From 9d41f9821c652a08d3cd551da3ec4cb31d6677db Mon Sep 17 00:00:00 2001
From: Yann Esposito
All endofunctors of \(\C\) form the category \(\E_\C\) of endofunctors of \(\C\).
- -F a -> G a
.If \(\C\) is small (\(\hom{\C}\) is a set). All functors from \(\C\) to some category \(\D\) form the category \(\mathrm{Func}(\C,\D)\).
+ +F a -> G a
\(\mathrm{Func}(\C,\C)\) is the category of endofunctors of \(\C\).
+data Tree a = Empty | Node a [Tree a]
toTree :: [a] -> Tree a
toTree [] = Empty
@@ -676,7 +682,7 @@ toTree (x:xs) = Node x [toTree xs]
data Tree a = Empty | Node a [Tree a]
toList :: Tree a -> [a]
toList Empty = []
@@ -691,17 +697,29 @@ toList (Node x l) = [x] ++ concat (map toList l)
-toMaybe :: [a] -> Maybe a ; mToList :: Maybe a -> [a]
-toMaybe [] = Nothing ; mToList Nothing = []
-toMaybe (x:xs) = Just x ; mToList Just x = [x]
-
+toMaybe :: [a] -> Maybe a
+toMaybe [] = Nothing
+toMaybe (x:xs) = Just x
toMaybe
is a natural transformation. It is also a morphism from []
to Maybe
in the Category of \(\) endofunctors.
mToList :: Maybe a -> [a]
+mToList Nothing = []
+mToList Just x = [x]
+
+toMaybe
is a natural transformation. It is also a morphism from []
to Maybe
in the Category of \(\) endofunctors.
All endofunctors of \(\C\) form the category \(\E_\C\) of endofunctors of \(\C\).
- -F a -> G a
.If \(\C\) is small (\(\hom{\C}\) is a set). All functors from \(\C\) to some category \(\D\) form the category \(\mathrm{Func}(\C,\D)\).
+ +F a -> G a
\(\mathrm{Func}(\C,\C)\) is the category of endofunctors of \(\C\).
diff --git a/categories/30_How/170_Category_of_Functors.md b/categories/30_How/170_Category_of_Functors.md new file mode 100644 index 0000000..aac320b --- /dev/null +++ b/categories/30_How/170_Category_of_Functors.md @@ -0,0 +1,18 @@ +Category of Functors +------------------------ + +If \\(\\C\\) is _small_ (\\(\\hom{\\C}\\) is a set). +All functors from \\(\\C\\) to some category \\(\\D\\) +form the category \\(\\mathrm{Func}(\\C,\\D)\\). + + + +- \\(\\ob{\\mathrm{Func}(\\C,\\D)}\\): Functors \\(F:\\C→\\D\\) +- \\(\\hom{\\mathrm{Func}(\\C,\\D)}\\): _natural transformations_ + - η familly \\(η\_X\\in\\hom{\\D}\\) for \\(X\\in\\ob{\\C}\\) s.t. + - ex: between Haskel functors; `F a -> G a` + Rearragement functions only. +- ∘: Functor composition + + +\\(\\mathrm{Func}(\\C,\\C)\\) is the category of endofunctors of \\(\\C\\). diff --git a/categories/30_How/170_Natural_Transformation_Examples.html b/categories/30_How/180_Natural_Transformation_Examples_1_4.html similarity index 87% rename from categories/30_How/170_Natural_Transformation_Examples.html rename to categories/30_How/180_Natural_Transformation_Examples_1_4.html index 9cd45aa..1a7b14c 100644 --- a/categories/30_How/170_Natural_Transformation_Examples.html +++ b/categories/30_How/180_Natural_Transformation_Examples_1_4.html @@ -1,4 +1,4 @@ -data Tree a = Empty | Node a [Tree a] toTree :: [a] -> Tree a toTree [] = Empty diff --git a/categories/30_How/170_Natural_Transformation_Examples.md b/categories/30_How/180_Natural_Transformation_Examples_1_4.md similarity index 94% rename from categories/30_How/170_Natural_Transformation_Examples.md rename to categories/30_How/180_Natural_Transformation_Examples_1_4.md index 71dca39..9a6bd3d 100644 --- a/categories/30_How/170_Natural_Transformation_Examples.md +++ b/categories/30_How/180_Natural_Transformation_Examples_1_4.md @@ -1,4 +1,4 @@ -Natural Transformation Examples (1/3) +Natural Transformation Examples (1/4) ------------------------------------
