the above data
gap> id_a := IdentityMorphism( C.a );
id(a):C.a -≻ C.a
gap> delta = CocartesianCodiagonal( a, 2 );
true
gap> CellAsEvaluatableString( delta, [ "C", "UC" ] );
"MorphismConstructor( UC,
        ObjectConstructor( UC,
                Pair( 2,
                      [ ObjectConstructor( C, 1 ),
                        ObjectConstructor( C, 1 ) ] ) ),
        Pair( [ 0, 0 ],
              [ IdentityMorphism( C, ObjectConstructor( C, 1 ) ),
                IdentityMorphism( C, ObjectConstructor( C, 1 ) ) ] ),
        ObjectConstructor( UC,
                Pair( 1, [ ObjectConstructor( C, 1 ) ] ) ) )"
gap> LUC := LazyCategory( UC );
LazyCategory(
FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) ) )
gap> Emb := EmbeddingFunctorOfUnderlyingCategory( LUC );
Embedding functor into lazy category
gap> Display( Emb );
Embedding functor into lazy category:

FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )
  |
  V
LazyCategory(
FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) ) )
gap> F := PreCompose( EmbeddingOfUnderlyingCategory( UC ), Emb );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category
gap> Display( F );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category:

PathCategory( FinQuiver( "Q(a)[]" ) )
  |
  V
LazyCategory(
FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) ) )
gap> G := ExtendFunctorToFiniteStrictCoproductCompletion( F );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) )
gap> Display( G );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) ):

FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) )
  |
  V
LazyCategory(
FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) ) )
gap> Gdelta := G( delta );
<A morphism in LazyCategory(
 FiniteStrictCoproductCompletion( PathCategory( FinQuiver( "Q(a)[]" ) ) ) )>
gap> IsWellDefined( Gdelta );
true

1.4-4 Homomorphism structure
gap> LoadPackage( "FiniteCocompletions" );
true
gap> T := TerminalCategoryWithMultipleObjects( );
TerminalCategoryWithMultipleObjects( )
gap> sFinSets := FiniteStrictCoproductCompletion( T );
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
gap> Display( sFinSets );
A CAP category with name
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) ):

35 primitive operations were used to derive 172 operations for this category
which algorithmically
* IsCategoryWithDecidableColifts
* IsCategoryWithDecidableLifts
* IsEquippedWithHomomorphismStructure
* IsFiniteCompleteCategory
* IsDistributiveCategory
and furthermore mathematically
* IsStrictCocartesianCategory
gap> t := TerminalObject( sFinSets );
<An object in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> IsTerminal( t );
true
gap> IsInitial( t );
false
gap> Display( t );
[ 1, [ TerminalObject ] ]

An object in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> i := InitialObject( sFinSets );
<An object in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> IsTerminal( i );
false
gap> IsInitial( i );
true
gap> Display( i );
[ 0, [  ] ]

An object in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> UniversalMorphismFromInitialObject(t) = UniversalMorphismIntoTerminalObject(i);
true
gap> A := [ 3, [ "A0" / T, "A1" / T, "A2" / T ] ] / sFinSets;
<An object in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> Display( A );
[ 3, [ A0, A1, A2 ] ]

An object in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> B := [ 2, [ "B0" / T, "B1" / T ] ] / sFinSets;
<An object in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> Display( B );
[ 2, [ B0, B1 ] ]

An object in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> piA := ProjectionInFactorOfDirectProduct( [ A, B ], 1 );
<A morphism in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> Display( piA );
{ 0,..., 5 } ⱶ[ 0, 1, 2, 0, 1, 2 ]→ { 0, 1, 2 }

[ ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct ]

A morphism in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> piB := ProjectionInFactorOfDirectProduct( [ A, B ], 2 );
<A morphism in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> Display( piB );
{ 0,..., 5 } ⱶ[ 0, 0, 0, 1, 1, 1 ]→ { 0, 1 }

[ ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct,
  ProjectionInFactorOfDirectProductWithGivenDirectProduct ]

A morphism in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> IsOne( UniversalMorphismIntoDirectProduct( [ piA, piB ] ) );
true
gap> I := HomStructure( sFinSets );
<An object in
 FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>
gap> Display( I );
[ 1, [ An object in TerminalCategoryWithSingleObject( ) ] ]

An object in
FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )
given by the above data
gap> U := ObjectDatum( I )[2][1];
<A zero object in TerminalCategoryWithSingleObject( )>
gap> HomAB := HomStructure( A, B );
<An object in
 FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>
gap> L := ObjectDatum( HomAB );
[ 8, [ <A zero object in TerminalCategoryWithSingleObject( )>,
       <A zero object in TerminalCategoryWithSingleObject( )>,
       <A zero object in TerminalCategoryWithSingleObject( )>,
       <A zero object in TerminalCategoryWithSingleObject( )>,
       <A zero object in TerminalCategoryWithSingleObject( )>,
       <A zero object in TerminalCategoryWithSingleObject( )>,
       <A zero object in TerminalCategoryWithSingleObject( )>,
       <A zero object in TerminalCategoryWithSingleObject( )> ] ]
gap> homAB := List( MorphismsOfExternalHom( A, B ), HomStructure );
[ <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>,
  <A morphism in
   FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )> ]
gap> List( homAB, IsWellDefined );
[ true, true, true, true, true, true, true, true ]
gap> List( homAB, m -> HomStructure( HomStructure( A, B, m ) ) ) = homAB;
true
gap> alpha := HomStructure( A, B, homAB[6] );
<A morphism in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> Display( alpha );
{ 0, 1, 2 } ⱶ[ 1, 0, 1 ]→ { 0, 1 }

[ InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism,
  InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism,
  InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism ]

A morphism in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> gamma := HomStructure( A, B, homAB[2] );
<A morphism in
 FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )>
gap> Display( gamma );
{ 0, 1, 2 } ⱶ[ 1, 0, 0 ]→ { 0, 1 }

[ InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism,
  InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism,
  InterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism ]

A morphism in
FiniteStrictCoproductCompletion( TerminalCategoryWithMultipleObjects( ) )
given by the above data
gap> hom_alpha_gamma := HomStructure( alpha, gamma );
<A morphism in
 FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )>
gap> Display( hom_alpha_gamma );
{ 0,..., 8 } ⱶ[ 7, 5, 5, 2, 0, 0, 2, 0, 0 ]→ { 0,..., 7 }

[ A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ),
  A morphism in TerminalCategoryWithSingleObject( ) ]

A morphism in
FiniteStrictCoproductCompletion( TerminalCategoryWithSingleObject( ) )
given by the above data
gap> IsWellDefined( hom_alpha_gamma );
true

1.4-5 Reflexive pair
gap> LoadPackage( "FiniteCocompletions" );
true
gap> LoadPackage( "Algebroids" );
true
gap> LoadPackage( "LazyCategories", ">= 2023.01-02" );
true
gap> Q := RightQuiver( "Q(A,B)[f:A->B,g:A->B]" );
Q(A,B)[f:A->B,g:A->B]
gap> C := FreeCategory( Q );
FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) )
gap> SetName( C.A, "C.A" ); SetName( C.B, "C.B" );
gap> UC := FiniteStrictCoproductCompletion( C );
FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )
gap> A := UC.A;
<An object in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> B := UC.B;
<An object in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> Display( A );
[ 1, [ C.A ] ]

An object in FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )
given by the above data
gap> Display( B );
[ 1, [ C.B ] ]

An object in FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )
given by the above data
gap> f := UC.f;
<A morphism in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> g := UC.g;
<A morphism in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> MorphismDatum( f );
[ [ 0 ], [ (A)-[(f)]->(B) ] ]
gap> MorphismDatum( g );
[ [ 0 ], [ (A)-[(g)]->(B) ] ]
gap> IsSplitEpimorphism( f );
false
gap> IsSplitEpimorphism( g );
false
gap> AuB := Coproduct( A, B );
<An object in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> Display( AuB );
[ 2, [ C.A, C.B ] ]

An object in FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )
given by the above data
gap> HomStructure( AuB, B );
|2|
gap> hom := MorphismsOfExternalHom( AuB, B );
[ <A morphism in FiniteStrictCoproductCompletion(
   FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>,
  <A morphism in FiniteStrictCoproductCompletion(
   FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )> ]
gap> ff := hom[1];
<A morphism in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> gg := hom[2];
<A morphism in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> ff = gg;
false
gap> IsSplitEpimorphism( ff );
true
gap> IsSplitEpimorphism( gg );
true
gap> MorphismDatum( ff );
[ [ 0, 0 ], [ (A)-[(f)]->(B), (B)-[(B)]->(B) ] ]
gap> MorphismDatum( gg );
[ [ 0, 0 ], [ (A)-[(g)]->(B), (B)-[(B)]->(B) ] ]
gap> HomStructure( B, AuB );
|1|
gap> ii := MorphismsOfExternalHom( B, AuB )[1];
<A morphism in FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )>
gap> ii = InjectionOfCofactorOfCoproduct( [ A, B ], 2 );
true
gap> f = ComponentOfMorphismFromCoproduct( ff, [ A, B ], 1 );
true
gap> IsOne( ComponentOfMorphismFromCoproduct( ff, [ A, B ], 2 ) );
true
gap> g = ComponentOfMorphismFromCoproduct( gg, [ A, B ], 1  );
true
gap> IsOne( ComponentOfMorphismFromCoproduct( gg, [ A, B ], 2 ) );
true
gap> LUC := LazyCategory( UC );
LazyCategory( FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) ) )
gap> Emb := EmbeddingFunctorOfUnderlyingCategory( LUC );
Embedding functor into lazy category
gap> Display( Emb );
Embedding functor into lazy category:

FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )
  |
  V
LazyCategory( FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) ) )
gap> F := PreCompose( EmbeddingOfUnderlyingCategory( UC ), Emb );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category
gap> Display( F );
Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category:

FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) )
  |
  V
LazyCategory( FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) ) )
gap> G := ExtendFunctorToFiniteStrictCoproductCompletion( F );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) )
gap> Display( G );
Extension to FiniteStrictCoproductCompletion( Source( Precomposition of
Embedding functor into a finite strict coproduct cocompletion category and
Embedding functor into lazy category ) ):

FiniteStrictCoproductCompletion(
FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) )
  |
  V
LazyCategory( FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) ) )
gap> Gff := G( ff );
<A morphism in LazyCategory( FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) ) )>
gap> IsWellDefined( Gff );
true
gap> Ggg := G( gg );
<A morphism in LazyCategory( FiniteStrictCoproductCompletion(
 FreeCategory( RightQuiver( "Q(A,B)[f:A->B,g:A->B]" ) ) ) )>
gap> IsWellDefined( Ggg );
true
gap> MorphismDatum( EvaluatedCell( Gff ) );
[ [ 0, 0 ], [ (A)-[(f)]->(B), (B)-[(B)]->(B) ] ]
gap> MorphismDatum( EvaluatedCell( Ggg ) );
[ [ 0, 0 ], [ (A)-[(g)]->(B), (B)-[(B)]->(B) ] ]
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