better graphs
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Yann Esposito
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1 changed files with 134 additions and 30 deletions
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@ -71,6 +71,21 @@ And will discuss about some categories.
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> - for all triplet of morphisms \\(h:A->B\\), \\(g:B->C\\) and \\(f:C->D\\)
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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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> \\( (f∘g)∘h = f∘(g∘h) \\)
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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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<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"]
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B -> D [label="h∘g",style="bold",fontcolor="yellow",color="yellow"]
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A -> D [label="(h∘g)∘f=h∘(g∘f)",style="bold",color="red",fontcolor="red"]
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</graph>
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### Representation of Category
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### Representation of Category
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Representing Category is not just a game.
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Representing Category is not just a game.
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@ -121,9 +136,9 @@ g -> C
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fg [label="", fixedsize="false", width=0,height=0,shape=none];
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fg [label="", fixedsize="false", width=0,height=0,shape=none];
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AC [label="", fixedsize="false", width=0,height=0,shape=none];
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AC [label="", fixedsize="false", width=0,height=0,shape=none];
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f -> fg [color="#b58900",style=dashed,arrowhead=None]
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f -> fg [color="red",style=dashed,arrowhead=None]
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fg -> g [color="#b58900",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",colorlabel="#b58900",color="#b58900",style=bold]
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fg -> AC [label="h=g∘f",fontcolor="red",color="red",style=bold]
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A -> AC [label="h",arrowhead=None]
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A -> AC [label="h",arrowhead=None]
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AC -> C
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AC -> C
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@ -140,44 +155,47 @@ we just double the number morphisms between different objects.
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<graph title="Naïve Category Representation Mess">
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<graph title="Naïve Category Representation Mess">
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f[label="", fixedsize="false", width=0,height=0,shape=none];
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A[pos="0,0!"]
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B[pos="4,0!"]
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C[pos="8,0!"]
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f[pos="2,1!",label="", fixedsize="false", width=0,height=0,shape=none];
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A -> f[label="f", arrowhead=None]
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A -> f[label="f", arrowhead=None]
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f -> B
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f -> B
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fp[label="", fixedsize="false", width=0,height=0,shape=none];
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fp[pos="2,0.5!",label="", fixedsize="false", width=0,height=0,shape=none];
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A -> fp[label="f'", arrowhead=None]
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A -> fp[label="f'", arrowhead=None]
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fp -> B
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fp -> B
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g[label="", fixedsize="false", width=0,height=0,shape=none];
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g[pos="6,0.5!",label="", fixedsize="false", width=0,height=0,shape=none];
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B -> g[label="g", arrowhead=None]
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B -> g[label="g", arrowhead=None]
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g -> C
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g -> C
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gp[label="", fixedsize="false", width=0,height=0,shape=none];
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gp[pos="6,1!",label="", fixedsize="false", width=0,height=0,shape=none];
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B -> gp[label="g'", arrowhead=None]
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B -> gp[label="g'", arrowhead=None]
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gp -> C
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gp -> C
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fg[label="", fixedsize="false", width=0,height=0,shape=none];
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fg[pos="6,0!",label="", fixedsize="false", width=0,height=0,shape=none];
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fpg[label="", fixedsize="false", width=0,height=0,shape=none];
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fpg[pos="2.5,-1.5!",label="", fixedsize="false", width=0,height=0,shape=none];
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fgp[label="", fixedsize="false", width=0,height=0,shape=none];
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fgp[pos="5.5,-1.5!",label="", fixedsize="false", width=0,height=0,shape=none];
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fpgp[label="", fixedsize="false", width=0,height=0,shape=none];
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fpgp[pos="2,0!",label="", fixedsize="false", width=0,height=0,shape=none];
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AC[label="", fixedsize="false", width=0,height=0,shape=none];
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AC[pos="4,-1!",label="", fixedsize="false", width=0,height=0,shape=none];
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ApCp[label="", fixedsize="false", width=0,height=0,shape=none];
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ApCp[pos="4,-3!",label="", fixedsize="false", width=0,height=0,shape=none];
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f -> fg [color="#b58900",style=dashed,arrowhead=None]
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f -> fg [color="red",style=dashed,arrowhead=None]
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fg -> g [color="#b58900",style=dashed,arrowhead=None]
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fg -> g [color="red",style=dashed,arrowhead=None]
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fg -> AC [color="#b58900",style=bold,label="h=g∘f"]
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fg -> AC [color="red",style=bold,fontcolor="red",label="h=g∘f"]
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fp -> fpgp [color="#d33682",style=dashed,arrowhead=None]
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fp -> fpgp [color="yellow",style=dashed,arrowhead=None]