`toMaybe` is a natural transformation. @@ -14,7 +12,4 @@ It is also a morphism from `[]` to `Maybe` in the Category of \\(\Hask\\) endofu diff --git a/categories/30_How/210_Natural_Transformation_Examples_4_4.html b/categories/30_How/210_Natural_Transformation_Examples_4_4.html new file mode 100644 index 0000000..f6ec1df --- /dev/null +++ b/categories/30_How/210_Natural_Transformation_Examples_4_4.html @@ -0,0 +1,12 @@ +data Tree a = Empty | Node a [Tree a] diff --git a/categories/30_How/190_Another_Nat_Trans_Example.html b/categories/30_How/190_Another_Nat_Trans_Example.html deleted file mode 100644 index d8807ad..0000000 --- a/categories/30_How/190_Another_Nat_Trans_Example.html +++ /dev/null @@ -1,14 +0,0 @@ -
-Natural Transformation Examples (3/3)
---toMaybe :: [a] -> Maybe a ; mToList :: Maybe a -> [a] -toMaybe [] = Nothing ; mToList Nothing = [] -toMaybe (x:xs) = Just x ; mToList Just x = [x] -
- - - - diff --git a/categories/30_How/180_Natural_Transformation_Examples.html b/categories/30_How/190_Natural_Transformation_Examples_2_4.html similarity index 90% rename from categories/30_How/180_Natural_Transformation_Examples.html rename to categories/30_How/190_Natural_Transformation_Examples_2_4.html index 30962c3..63306ec 100644 --- a/categories/30_How/180_Natural_Transformation_Examples.html +++ b/categories/30_How/190_Natural_Transformation_Examples_2_4.html @@ -1,4 +1,4 @@ -
toMaybe
is a natural transformation. It is also a morphism from[]
toMaybe
in the Category of \(\) endofunctors.Natural Transformation Examples (2/3)
+Natural Transformation Examples (2/4)
data Tree a = Empty | Node a [Tree a] toList :: Tree a -> [a] toList Empty = [] diff --git a/categories/30_How/180_Natural_Transformation_Examples.md b/categories/30_How/190_Natural_Transformation_Examples_2_4.md similarity index 95% rename from categories/30_How/180_Natural_Transformation_Examples.md rename to categories/30_How/190_Natural_Transformation_Examples_2_4.md index c5274cd..bd3212a 100644 --- a/categories/30_How/180_Natural_Transformation_Examples.md +++ b/categories/30_How/190_Natural_Transformation_Examples_2_4.md @@ -1,4 +1,4 @@ -Natural Transformation Examples (2/3) +Natural Transformation Examples (2/4) ------------------------------------
+data Tree a = Empty | Node a [Tree a] diff --git a/categories/30_How/200_Natural_Transformation_Examples_3_4.html b/categories/30_How/200_Natural_Transformation_Examples_3_4.html new file mode 100644 index 0000000..ee5ebc9 --- /dev/null +++ b/categories/30_How/200_Natural_Transformation_Examples_3_4.html @@ -0,0 +1,12 @@ +
+Natural Transformation Examples (3/4)
++toMaybe :: [a] -> Maybe a +toMaybe [] = Nothing +toMaybe (x:xs) = Just x
