metapost for better rendering
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Yann Esposito
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0b244e6615
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556ac267fb
1 changed files with 141 additions and 89 deletions
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@ -60,7 +60,7 @@ And will discuss about some categories.
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> - For every pair of objects \\((A,B)\\) a set \\(\hom(A,B)\\)
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> of _morphisms_ \\(f:A→B\\) (Another notation for \\(f\in \hom(A,B)\\))
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> - A composition operator \\(∘\\)
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> which associate to each couple \\(g:A→B\\), \\(f:B→C\\) another morphism \\(f∘g:A→C\\).
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> which associate to each couple \\(f:A→B\\), \\(g:B→C\\) another morphism \\(g∘f:A→C\\).
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>
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> With the following properties
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>
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@ -68,23 +68,45 @@ And will discuss about some categories.
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> \\(id_x:x→x\\)
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> s.t. for any morphism \\(f:A->B\\),
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> \\(id_A∘f = f = f∘id_B\\)
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> - for all triplet of morphisms \\(h:A->B\\), \\(g:B->C\\) and \\(f:C->D\\)
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> \\( (f∘g)∘h = f∘(g∘h) \\)
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> - for all triplet of morphisms \\(f:A->B\\), \\(g:B->C\\) and \\(h:C->D\\)
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> \\( (h∘g)∘f = h∘(g∘f) \\)
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<mpost title="Identity is left and right neutral element for ∘">
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z0=origin;
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z1=(2gu,0);
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<graph title="Identity is left and right neutral element for ∘">
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A -> A [label="idA"]
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B -> B [label="idB"]
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A -> B [label="f∘idA=f=idB∘f"]
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</graph>
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drawState(z0,btex $A$ etex);
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drawState(z1,btex $B$ etex);
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<graph title="Associativity">
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A -> B [label="f"]
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B -> C [label="g"]
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C -> D [label="h"]
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A -> C [label="g∘f",style="bold",fontcolor="cyan",color="cyan",constraint="false"]
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B -> D [label="h∘g",style="bold",fontcolor="yellow",color="yellow",constraint="false"]
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A -> D [label="(h∘g)∘f=h∘(g∘f)",style="bold",color="red",fontcolor="red",constraint="false"]
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</graph>
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drawLoop(z0,btex $id_A$ etex);
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drawLoop(z1,btex $id_B$ etex);
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drawEdge(z0,z1,btex $f\circ id_A = f = id_B\circ f$ etex);
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</mpost>
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<mpost title="Associativity">
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z0=origin;
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z1=(gu,0);
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z2=(2gu,0);
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z3=(3gu,0);
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drawState(z0,btex $A$ etex);
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drawState(z1,btex $B$ etex);
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drawState(z2,btex $C$ etex);
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drawState(z3,btex $D$ etex);
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drawEdge(z0,z1,btex $f$ etex);
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drawEdge(z1,z2,btex $g$ etex);
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drawEdge(z2,z3,btex $h$ etex);
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drawoptions(withcolor blue);
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drawEdgeWithAngle(z0,z2,btex $g\circ f$ etex,30);
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drawoptions(withcolor yellow);
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drawEdgeWithAngle(z1,z3,btex $h\circ g$ etex,-30);
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drawoptions(withcolor red);
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drawEdgeWithAngle(z0,z3,btex $(h\circ g)\circ f = h\circ (g\circ f)$ etex,45);
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</mpost>
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### Representation of Category
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@ -96,17 +118,24 @@ A naïve representation (which can work in many cases) is to represent
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a specific category as a directed graph.
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Here is a first example of the representation of a category:
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<graph title="First Naïve Category Representation">
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<mpost title="First Naïve Category Representation">
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z0=origin;
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z1=2/3(gu,gu);
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z2=(4/3gu,0);
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A -> B [label="f"]
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B -> C [label="g"]
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A -> C [label="h"]
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drawState(z0,btex $A$ etex);
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drawState(z1,btex $B$ etex);
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drawState(z2,btex $C$ etex);
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A -> A [label="idA"]
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B -> B [label="idB"]
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C -> C [label="idC"]
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drawEdge(z0,z1,btex $f$ etex);
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drawEdge(z1,z2,btex $g$ etex);
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drawEdge(z0,z2,btex $h$ etex);
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</graph>
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drawLoop(z0,btex $id_A$ etex);
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drawLoop(z1,btex $id_B$ etex);
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drawLoop(z2,btex $id_C$ etex);
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</mpost>
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From this graph we can conclude without any ambiguity that:
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@ -122,33 +151,35 @@ What is \\(∘\\)?
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Now, we can add informations to our previous representation.
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We simply add a relation between 3 arrows that represent the composition.
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<graph title="Naïve Category Representation">
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<mpost title="Naïve Category Representation">
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z0=origin;
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z1=(gu,gu);
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z2=(2gu,0);
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A[pos="0,0!"]
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B[pos="1.5,1.5!"]
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C[pos="3,0!"]
