Re: [Haskell-beginners] Category question

29 May 2012


      On Mon, May 28, 2012 at 06:50:34PM +0200, Manfred Lotz wrote:
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On Mon, 28 May 2012 10:57:11 -0400
Brent Yorgey  wrote:
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On Mon, May 28, 2012 at 04:14:40PM +0200, Manfred Lotz wrote:
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For me id: A -> A could be defined by: A morphism id: A -> A is
called identity morphism iff for all x of A we have  id(x) = x.
This is not actually a valid definition; the notation id(x) = x does
not make sense.  It seems you are assuming that morphisms represent
some sort of function, but that is only true in certain special
categories.
Ok, it is a valid definition only in a certain context. In the
far wider context of category theory this indeed makes no sense.
Right.
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In 'Conceptual Mathematics' by F. William Lawvere, Stephen H. Schanuel
they define an identity map with fa = a for each a in A.
Then on page 17 they define category and say
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Identity Maps: (one per object) 1A: A -> A
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Rules for a category
1. The identity laws:
where they say g . 1A = g and 1B . f = f
2. associatlve laws
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It seems that this definition of a category is not as general as it
could be. Here 1. is something which follows easily from the definition
of an identity map.
I am guessing (though I have not looked at 'Conceptual Mathematics' in
detail) that they use 'an identity map with fa = a for each a in A'
simply as an *example* to help build intuition; then on page 17 they
generalize this example to the fully abstract definition of a
category. It does seem unfortunate that they continue to use the name
'identity map', because morphisms/arrows are more general than 'maps'
(to me, 'map' is synonymous with 'function').

-Brent