+ + + + diff --git a/categories/30_How/190_Another_Nat_Trans_Example.md b/categories/30_How/200_Natural_Transformation_Examples_3_4.md similarity index 55% rename from categories/30_How/190_Another_Nat_Trans_Example.md rename to categories/30_How/200_Natural_Transformation_Examples_3_4.md index 85f4cb5..42040a7 100644 --- a/categories/30_How/190_Another_Nat_Trans_Example.md +++ b/categories/30_How/200_Natural_Transformation_Examples_3_4.md @@ -1,11 +1,9 @@ -Natural Transformation Examples (3/3) +Natural Transformation Examples (3/4) ------------------------------------- -
toMaybe
is a natural transformation. It is also a morphism from[]
toMaybe
in the Category of \(\) endofunctors.-toMaybe :: [a] -> Maybe a ; mToList :: Maybe a -> [a] -toMaybe [] = Nothing ; mToList Nothing = [] -toMaybe (x:xs) = Just x ; mToList Just x = [x] -
toMaybe :: [a] -> Maybe a +toMaybe [] = Nothing +toMaybe (x:xs) = Just x
Natural Transformation Examples (4/4)
++ +mToList :: Maybe a -> [a] +mToList Nothing = [] +mToList Just x = [x]
+ + + + diff --git a/categories/30_How/210_Natural_Transformation_Examples_4_4.md b/categories/30_How/210_Natural_Transformation_Examples_4_4.md new file mode 100644 index 0000000..519ea17 --- /dev/null +++ b/categories/30_How/210_Natural_Transformation_Examples_4_4.md @@ -0,0 +1,18 @@ +Natural Transformation Examples (4/4) +------------------------------------- + +
toMaybe
is a natural transformation. It is also a morphism from[]
toMaybe
in the Category of \(\) endofunctors.+ + +`toMaybe` is a natural transformation. +It is also a morphism from `[]` to `Maybe` in the Category of \\(\Hask\\) endofunctors. + + + diff --git a/categories/30_How/210_Monads.html b/categories/30_How/220_Monads.html similarity index 100% rename from categories/30_How/210_Monads.html rename to categories/30_How/220_Monads.html diff --git a/categories/30_How/210_Monads.md b/categories/30_How/220_Monads.md similarity index 100% rename from categories/30_How/210_Monads.md rename to categories/30_How/220_Monads.md diff --git a/categories/30_How/220_Monads_are_just_monoids_1_4.html b/categories/30_How/230_Monads_are_just_monoids_1_4.html similarity index 100% rename from categories/30_How/220_Monads_are_just_monoids_1_4.html rename to categories/30_How/230_Monads_are_just_monoids_1_4.html diff --git a/categories/30_How/220_Monads_are_just_monoids_1_4.md b/categories/30_How/230_Monads_are_just_monoids_1_4.md similarity index 100% rename from categories/30_How/220_Monads_are_just_monoids_1_4.md rename to categories/30_How/230_Monads_are_just_monoids_1_4.md diff --git a/categories/30_How/230_Monads_are_just_Monoids_2_4.html b/categories/30_How/240_Monads_are_just_Monoids_2_4.html similarity index 100% rename from categories/30_How/230_Monads_are_just_Monoids_2_4.html rename to categories/30_How/240_Monads_are_just_Monoids_2_4.html diff --git a/categories/30_How/230_Monads_are_just_Monoids_2_4.md b/categories/30_How/240_Monads_are_just_Monoids_2_4.md similarity index 100% rename from categories/30_How/230_Monads_are_just_Monoids_2_4.md rename to categories/30_How/240_Monads_are_just_Monoids_2_4.md diff --git a/categories/30_How/240_Monads_are_just_Monoids_3_4.html b/categories/30_How/250_Monads_are_just_Monoids_3_4.html similarity index 100% rename from categories/30_How/240_Monads_are_just_Monoids_3_4.html rename to categories/30_How/250_Monads_are_just_Monoids_3_4.html