From: Chris <chris.barker@nyu.edu>
Date: Mon, 16 Mar 2015 15:59:49 +0000 (-0400)
Subject: edits
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=84fa57920eb828f2ba7c124378c7ca79b3c4bd3c

edits
---

84fa57920eb828f2ba7c124378c7ca79b3c4bd3c
diff --cc topics/_week7_monads.mdwn
index a751da29,e8785bbd..50b2cb89
--- a/topics/_week7_monads.mdwn
+++ b/topics/_week7_monads.mdwn
@@@ -10,44 -12,10 +10,43 @@@ monsters, monads are burritos.  We're p
  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.
  
 -<table border=2>x</table>
 +### Boxes: type expressions with one free type variable
 +
 +Recall that we've been using lower-case Greek letters
 +<code>&alpha;, &beta;, &gamma;, ...</code> 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 ≡ Int
 +    P ≡ α -> α
 +    P ≡ ∀α. α -> α
 +    P ≡ ∀α. α -> β 
 +
 +etc.
 +
 +A box type will be a type expression that contains exactly one free
 +type variable.  Some examples (using OCaml's type conventions):
 +
 +    α Maybe
 +    α List
 +    (α, P) Tree    (assuming P contains no free type variables)
 +    (α, α) Tree
 +
 +The idea is that whatever type the free type variable α might be,
 +the boxed type will be a box that "contains" an object of type α.
 +For instance, if `α List` is our box type, and α 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 `α List`, and `α == Int`, we
 +would write
 +
- <code><table border>Int</table></code>
- 
++<table border>Int</table>
 +