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### 2 Finitely presented categories generated by enhanced quivers

#### 2.1 Properties

##### 2.1-1 IsCommutative
 ‣ IsCommutative( C ) ( property )

Returns: true or false

Check whether the finitely presented category C is commutative.

##### 2.1-2 IsCounitary
 ‣ IsCounitary( B ) ( property )

Returns: true or false

Check whether B is counitary.

##### 2.1-3 IsCoassociative
 ‣ IsCoassociative( B ) ( property )

Returns: true or false

Check whether B is coassociative.

#### 2.2 Attributes

##### 2.2-1 UnderlyingQuiver
 ‣ UnderlyingQuiver( C ) ( attribute )

Returns: a QPA quiver

The quiver underlying the finitely presented category C.

##### 2.2-2 UnderlyingQuiverAlgebra
 ‣ UnderlyingQuiverAlgebra( C ) ( attribute )

Returns: a QPA path algebra

The quiver algebra (=path algebra with relations) underlying the finitely presented category C.

##### 2.2-3 Size
 ‣ Size( C ) ( attribute )

Returns: a nonnegative integer

The number of morphisms in the finitely presented category C.

##### 2.2-4 BasisPathsByVertexIndex
 ‣ BasisPathsByVertexIndex( C ) ( attribute )

Returns: a matrix of basis paths of a QPA path algebra

The matrix of basis paths of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C, indexed by the vertex indices of source and target of the path.

##### 2.2-5 BasisMorphismsByVertexIndex
 ‣ BasisMorphismsByVertexIndex( A ) ( attribute )

Returns: a matrix of basis morphisms

The matrix of basis morphisms of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C, indexed by the vertex indices of source and target of the morphism.

##### 2.2-6 HomStructureOnBasisPaths
 ‣ HomStructureOnBasisPaths( C ) ( attribute )

Returns: a six-dimensional matrix of matrices

The hom structure on basis paths of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C: HomStructureOnBasisPaths( A )[ v_index ][ w_index ][ v'_index ][ w'_index ][ basis_path_1_index ][ basis_path_2_index ] = [ Hom(v,w) -> Hom(v',w'): x -> basis_path_1 * x * basis_path_2 ] for basis_path_1: v' -> v and basis_path_2: w -> w'.

##### 2.2-7 AssignSetOfObjects
 ‣ AssignSetOfObjects( C, label ) ( operation )

Returns: nothing

Assigns the objects of the finitely presented category C to global variables. Names of the variables are the concatenation of label with the names of the defining vertices.

##### 2.2-8 SetOfGeneratingMorphisms
 ‣ SetOfGeneratingMorphisms( C, obj_1, obj_2 ) ( operation )

Returns: a list

The subset of the generating morphisms that start at obj_1 and ends at obj_2.

##### 2.2-9 SetOfGeneratingMorphisms
 ‣ SetOfGeneratingMorphisms( obj_1, obj_2 ) ( operation )

Returns: a list

The subset of the generating morphisms that start at obj_1 and ends at obj_2.

##### 2.2-10 SetOfGeneratingMorphisms
 ‣ SetOfGeneratingMorphisms( C, i, j ) ( operation )

Returns: a list

Delegates to SetOfGeneratingMorphisms( C, SetOfObjects(C)[i], SetOfObjects(C)[j] ).

##### 2.2-11 AssignSetOfGeneratingMorphisms
 ‣ AssignSetOfGeneratingMorphisms( C, label ) ( operation )

Returns: nothing

Assigns the generating morphisms of the finitely presented category C to global variables. Names of the variables are the concatenation of label with the names of the defining arrows.

##### 2.2-12 RelationsOfFpCategory
 ‣ RelationsOfFpCategory( C ) ( attribute )

Returns: a QPA path algebra

The relations of the finitely presented category C corresponding to RelationsOfAlgebra( UnderlyingQuiverAlgebra( C ) ).

