From: Chris
Date: Mon, 16 Mar 2015 15:57:42 +0000 (-0400)
Subject: monads
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=739888c429c1c16c40941d644cf218dd07f4fa27
monads
---
diff --git a/topics/_week7_monads.mdwn b/topics/_week7_monads.mdwn
index 21ac2f59..a751da29 100644
--- a/topics/_week7_monads.mdwn
+++ b/topics/_week7_monads.mdwn
@@ -1,3 +1,5 @@
+
+
Monads
======
@@ -8,10 +10,44 @@ monsters, monads are burritos. We're part of the backlash that
prefers to say that monads are monads.
The closest we will come to metaphorical talk is to suggest that
-monadic types place objects inside of boxes, and that monads wrap and
-unwrap boxes to expose or enclose the objects inside of them. In any
-case, the emphasis will be on starting with the abstract structure of
-monads, followed by instances of monads from the philosophical and
+monadic types place objects inside of *boxes*, and that monads wrap
+and unwrap boxes to expose or enclose the objects inside of them. In
+any case, the emphasis will be on starting with the abstract structure
+of monads, followed by instances of monads from the philosophical and
linguistics literature.
-To
+### Boxes: type expressions with one free type variable
+
+Recall that we've been using lower-case Greek letters
+`α, β, γ, ...`

to represent types. We'll
+use `P`, `Q`, `R`, and `S` as metavariables over type schemas, where a
+type schema is a type expression that may or may not contain unbound
+type variables. For instance, we might have
+
+ P â¡ Int
+ P â¡ Î± -> Î±
+ P â¡ âÎ±. Î± -> Î±
+ P â¡ âÎ±. Î± -> Î²
+
+etc.
+
+A box type will be a type expression that contains exactly one free
+type variable. Some examples (using OCaml's type conventions):
+
+ Î± Maybe
+ Î± List
+ (Î±, P) Tree (assuming P contains no free type variables)
+ (Î±, Î±) Tree
+
+The idea is that whatever type the free type variable Î± might be,
+the boxed type will be a box that "contains" an object of type Î±.
+For instance, if `Î± List` is our box type, and Î± is the basic type
+Int, then in this context, `Int List` is the type of a boxed integer.
+
+We'll often write box types as a box containing the value of the free
+type variable. So if our box type is `Î± List`, and `Î± == Int`, we
+would write
+
+`Int`

+
+