[Haskell-cafe] Pierce on type theory and category theory

Dan Weston westondan at imageworks.com
Tue Sep 25 19:03:20 EDT 2007

The absolute easiest, clearest, and most entertaining book on Category 
Theory (which I highly recommend) is:

Conceptual Mathematics: A First Introduction to Categories (Paperback)
by F. William Lawvere and Stephen Hoel Schanuel, $25

It literally reads like a series of college lectures (called Article), 
each followed by study sections (called Sessions), complete with fake 
dialog with fake students asking the questions that are probably on your 
mind while reading the Articles.

It has a tone so conversational that working mathematicians will run 
screaming, but I have found it highly enlightening. With hardly a proff 
in sight, this is definitely the starter book par excellence of Category 

Here is a random excerpt on page 25, a dialog between Chad and the 

PROFESSOR: Is this correct? Not quite, because we are supposed to draw 
two maps, and the thing draw for h . g is not a map; one of the points 
of the domain of h . g has been left without an assigned output. This 
deficiency won't matter for the next step, because that information is 
going to get lost anyhow, but it belongs in this step and it is 
incorrect to omit it. Chad's trouble was that in drawing h . g, he 
notices that the last arrow would be irrelevant to the composite (h . g) 
. f, so he left it out.

CHAD: It seems the principle is like in multiplication, where the order 
in which you do things doesn't matter; you get the same answer.

PROFESSOR: I am glad you mention order. Let me give you an example...

I defy you to be scared by this book. There are an obscene amount of 
diagrams illustrating the points as well, along with numerous (and 
redundant) exercises to make sure you are getting it all.

Dan Weston

Seth Gordon wrote:
> Are Benjamin C. Pierce's _Types and Programming Languages_ and/or _Basic 
> Category Theory for Computer Scientists_ suitable for self-study?
> (Do they have problem sets that can be checked by either looking up 
> answers in The Back of the Book, or by trying to compile/run some code 
> that the student writes?)
> FWIW, my formal math education--well, all the formal math education that 
> I can remember at this point--stopped at freshman calculus plus one 
> semester of statistics.
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