##### 2.2-13 OppositeFpCategory
 ‣ OppositeFpCategory( C ) ( attribute )

Returns: a CAP category

The finitely presented category defined by the opposite of the underlying quiver with relations.

##### 2.2-14 Antipode
 ‣ Antipode( B ) ( attribute )

Returns: a CAP functor

The antipode of the Hopf finitely presented category B.

##### 2.2-15 UnderlyingVertex
 ‣ UnderlyingVertex( obj ) ( attribute )

Returns: a vertex in a QPA quiver

The vertex of the quiver underlying the object obj in a finitely presented category.

##### 2.2-16 UnderlyingQuiverAlgebraElement
 ‣ UnderlyingQuiverAlgebraElement( mor ) ( attribute )

Returns: an element in a QPA path algebra

The quiver algebra element underlying the morphism mor in a finitely presented category.

##### 2.2-17 UnderlyingAlgebra
 ‣ UnderlyingAlgebra( C ) ( attribute )

Returns: a ring

The underlying algebra of the finitely presented category C.

##### 2.2-18 Parity
 ‣ Parity( C ) ( attribute )

Returns: a string ("left" or "right")

The parity of the finitely presented category C.

##### 2.2-19 POW
 ‣ POW( C, n ) ( operation )

Returns: a CAP category

The n-th power of the finitely presented category C. Admissible values for n are $$0,1,2$$.

##### 2.2-20 NerveTruncatedInDegree2AsFunctor
 ‣ NerveTruncatedInDegree2AsFunctor( C ) ( attribute )

Returns: a CAP functor

The input is a finitely presented category C equipped with a homomorphism structure with values in the skeletal category SkeletalFinSets of finite sets. The output is the nerve of B truncated in degree $$2$$, as a presheaf on SimplicialCategoryTruncatedInDegree($$2$$) with values in SkeletalFinSets.

We compute the nerve of the full subcategory of the simplicial category $$\Delta$$ on the objects $$[0], [1], [2]$$.

gap> Delta2 := SimplicialCategoryTruncatedInDegree( 2 );
FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations
gap> RelationsOfFpCategory( Delta2 );
[ [ (s*id), (C0) ], [ (t*id), (C0) ],
[ (ps*is), (C1) ], [ (pt*it), (C1) ],
[ (is*id), (it*id) ], [ (pt*is), (id*t) ],
[ (ps*it), (id*s) ], [ (s*pt), (t*ps) ],
[ (s*mu), (s*ps) ], [ (t*mu), (t*pt) ],
[ (mu*is), (C1) ], [ (mu*it), (C1) ] ]
gap> Size( Delta2 );
31
gap> N := NerveTruncatedInDegree2AsFunctor( Delta2 );
Functor from FreeCategory( RightQuiver(
"Delta_op(C0,C1,C2)[id:C0->C1,s:C1->C0,t:C1->C0,
is:C1->C2,it:C1->C2,
ps:C2->C1,pt:C2->C1,mu:C2->C1]" ) ) / relations
-> SkeletalFinSets
gap> Delta2op := SourceOfFunctor( N );
FreeCategory( RightQuiver(
"Delta_op(C0,C1,C2)[id:C0->C1,s:C1->C0,t:C1->C0,
is:C1->C2,it:C1->C2,
ps:C2->C1,pt:C2->C1,mu:C2->C1]" ) ) / relations
gap> N( Delta2op.C0 );
|3|
gap> Display( N( Delta2op.C0 ) );
{ 0, 1, 2 }
gap> N( Delta2op.C1 );
|31|
gap> Display( N( Delta2op.C1 ) );
{ 0,..., 30 }
gap> N( Delta2op.C2 );
|393|
gap> Display( N( Delta2op.C2 ) );
{ 0,..., 392 }
gap> N( Delta2op.id );
|3| → |31|
gap> Display( N( Delta2op.id ) );
{ 0, 1, 2 } ⱶ[ 0, 5, 21 ]→ { 0,..., 30 }


##### 2.2-21 YonedaFibrationAsNaturalTransformation
 ‣ YonedaFibrationAsNaturalTransformation( B ) ( attribute )

Returns: a CAP natural transformation

The input is a finitely presented category B. The output is a natural morphism. Its source is the functor $$B \to H, c \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,c), \psi \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,\psi)$$. Its targe is the constant functor of $$0$$-cells $$B \to H, c \mapsto B_0, \psi \mapsto \mathrm{id}_{B_0}$$.

