Functor: structure preserving mapping between categories
+- isomorphism: \(f:A→B\) which can be "undone" i.e.
+
\(∃g:B→A\), \(g∘f=id_A\) & \(f∘g=id_B\)
+
in this case, \(A\) & \(B\) are isomorphic.
+ - Dual: reverse direction of arrows of \(\C\)
+
- Initial: \(Z\in\ob{\C}\) s.t. \(∀Y∈\ob{\C}, \#\hom{Z,Y}=1\)
+
Unique ("up to isormophism")
+ - Terminal: \(T\in\ob{\C}\) s.t. \(T\) is initial in the dual of \(\C\)
+
- Functor: structure preserving mapping between categories
- ...
diff --git a/categories/20_What/200_Functor.html b/categories/20_What/200_Functor.html
index bcbaefa..6e05d16 100644
--- a/categories/20_What/200_Functor.html
+++ b/categories/20_What/200_Functor.html
@@ -2,14 +2,14 @@
A functor is a mapping between two categories.
Let \(\C\) and \(\D\) be two categories.
-A functor \(\F\) from \(\C\) to \(\D\):
+A functor \(\F\) from \(\C\) to \(\D\):
- - Associate objects: \(A\in\ob{\C}\) to \(\F(A)\)\( \in\ob{\D}\)
- - Associate morphisms: \(f:A\to B\) to \(\F(f) : \F(A) \to \F(B)\)
+
- Associate objects: \(A\in\ob{\C}\) to \(\F(A)\in\ob{\D}\)
+ - Associate morphisms: \(f:A\to B\) to \(\F(f) : \F(A) \to \F(B)\)
such that
- - \( \F (\id_X) = \id_{\F(X)} \),
- - \( \F (g \circ_\C f) = \F(g) \circ_\D \F(f) \)
+ - \( \F (\)\(\id_X\)\() = \)\(\id\)\(\vphantom{\id}_{\F(}\)\(\vphantom{\id}_X\)\(\vphantom{\id}_{)} \),
+ - \( \F (\)\(g∘f\)\() = \)\( \F(\)\(g\)\() \)\(\circ\)\( \F(\)\(f\)\() \)
diff --git a/categories/30_How/040_Haskell_Types.html b/categories/30_How/040_Haskell_Types.html
index e4cdc09..2956067 100644
--- a/categories/30_How/040_Haskell_Types.html
+++ b/categories/30_How/040_Haskell_Types.html
@@ -1,14 +1,16 @@
Haskell Types
We can make function that can work for all type parameter. Such function can only work with the topology induced by the type. We know such function won't work on the elements.
-Sometimes, the type determine a lot about the function:
+Sometimes, the type determine a lot about the function★:
fst :: (a,b) -> a -- Only one choice
snd :: (a,b) -> b -- Only one choice
f :: a -> [a] -- Many choices
-- Possibilities: f x=[], or [x], or [x,x] or [x,...,x]
? :: [a] -> [a] -- Many choices
--- can only duplicate/remove/reorder elements
+-- can only rearrange: duplicate/remove/reorder elements
-- for example: the type of addOne isn't [a] -> [a]
-addOne l = map (+1) l
-
+addOne l = map (+1) l
+-- The (+1) force a to be a Num.
+
+
★:Theorems for free!, Philip Wadler, 1989
diff --git a/categories/30_How/040_Haskell_Types.md b/categories/30_How/040_Haskell_Types.md
index 2bd72dd..27e790d 100644
--- a/categories/30_How/040_Haskell_Types.md
+++ b/categories/30_How/040_Haskell_Types.md
@@ -5,7 +5,7 @@ We can make function that can work for _all_ type parameter.
Such function can only work with the _topology_ induced by the type.
We know such function won't work _on_ the elements.
-Sometimes, the type determine a lot about the function:
+Sometimes, the type determine a lot about the function★:
fst :: (a,b) -> a -- Only one choice
snd :: (a,b) -> b -- Only one choice
@@ -13,6 +13,9 @@ f :: a -> [a] -- Many choices
-- Possibilities: f x=[], or [x], or [x,x] or [x,...,x]
? :: [a] -> [a] -- Many choices
--- can only duplicate/remove/reorder elements
+-- can only rearrange: duplicate/remove/reorder elements
-- for example: the type of addOne isn't [a] -> [a]
-addOne l = map (+1) l
+addOne l = map (+1) l
+-- The (+1) force a to be a Num.
+
+★:Theorems for free!, Philip Wadler, 1989
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diff --git a/themes/style/y/main.css b/themes/style/y/main.css
index a05bb5d..d22b128 100644
--- a/themes/style/y/main.css
+++ b/themes/style/y/main.css
@@ -68,10 +68,18 @@ body.deck-container {
color: #6c71c4; }
.deck-container .blue {
color: #268bd2; }
+ .deck-container .backblue {
+ border: solid 2px #268bd2;
+ border-radius: 2px;
+ background-color: rgba(38, 139, 210, 0.2); }
.deck-container .cyan {
color: #2aa198; }
.deck-container .green {
color: #859900; }
+ .deck-container .backgreen {
+ border: solid 2px #859900;
+ border-radius: 2px;
+ background-color: rgba(133, 153, 0, 0.2); }
.deck-container div {
font-family: "ComputerModernSansSerif", Helvetica, sans-serif; }
.deck-container .and {
@@ -260,6 +268,8 @@ body.deck-container {
margin-top: 0; }
.deck-container .small {
font-size: 0.8em; }
+ .deck-container .smaller {
+ font-size: 0.6em; }
.deck-container .sc {
text-transform: uppercase;
font-size: 0.8em; }
@@ -469,10 +479,9 @@ body.deck-container {
margin-bottom: 0; }
.deck-container img, .deck-container .likeimg {
max-width: 80%;
- border: 2px solid #586e75;
- background-color: #073642;
- padding: 0.5em;
- box-shadow: 0 0 10px #002b36 inset;
+ border: 2px solid #073642;
+ padding: 0.3em;
+ box-shadow: 0 0 0.5em black inset;
border-radius: 0.5em;
text-align: center; }
.deck-container a:hover img {
diff --git a/themes/style/y/main.sass b/themes/style/y/main.sass
index 48f0c9d..219ca3e 100644
--- a/themes/style/y/main.sass
+++ b/themes/style/y/main.sass
@@ -213,10 +213,18 @@ body.deck-container
color: $violet
.blue
color: $blue
+ .backblue
+ border: solid 2px $blue
+ border-radius: 2px
+ background-color: rgba(38,139,210,.2)
.cyan
color: $cyan
.green
color: $green
+ .backgreen
+ border: solid 2px $green
+ border-radius: 2px
+ background-color: rgba(133,153,0,.2)
font-family: "ComputerModernSansSerif",Helvetica,sans-serif
div
@@ -399,6 +407,8 @@ body.deck-container
// -- TYPOGRAPHY --
.small
font-size: 0.8em
+ .smaller
+ font-size: 0.6em
.sc
+sc
@@ -627,10 +637,9 @@ body.deck-container
img,.likeimg
max-width: 80%
- border: 2px solid $borderColor
- background-color: $imageBackgroundColor
- padding: 0.5em
- box-shadow: 0 0 10px $base03 inset
+ border: 2px solid $base02
+ padding: 0.3em
+ box-shadow: 0 0 .5em #000 inset
border-radius: 0.5em
text-align: center
a:hover img
+
+
+