diff --git a/categories/30_How/240_Monads_are_just_Monoids_3_4.md b/categories/30_How/250_Monads_are_just_Monoids_3_4.md similarity index 100% rename from categories/30_How/240_Monads_are_just_Monoids_3_4.md rename to categories/30_How/250_Monads_are_just_Monoids_3_4.md diff --git a/categories/30_How/250_Monads_are_just_Monoids_4_4.html b/categories/30_How/260_Monads_are_just_Monoids_4_4.html similarity index 100% rename from categories/30_How/250_Monads_are_just_Monoids_4_4.html rename to categories/30_How/260_Monads_are_just_Monoids_4_4.html diff --git a/categories/30_How/250_Monads_are_just_Monoids_4_4.md b/categories/30_How/260_Monads_are_just_Monoids_4_4.md similarity index 100% rename from categories/30_How/250_Monads_are_just_Monoids_4_4.md rename to categories/30_How/260_Monads_are_just_Monoids_4_4.md diff --git a/categories/30_How/260_Monads_generalize_composition.html b/categories/30_How/270_Monads_generalize_composition.html similarity index 100% rename from categories/30_How/260_Monads_generalize_composition.html rename to categories/30_How/270_Monads_generalize_composition.html diff --git a/categories/30_How/260_Monads_generalize_composition.md b/categories/30_How/270_Monads_generalize_composition.md similarity index 100% rename from categories/30_How/260_Monads_generalize_composition.md rename to categories/30_How/270_Monads_generalize_composition.md diff --git a/categories/30_How/270_Monads_utility.html b/categories/30_How/280_Monads_utility.html similarity index 100% rename from categories/30_How/270_Monads_utility.html rename to categories/30_How/280_Monads_utility.html diff --git a/categories/img/mp/cat-hask-endofunctor.mp b/categories/img/mp/cat-hask-endofunctor.mp new file mode 100644 index 0000000..c799041 --- /dev/null +++ b/categories/img/mp/cat-hask-endofunctor.mp @@ -0,0 +1,108 @@ +% DPI 180 +pair int,bool,func,intlist,list; +resize(.8cm); +int=origin; +% drawoptions (withcolor baseOne withpen pencircle scaled 1.2); + +intlist=int shifted (gu,0); +func=int shifted (0,-.6gu); +bool=func shifted (gu,0); +list=func shifted (2u,0); + + +drawLoopPoint(int,int shifted (0,1.6u),"\mathtt{sqrt}"); +drawEdge(int,int,"\mathtt{id}"); +drawEdgeAngle(int,bool,"\mathtt{odd}",-30); +drawEdgeAngle(int,bool,"\mathtt{even}",30); +drawEdge(intlist,intlist,"\mathtt{tail}"); +drawEdge(intlist,int,"\mathtt{length}"); + +% ---------- F ---------------- + +pair decal; +pair fint,fbool,ffunc,fintlist,flist; +decal := (-1.2gu,-1.5gu); +fint=int shifted decal; +fintlist=intlist shifted decal; +ffunc= func shifted decal; +fbool= bool shifted decal; +flist= list shifted decal; + +drawLoopPoint(fint,fint shifted (0,1.9u),"\scriptstyle\mathtt{fmap\ sqrt}"); +drawEdge(fint,fint,"\scriptstyle\mathtt{fmap\ id}"); +drawEdgeAngle(fint,fbool,"\scriptstyle\mathtt{fmap\ odd}",-30); +drawEdgeAngle(fint,fbool,"\scriptstyle\mathtt{fmap\ even}",30); +drawEdge(fintlist,fintlist,"\scriptstyle\mathtt{fmap\ tail}"); +drawEdge(fintlist,fint,"\scriptstyle\mathtt{fmap\ length}"); + +drawblock(fint,fbool,"\mathtt{F}"); + +pair ff; +ff = .5[fint,fintlist] shifted (0,u); +path b; +b := block(origin,origin); +draw b scaled .5 shifted