d&i YonedaProjective

PackageInfo.g

@@ -11,7 +11,7 @@ SetPackageInfo( rec(
 PackageName := "CatReps",
 Subtitle := "Representations and cohomology of finite categories",
 Version := Maximum( [
-                   "2020.02.19", ## Mohamed's version
+                   "2020.02.20", ## Mohamed's version
                    ## this line prevents merge conflicts
                    "2020.01.01", ## Tibor's version
                    ## this line prevents merge conflicts

examples/ConcreteCategoryWithEndomorphismGroups.g

@@ -22,6 +22,9 @@ Perform( gmorphisms, Display );
 #! [ [ 1, 2, 3 ], [ [ 1, 4 ], [ 2, 5 ], [ 3, 6 ] ], [ 4, 5, 6 ] ]
 #! A morphism in subcategory given by:
 #! [ [ 4, 5, 6 ], [ [ 4, 5 ], [ 5, 6 ], [ 6, 4 ] ], [ 4, 5, 6 ] ]
+#! @EndExample
+
+#! @Example
 qc3c3 := RightQuiver( "q(2)[a:1->1,b:1->2,c:2->2]" );
 #! q(2)[a:1->1,b:1->2,c:2->2]
 HOMALG_MATRICES.PreferDenseMatrices := true;
@@ -596,4 +599,72 @@ Display( iso );
 #!  . . . . . . . . . . . 1 . . . .
 #! 
 #! An isomorphism in Category of matrices over GF(3)
+proj1 := YonedaProjective( CatReps, kq.1 );
+#! <A projective object in The category of functors: Bialgebroid
+#!  generated by the right quiver q(2)[a:1->1,b:1->2,c:2->2]
+#!  -> Category of matrices over GF(3)>
+Display( proj1 );
+#! An object in The category of functors: Bialgebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of (1):
+#! A vector space object over GF(3) of dimension 3
+#! 
+#! Image of (2):
+#! A vector space object over GF(3) of dimension 3
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#!  . 1 .
+#!  . . 1
+#!  1 . .
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#!  1 . .
+#!  . 1 .
+#!  . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  . 1 .
+#!  . . 1
+#!  1 . .
+#! 
+#! A morphism in Category of matrices over GF(3)
+proj2 := YonedaProjective( CatReps, kq.2 );
+#! <A projective object in The category of functors: Bialgebroid
+#!  generated by the right quiver q(2)[a:1->1,b:1->2,c:2->2]
+#!  -> Category of matrices over GF(3)>
+Display( proj2 );
+#! An object in The category of functors: Bialgebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of (1):
+#! A vector space object over GF(3) of dimension 0
+#! 
+#! Image of (2):
+#! A vector space object over GF(3) of dimension 3
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#! (an empty 0 x 0 matrix)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#! (an empty 0 x 3 matrix)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  . 1 .
+#!  . . 1
+#!  1 . .
+#! 
+#! A morphism in Category of matrices over GF(3)
 #! @EndExample

gap/CatRepsWithCAP.gd

@@ -115,6 +115,15 @@ DeclareOperation( "EmbeddingOfSubRepresentation",
 DeclareOperation( "WeakDirectSumDecomposition",
         [ IsCapCategoryObjectInHomCategory ] );
 
+#! @Description
+#!  Return Yoneda's projective representation given by the object <A>o</A>,
+#!  i.e., the submodule of the category algebra consisting of all arrows
+#!  starting at <A>o</A>.
+#! @Arguments H, o
+#! @Returns IsCapCategoryObjectInHomCategory
+DeclareOperation( "YonedaProjective",
+        [ IsCapHomCategory, IsCapCategoryObject ] );
+
 ####################################
 #
 #! @Section Tools

gap/CatRepsWithCAP.gi

@@ -189,7 +189,80 @@ InstallMethod( WeakDirectSumDecomposition,
     k := CommutativeRingOfLinearCategory( kq );
 
     d := List( d, eta -> List( [ 1 .. Length( eta ) ], i -> VectorSpaceMorphism( VectorSpaceObject( Length( eta[i] ), k ), eta[i], F( kq.(i) ) ) ) );
-
+    
     return List( d, eta -> EmbeddingOfSubRepresentation( eta, F ) );
 
 end );
+
+##
+InstallMethod( YonedaProjective,
+        "for a Hom-category and a CAP object",
+        [ IsCapHomCategory, IsCapCategoryObject ],
+
+  function( CatReps, o )
+    local kq, k, basis_list, A, basis, dimensions, a, arrows, matrices,
+          source, target, dim_source, dim_target, b, b_source, b_target,
+          matrix, b_a_path, b_a, coeffs, yproj;
+
+    kq := Source( CatReps );
+
+    o := Position( SetOfObjects( kq ), o );
+
+    k := CommutativeRingOfLinearCategory( CatReps );
+
+    ## code from QPA2/lib/special-representations.gi
+
+    A := UnderlyingQuiverAlgebra( kq );
+
+    basis_list := BasisOfProjectives( A );
+    basis := basis_list[ o ];
+
+    dimensions := List( basis, Length );
+    arrows := Arrows( QuiverOfAlgebra( A ) );
+    matrices := [ ];
+
+    for a in arrows do
+
+        source := VertexIndex( Source( a ) );
+        target := VertexIndex( Target( a ) );
+
+        dim_source := dimensions[ source ];
+        dim_target := dimensions[ target ];
+
+        if dim_source = 0 or dim_target = 0 then
+
+            matrix := HomalgZeroMatrix( dim_source, dim_target, k );
+
+        else
+
+            b_source := List( basis[ source ], b -> Paths(b)[ 1 ] );
+            b_target := List( basis[ target ], b -> Paths(b)[ 1 ] );
+
+            matrix := [ ];
+
+            for b in b_source do
+                b_a_path := ComposePaths( b, a );
+                b_a := PathAsAlgebraElement( A, b_a_path );
+                coeffs := CoefficientsOfPaths( b_target, b_a );
+                Add( matrix, coeffs );
+            od;
+
+            matrix := HomalgMatrix( matrix, dim_source, dim_target, k );
+
+            matrix := VectorSpaceMorphism( VectorSpaceObject( dim_source, k ), matrix, VectorSpaceObject( dim_target, k ) );
+
+        fi;
+
+        Add( matrices, matrix );
+
+    od;
+
+    dimensions := List( dimensions, dim -> VectorSpaceObject( dim, k ) );
+
+    yproj := AsObjectInHomCategory( kq, dimensions, matrices );
+
+    SetIsProjective( yproj, true );
+
+    return yproj;
+
+end );