##### 2.2-22 YonedaProjectionAsNaturalEpimorphism
 ‣ YonedaProjectionAsNaturalEpimorphism( B ) ( attribute )

Returns: a CAP natural transformation

The input is a finitely presented category B. The output is a natural epimorphism. Its source is the functor $$B \to H, c \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(a,b) \times \mathrm{Hom}(b,c), \psi \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(1_a,1_b) \times \mathrm{Hom}(b,\psi)$$. Its target is the functor $$B \to H, c \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,c), \psi \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,\psi)$$.

##### 2.2-23 YonedaCompositionAsNaturalEpimorphism
 ‣ YonedaCompositionAsNaturalEpimorphism( B ) ( attribute )

Returns: a CAP natural transformation

The input is a finitely presented category B. The output is a natural epimorphism. Its source is the functor $$B \to H, c \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(a,b) \times \mathrm{Hom}(b,c), \psi \mapsto \sqcup_{a,b\in B} \mathrm{Hom}(1_a,1_b) \times \mathrm{Hom}(b,\psi)$$. Its target is the functor $$B \to H, c \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,c), \psi \mapsto \sqcup_{a\in B} \mathrm{Hom}(a,\psi)$$.

We compute the Yoneda composition natural epimorphism of the full subcategory of the simplicial category $$\Delta$$ on the objects $$[0], [1], [2]$$.

gap> Delta2 := SimplicialCategoryTruncatedInDegree( 2 );
FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations
gap> RelationsOfFpCategory( Delta2 );
[ [ (s*id), (C0) ], [ (t*id), (C0) ],
[ (ps*is), (C1) ], [ (pt*it), (C1) ],
[ (is*id), (it*id) ], [ (pt*is), (id*t) ],
[ (ps*it), (id*s) ], [ (s*pt), (t*ps) ],
[ (s*mu), (s*ps) ], [ (t*mu), (t*pt) ],
[ (mu*is), (C1) ], [ (mu*it), (C1) ] ]
gap> Size( Delta2 );
31
gap> Ymu := YonedaCompositionAsNaturalEpimorphism( Delta2 );
Natural transformation from
Functor from FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations ->
SkeletalFinSets
->
Functor from FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations ->
SkeletalFinSets
gap> Ymu := YonedaProjectionAsNaturalEpimorphism( Delta2 );
Natural transformation from
Functor from FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations ->
SkeletalFinSets
->
Functor from FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations ->
SkeletalFinSets
gap> Ys := YonedaFibrationAsNaturalTransformation( Delta2 );
Natural transformation from
Functor from FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations ->
SkeletalFinSets
->
Functor from FreeCategory( RightQuiver(
"Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1,
is:C2->C1,it:C2->C1,
ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations ->
SkeletalFinSets


##### 2.2-24 TruthMorphismOfTrueToSieveFunctor
 ‣ TruthMorphismOfTrueToSieveFunctor( B ) ( attribute )

Returns: a CAP functor

Return the truth morphism of true from terminal functor to the functor of sieves from OppositeFpCategory( B ) to RangeCategoryOfHomomorphismStructure( B ).

