(Off-topic) Question about categories

Daniel Yokomiso daniel_yokomiso at yahoo.com.br
Fri Sep 19 09:01:12 EDT 2003

----- Original Message ----- 
From: "Graham Klyne" <gk at ninebynine.org>
To: "Haskell Cafe" <haskell-cafe at haskell.org>
Sent: Thursday, September 18, 2003 9:44 AM
Subject: (Off-topic) Question about categories

> (I'm asking here because I believe there is some overlap between category
> theory and functional programming cognoscenti...)
> It has been suggested to me that categories may be a useful framework
> useful than set theory) for conceptualizing things for which there is no
> well-defined equality relationship.  (The discussion is about resources in
> WWW.)  I've done a little digging about category theory, and find some
> relatively approachable material at:
>    http://www.wikipedia.org/wiki/Category_theory
>    http://www.wikipedia.org/wiki/Class_(set_theory)
> and nearby.
> But I'm hitting a mental block with this (and other places I've look don't
> add anything I can grok).  The definition of a category depends on the
> definition of a morphism, and in particular the existence of an identity
> morphism for every object in a category.  The definition of an morphism is
> in terms of equality of compositions of morphisms:
>      for f : A -> B we have Id[B]. f = f = f . Id[A]
> My problem is this:  how does it make sense to define an equality of
> morphisms without some well-defined concept of equality on the underlying
> objects to which they apply?  That is, given object X and an object Y, it
> is possible to examine them and determine whether or not they are the same
> object.  And if the underlying objects have such a concept of equality,
> what prevents them from being sets or members of sets?  But categories are
> presented as being more general than sets.
> Does anyone see the cause of my non-comprehension here?
> #g
> ------------
> Graham Klyne
> GK at NineByNine.org


    There's some additional links about category theory in this LtU
discussion: http://lambda.weblogs.com/discuss/msgReader$979. Also there's
this presentation "Category Theory for Beginners"
http://www.cs.toronto.edu/~sme/presentations/cat101.pdf. HTH.

    Best regards,
    Daniel Yokomiso.

"Honesty is the best policy, but insanity is a better defense."

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