Re: [Haskell-beginners] Category question

On Mon, 28 May 2012 10:04:33 -0400 Brent Yorgey wrote:

On Mon, May 28, 2012 at 02:08:14PM +0200, Manfred Lotz wrote:

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On Mon, 28 May 2012 13:34:01 +0200 Alessandro Pezzoni wrote:

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On Mon, May 28, 2012 at 11:54:11AM +0200, Manfred Lotz wrote:

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In the definition of a (mathematical) category it is said (among other things), that for any object A there exists an identity morphism:

idA: A -> A and if f: A -> B for two objects A, B then

idB . f = f and f . idA = f

must hold.

My question: Because I cannot think of any counterexample for the last statement I would like to know if I just could omit this from the definition and formulate this as a small theorem.

Or does there exist a counterexample where all conditions of a category hold but there exist two objects A, and B where we have idB . f <> f and/or f .idA <> f?

When you ask that idB . f = f and f . idA = f you are basically defining a left and a right identity, respectively.

If I get your question correctly, you are asking if you can drop the axiom (requirement) of existence of an identity morphism for every object and deduce it from the other axioms, i.e. that the composition of morphisms is always well defined and that it is associative.

No, I do not want to drop the requirement of existence of an identity morphism. I only want to drop the axion that idB .f = f and f . idA = f do hold because IMHO this follows readily from the definition of what an identity morphism is all about.

"Follows readily from the definition of what an identity morphism is all about" -- and what exactly is that defintion?

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. IMHO, from this the following follows readily.

In fact, the definition is precisely that

idB . f = f and f . idA = f

This is not an "extra" requirement on identity morphisms. It is simply the definition.

I agree that I could define id by these two statements. My point is that in the books about category theory those two statements are stated as axioms, and id is (in many books) just self understood or defined as I have defined it above. If in a book about category the author would say that for each object A there must exist a morphism id: A -> A (called identity morphism) which is defined by idB . f = f and f . idA = f then this would be clearer (and better, IMHO). -- Manfred -- Manfred