ff; +label.top(btex $\mathtt{F\ F}$ etex, blockLabelPosition(b)) scaled .5 shifted ff; +draw b scaled .25 shifted ff; +label.top(btex $\mathtt{F\ F\ F}$ etex, blockLabelPosition(b)) scaled .25 shifted ff; + +% ------------ G -------------- +pair gdecal; +pair gint,gbool,gfunc,gintlist,glist; +gdecal := (1.2gu,-1.5gu); +gint=int shifted gdecal; +gintlist=intlist shifted gdecal; +gfunc= func shifted gdecal; +gbool= bool shifted gdecal; +glist= list shifted gdecal; + +drawLoopPoint(gint,gint shifted (0,1.9u),"\scriptstyle\mathtt{fmap\ sqrt}"); +drawEdge(gint,gint,"\scriptstyle\mathtt{fmap\ id}"); +drawEdgeAngle(gint,gbool,"\scriptstyle\mathtt{fmap\ odd}",-30); +drawEdgeAngle(gint,gbool,"\scriptstyle\mathtt{fmap\ even}",30); +drawEdge(gintlist,gintlist,"\scriptstyle\mathtt{fmap\ tail}"); +drawEdge(gintlist,gint,"\scriptstyle\mathtt{fmap\ length}"); + +drawblock(gint,gbool,"\mathtt{G}"); + +pair gf; +gf = .5[gint,gintlist] shifted (0,u); +path b; +b := block(origin,origin); +draw b scaled .5 shifted gf; +label.top(btex $\mathtt{G\ G}$ etex, blockLabelPosition(b)) scaled .5 shifted gf; +draw b scaled .25 shifted gf; +label.top(btex $\mathtt{G\ G\ G}$ etex, blockLabelPosition(b)) scaled .25 shifted gf; + + +drawState(int,"\mathtt{Int}"); +drawState(intlist,"\mathtt{[Int]}"); +drawState(func,"\mathtt{a\rightarrow b}"); +drawState(list,"\mathtt{[a]}"); +drawState(bool,"\mathtt{Bool}"); + +drawState(fint,"\mathtt{F\ Int}"); +drawState(fintlist,"\mathtt{F\ [Int]}"); +drawState(ffunc,"\mathtt{F\ a\rightarrow b}"); +drawState(flist,"\mathtt{F\ [a]}"); +drawState(fbool,"\mathtt{F\ Bool}"); + +drawState(gint,"\mathtt{G\ Int}"); +drawState(gintlist,"\mathtt{G\ [Int]}"); +drawState(gfunc,"\mathtt{G\ a\rightarrow b}"); +drawState(glist,"\mathtt{G\ [a]}"); +drawState(gbool,"\mathtt{G\ Bool}"); + +drawoptions(withcolor yellow); +label.top(btex $I$ etex, .5[int,intlist] shifted (0,u)); + +drawoptions(withpen pencircle scaled 2bp withcolor yellow); +ahlength:=.3u; + +drawEdgeAngle(.5[int,func] shifted (-.5u,0), ff shifted (0,2.2u), "?", -30); +drawEdgeAngle(ff shifted (-u,2.2u), .5[int,func] shifted (-.5u,u), "?", 30); +drawEdgeAngle(.5[intlist,bool] shifted (.5u,0), gf shifted (0,2.2u), "?", 30); +drawEdgeAngle(gf shifted (u,2.2u), .5[intlist,bool] shifted (.5u,u), "?", -30); + +drawEdgeAngle(.5[fintlist,fbool] shifted (1.1u,u), .5[gint,gfunc] shifted (-1.1u,u), "?", 30); +drawEdgeAngle(.5[gint,gfunc] shifted (-1.1u,u), .5[fintlist,fbool] shifted (1.1u,u), "?", 30); diff --git a/categories/img/mp/cat-hask-endofunctor.png b/categories/img/mp/cat-hask-endofunctor.png new file mode 100644 index 0000000000000000000000000000000000000000..12bd2477f93c511c3f25db3385c2ac135215770b GIT binary patch literal 889470 zcmagF1yoz#wmnRXOK=JVm*NtvxVsf7F2%hqS{#DAyA~3lKyfSX4#8TAv`{R#yUU;V z-tF)9-S>_0Wt`-kaYjaR_Fl5*nse<)_1E$M3?K#q0s=r$LFNqt0#Y~v0wO*x%F`Xl zzGo2v0y?vuw6waNg#`kF=daX68}*d%7hw~Ze)E_g(m(fp&(>nbRM7WMHI_wAThvdZ zHJ9!~lOe>ji=xgC2-0B2XU*3`!-&$GWWz^FA&H@7VY-w*P|5LO6hFE1-uUD(F35u# zK#PJ95cRCE5aSBWaj};$33}x#PtVEoX%`Mhe;gXrkU&77ED!R>c`3w@jEVq5NSILG z#6(CiLYN%ED}1^rKmToList :: Maybe a -> [a] +mToList Nothing = [] +mToList Just x = [x]
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