gap> q := RightQuiver( "q(1)[a:1->1]" );
q(1)[a:1->1]
gap> Fq := FreeCategory( q );
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
gap> M0 := Fq / [ [ Fq.a^3, Fq.1 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0 := SieveFunctor( M0 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
-> SkeletalFinSets
gap> M0op := AsCapCategory( Source( S0 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0( M0op.1 );
|2|
gap> Display( S0( M0op.1 ) );
{ 0, 1 }
gap> S0( M0op.a );
|2| → |2|
gap> Display( S0( M0op.a ) );
{ 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }
gap> M1 := Fq / [ [ Fq.a^3, Fq.a ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1 := SieveFunctor( M1 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
-> SkeletalFinSets
gap> M1op := AsCapCategory( Source( S1 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1( M1op.1 );
|3|
gap> Display( S1( M1op.1 ) );
{ 0, 1, 2 }
gap> S1( M1op.a );
|3| → |3|
gap> Display( S1( M1op.a ) );
{ 0, 1, 2 } ⱶ[ 0, 2, 2 ]→ { 0, 1, 2 }
gap> M2 := Fq / [ [ Fq.a^3, Fq.a^2 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2 := SieveFunctor( M2 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> M2op := AsCapCategory( Source( S2 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2( M2op.1 );
|4|
gap> Display( S2( M2op.1 ) );
{ 0 ,..., 3 }
gap> S2( M2op.a );
|4| → |4|
gap> Display( S2( M2op.a ) );
{ 0,..., 3 } ⱶ[ 0, 2, 3, 3 ]→ { 0,..., 3 }
gap> eta2 := TruthMorphismOfTrueToSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> eta2( M2op.1 );
|1| → |4|
gap> Display( eta2( M2op.1 ) );
{ 0 } ⱶ[ 3 ]→ { 0,..., 3 }
gap> iota2 := EmbeddingOfSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> iota2( M2op.1 );
|4| → |8|
gap> Display( iota2( M2op.1 ) );
{ 0,..., 3 } ⱶ[ 0, 4, 6, 7 ]→ { 0,..., 7 }


##### 2.2-25 SieveFunctor
 ‣ SieveFunctor( B ) ( attribute )

Returns: a CAP functor

Return the functor of sieves from OppositeFpCategory( B ) to RangeCategoryOfHomomorphismStructure( B ).

gap> q := RightQuiver( "q(1)[a:1->1]" );
q(1)[a:1->1]
gap> Fq := FreeCategory( q );
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
gap> M0 := Fq / [ [ Fq.a^3, Fq.1 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0 := SieveFunctor( M0 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
-> SkeletalFinSets
gap> M0op := AsCapCategory( Source( S0 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0( M0op.1 );
|2|
gap> Display( S0( M0op.1 ) );
{ 0, 1 }
gap> S0( M0op.a );
|2| → |2|
gap> Display( S0( M0op.a ) );
{ 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }
gap> M1 := Fq / [ [ Fq.a^3, Fq.a ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1 := SieveFunctor( M1 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
-> SkeletalFinSets
gap> M1op := AsCapCategory( Source( S1 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1( M1op.1 );
|3|
gap> Display( S1( M1op.1 ) );
{ 0, 1, 2 }
gap> S1( M1op.a );
|3| → |3|
gap> Display( S1( M1op.a ) );
{ 0, 1, 2 } ⱶ[ 0, 2, 2 ]→ { 0, 1, 2 }
gap> M2 := Fq / [ [ Fq.a^3, Fq.a^2 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2 := SieveFunctor( M2 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> M2op := AsCapCategory( Source( S2 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2( M2op.1 );
|4|
gap> Display( S2( M2op.1 ) );
{ 0 ,..., 3 }
gap> S2( M2op.a );
|4| → |4|
gap> Display( S2( M2op.a ) );
{ 0,..., 3 } ⱶ[ 0, 2, 3, 3 ]→ { 0,..., 3 }
gap> eta2 := TruthMorphismOfTrueToSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> eta2( M2op.1 );
|1| → |4|
gap> Display( eta2( M2op.1 ) );
{ 0 } ⱶ[ 3 ]→ { 0,..., 3 }
gap> iota2 := EmbeddingOfSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> iota2( M2op.1 );
|4| → |8|
gap> Display( iota2( M2op.1 ) );
{ 0,..., 3 } ⱶ[ 0, 4, 6, 7 ]→ { 0,..., 7 }