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drawState(z0,btex $A$ etex);
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drawState(z1,btex $B$ etex);
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drawState(z2,btex $C$ etex);
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f[pos="0.75,0.75!",label="", fixedsize="false", width=0,height=0,shape=none];
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A -> f[label="f", arrowhead=None]
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f -> B
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drawEdge(z0,z1,btex $f$ etex);
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drawEdge(z1,z2,btex $g$ etex);
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drawEdge(z0,z2,btex $h$ etex);
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g[pos="2.25,0.75!",label="", fixedsize="false", width=0,height=0,shape=none];
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B -> g[label="g", arrowhead=None]
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g -> C
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drawoptions( withcolor red );
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z3=.6[z0,z1];
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z4=.6[z2,z1];
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fg [pos="1.5,0.75!",label="", fixedsize="false", width=0,height=0,shape=none];
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AC [pos="1.5,0!",label="", fixedsize="false", width=0,height=0,shape=none];
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draw z3 -- z4 dashed evenly;
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f -> fg [color="red",style=dashed,arrowhead=None]
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fg -> g [color="red",style=dashed,arrowhead=None]
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fg -> AC [label="h=g∘f",fontcolor="red",color="red",style=bold]
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z5 = .45[z3,z4];
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z6 = .55[z0,z2];
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path bigarrow;
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bigarrow := subpath (.1,.9) of z5 -- z6;
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drawarrow bigarrow withpen pencircle scaled 1.5;
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A -> AC [label="h",arrowhead=None]
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AC -> C
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label.rt(btex $h = g \circ f$ etex, .5[z5,z6]);
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</graph>
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</mpost>
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Now we have a complete representation.
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We don't have to represent \\(idX\\), we know there are there.
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@ -228,42 +259,49 @@ A -> C [label="h'\n=g'∘f\n=g∘f'"]
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Which can be a valid category by choosing ∘ appropriately?
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The identity morphisms are assumed.
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<graph title="Can be a valid category">
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A[label="★"]
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B[label="★"]
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C[label="★"]
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A -> B
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B -> C
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A -> C [constraint=false]
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</graph>
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<mpost title="This can be a valid category">
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z0=(0,0);
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z1=2(u,0);
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z2=2(2u,0);
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<graph title="Cannot be a category; no candidate for \(g∘f\).">
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A[label="★"]
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B[label="★"]
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C[label="★"]
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A -> B[label="f"]
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B -> C[label="g"]
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</graph>
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drawedge(z0,z1,"");
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drawedge(z1,z2,"");
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drawedgeangle(z0,z2,"",50);
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<graph title="Also a valid category">
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A[label="★",pos="0,50"]
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B[label="★",pos="50,50"]
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C[label="★",pos="25,0"]
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A -> B
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B -> A
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B -> C
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A -> C
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</graph>
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drawstate(z0);
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drawstate(z1);
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drawstate(z2);
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</mpost>
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<graph title="Not a category; no \(A→C\) while there exists \(A→B\) and \(B→C\)">
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A[pos="0,0"]
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B[pos="2,0"]
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C[pos="1,-1"]
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A -> B
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B -> C
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B -> A
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C -> A
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</graph>
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<mpost title="Cannot be a category; no candidate for g∘f.">
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z0=(0,0); z1=3(u,0); z2=3(2u,0);
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drawstate(z0); drawstate(z1); drawstate(z2);
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drawedge(z0,z1,btex $f$ etex);
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drawedge(z1,z2,btex $g$ etex);
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</mpost>
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<mpost title="Also a possible valid category">
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z0=(0,0); z1=(4u,0); z2=(2u,-3u);
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drawstate(z0); drawstate(z1); drawstate(z2);
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drawedgeangle(z0,z1,"",35);
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drawedgeangle(z1,z0,"",-145);
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drawedge(z0,z2,"");
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drawedge(z1,z2,"");
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</mpost>
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<mpost title="Not a category; no \(A→C\) while there exists \(A→B\) and \(B→C\)">
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z0=(0,0); z1=(gu,0); z2=(.5gu,-.75gu);
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drawState(z0,btex $A$ etex);
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drawState(z1,btex $B$ etex);
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drawState(z2,btex $C$ etex);
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drawEdgeWithAngle(z0,z1,"",35);
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drawEdgeWithAngle(z1,z0,"",-145);
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drawEdge(z1,z2,"");
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drawEdge(z2,z0,"");
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</mpost>
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<graph title="Cannot be a category ; no possible associative ∘<br/>\((h∘g)∘f=idB∘f=f≠h=h∘idA=h∘(g∘f)\)">
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A -> B[label="g"]
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@ -305,19 +343,33 @@ Typical examples:
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- `Integer` with `(*)` and `1`
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- Generalized by `Monoid a` with `(<>)` and `mempty`
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<graph title="Strings are Categories">
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<mpost title="Strings are Categories">
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u:=.5cm;
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def drawloop(expr a,b,l) =
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pair ba,ea;
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path circ,p,s;
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p:=a{1,1}..b..{1,-1}cycle;
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circ:= fullcircle scaled 6 shifted a;
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ba = circ intersectionpoint (subpath (0,1) of p);
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ea = circ intersectionpoint (subpath (1,2) of p);
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s:= ba{1,1}..b..{1,-1}ea;
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drawarrow s;
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label.top(l,b);
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enddef;
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★ -> ★[label="e"]
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★ -> ★[label="x"]
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★ -> ★[label="y"]
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★ -> ★[label="..."]
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pair A,B,C,D,E,F;
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A:=(0,0);
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B:=(0,u);
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C:=(0,2u);
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D:=(0,3u);
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E:=(0,4u);
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F:=(0,6u);
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</graph>
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metapost test
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<mpost title="metapost test">
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pair A, B, C;
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A:=(0,0); B:=(1cm,0); C:=(0,1cm);
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draw A--B--C;
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drawloop(A,B,btex $\varepsilon$ etex);
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drawloop(A,C,btex $a$ etex);
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drawloop(A,D,btex $b$ etex);
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drawloop(A,E,btex $ab$ etex);
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drawloop(A,F,btex $\omega$ etex);
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draw (0,4.8u)--(0,5.8u) dashed withdots;
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draw A withpen pencircle scaled 4bp;
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</mpost>
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