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fpgp -> gp [color="#d33682",style=dashed,arrowhead=None]
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fpgp -> gp [color="yellow",style=dashed,arrowhead=None]
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fpgp -> AC [color="#d33682",style=bold,label="h=g'∘f'"]
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fpgp -> AC [color="yellow",style=bold,fontcolor="yellow",label="h=g'∘f'"]
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fp -> fpg [color="#dc322f",style=dashed,arrowhead=None]
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fp -> fpg [color="blue",style=dashed,arrowhead=None]
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fpg -> g [color="#dc322f",style=dashed,arrowhead=None]
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fpg -> g [color="blue",style=dashed,arrowhead=None]
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fpg -> ApCp [color="#dc322f",style=bold,label="h'=g∘f'"]
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fpg -> ApCp [color="blue",style=bold,fontcolor="blue",label="h'=g∘f'"]
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f -> fgp [color="#268bd2",style=dashed,arrowhead=None]
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f -> fgp [color="violet",style=dashed,arrowhead=None]
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fgp -> gp [color="#268bd2",style=dashed,arrowhead=None]
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fgp -> gp [color="violet",style=dashed,arrowhead=None]
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fgp -> ApCp [color="#268bd2",style=bold,label="h'=g'∘f"]
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fgp -> ApCp [color="violet",style=bold,fontcolor="violet",label="h'=g'∘f"]
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A -> AC [label="h",arrowhead=None]
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A -> AC [label="h",arrowhead=None]
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AC -> C
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AC -> C
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@ -188,16 +206,102 @@ ApCp -> C
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</graph>
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</graph>
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In fact we could have made something equivalent and far easier to read.
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By removing the graphical representation of ∘ we could create a more readable representation.
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But the ∘ relation will be more hidden.
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<graph title="Less Graphic Category Representation">
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<graph title="Fewer Details Category Representation">
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A -> B[label="f"]
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A -> B[label="f"]
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A -> B[label="f'"]
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A -> B[label="f'"]
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B -> C[label="g"]
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B -> C[label="g"]
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B -> C[label="g'"]
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B -> C[label="g'"]
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A -> C [label="h=g∘f=g'∘f'"]
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A -> C [label="h\n=g∘f\n=g'∘f'"]
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A -> C [label="h'=g'∘f=g∘f'"]
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A -> C [label="h'\n=g'∘f\n=g∘f'"]
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</graph>
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### Examples
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Which can be a valid category by choosing ∘ appropriately?
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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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<graph title="Not a category; think about \(g∘f\).">
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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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<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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<graph title="Not a category; no \(A→C\) while there exists \(A→B\) and \(B→C\)">
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A -> B [constraint=false]
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B -> C [constraint=false]
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B -> A
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C -> A
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</graph>
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<graph title="Not a category; \((h∘g)∘f=idB∘f=f≠h=h∘idA=h∘(g∘f)\)">
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A -> B[label="g"]
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B -> A[label="f"]
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B -> A[label="h"]
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</graph>
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To continue to gain some intuition I will give some degenerated Category examples.
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### Monoids
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What are Monoids?
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Things that you can operate a list of in any evaluation order and obtain the same result.
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More precisely; let `l` be a list of elements of the monoid.
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then
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<code class="haskell">
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foldl (<>) e l = foldr (<>) e l
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</code>
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Where `(<>)` is the monoid operation.
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And `e` is the neutral element of the monoid.
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Equivalently:
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<code class="haskell">
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((e <> x) <> y) <> z = x <> ( y <> ( z <> e) )
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</code>
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Or another way of saying it is that `x <> y <> z` doesn't need any parenthesis.
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Because whatever the order of evaluation the result will be the same.
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Typical examples:
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- `String` with `(++)` and `""`
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- `Lists` with `(++)` and `[]`
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- `Data.Text` with `append` and `empty`
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- `Integer` with `(+)` and `0`
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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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★ -> ★[label="e"]
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★ -> ★[label="x"]
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★ -> ★[label="y"]
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★ -> ★[label="..."]
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</graph>
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</graph>
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