#### 2.3 Operations

##### 2.3-1 TrivialCategory
 ‣ TrivialCategory( str ) ( operation )

##### 2.3-2 \*
 ‣ \*( C, D ) ( operation )

##### 2.3-3 ElementaryTensor
 ‣ ElementaryTensor( a, b, T ) ( operation )

Returns: a morphism in a CAP category

Given an object a in a finitely presented category A and an object b in a finitely presented category B and the tensor product T of A and B, return the tensor product of a and b in T.

##### 2.3-4 ElementaryTensor
 ‣ ElementaryTensor( a, g, T ) ( operation )

Returns: a morphism in a CAP category

Given an object a in a finitely presented category A and a morphism g in a finitely presented category B and the tensor product T of A and B, return the tensor product of a and g in T.

##### 2.3-5 ElementaryTensor
 ‣ ElementaryTensor( f, b, T ) ( operation )

Returns: a morphism in a CAP category

Given a morphism f in a finitely presented category A and an object b in a finitely presented category B and the tensor product T of A and B, return the tensor product of f and b in T.

##### 2.3-6 QuiverVertexAsIdentityPath
 ‣ QuiverVertexAsIdentityPath( vertex ) ( operation )

Returns: a path

Simply returns vertex, but with the semantics of being an identity path.

#### 2.4 Constructors

##### 2.4-1 FreeCategory
 ‣ FreeCategory( q ) ( operation )
 ‣ Category( q, L ) ( operation )
 ‣ QuotientCategory( C, L ) ( operation )
 ‣ /( C, L ) ( operation )

Returns: a CAP category

Construct the finitely presented category generated by the quiver q, possibly modulo the relations L.

##### 2.4-2 ObjectInFpCategory
 ‣ ObjectInFpCategory( A, V ) ( operation )

Returns: an object in a CAP category

The constructor of objects in a finitely presented category C given a vertex V in the underlying quiver.

##### 2.4-3 MorphismInFpCategory
 ‣ MorphismInFpCategory( S, path, T ) ( operation )
 ‣ MorphismInFpCategory( A, path ) ( operation )

Returns: an object in a CAP category a morphism in a CAP category

Delegates to ObjectInFpCategory( C, V ). The constructor of morphisms in a finitely presented category C given the source S, the target T, and the underlying path path. If neither S nor T are provided they are read off from path.

##### 2.4-4 \/
 ‣ \/( path, A ) ( operation )

Returns: a morphism in a CAP category

Delegates to MorphismInFpCategory( path ).

#### 2.5 Global functions

##### 2.5-1 SimplicialCategoryTruncatedInDegree
 ‣ SimplicialCategoryTruncatedInDegree( n ) ( function )

Returns: a CAP category

The full subcategory of the simplicial category $$\Delta$$ on the objects $$[0], \ldots, [n]$$.

#### 2.6 GAP categories

##### 2.6-1 IsFpCategory
 ‣ IsFpCategory( arg ) ( filter )

Returns: true or false

The GAP category of finitely presented categories.

##### 2.6-2 IsMonoidAsCategory
 ‣ IsMonoidAsCategory( arg ) ( filter )

Returns: true or false

The GAP category of algebras.

##### 2.6-3 IsCellInFpCategory
 ‣ IsCellInFpCategory( arg ) ( filter )

Returns: true or false

The GAP category of cells in a finitely presented category.

##### 2.6-4 IsObjectInFpCategory
 ‣ IsObjectInFpCategory( arg ) ( filter )

Returns: true or false

The GAP category of objects in a finitely presented category.

##### 2.6-5 IsMorphismInFpCategory
 ‣ IsMorphismInFpCategory( arg ) ( filter )

Returns: true or false

The GAP category of morphisms in a finitely presented category.

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