Re: [Haskell-beginners] Category question

On Mon, 28 May 2012, Manfred Lotz wrote:

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.

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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).

This is exactly what category theory books do (or should) say. Do you have a particular example of a book which does not state things in this way?

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

... Identity Maps: (one per object) 1A: A -> A ... Rules for a category 1. The identity laws: where they say g . 1A = g and 1B . f = f 2. associatlve laws ...

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.

No. The point is that, by definition, a category, call it C, is a struct with two sets, Obj(C) and Mor(C), and further operations: 1. head: Mor(C) -> Obj(C) 2. tail: Mor(C) -> Obj(C) 3. id: Obj(C) -> Mor(C) 4. *: Mor(C) x Mor(C) -> Mor(C) where head and tail and id are everywhere defined single valued maps. They are all maps of sets. *, read "composition of morphisms" is a map of sets, with signature as displayed, but is not usually everywhere defined. We have then several "equational" axioms, which C is required to satisfy to be a category. (set theoretical note: We have, partly implicitly, ruled out categories which are not "small". See standard texts for this locus of difficulty.) By the axioms, any object b of C must have defined its associated identity morphism id[b]. For many categories, b will always be an actual set, and id[b] will be the unique map of sets defined by (id[b])(x) = x , for all x in b where (id[b])(x) is read "the result of applying id[b] to the element x of b". But, as explained, many categories have objects which are not sets. Indeed, often, no object is a set. The definition of category never mentions whether or not the objects are sets. And, as we have seen, there are many categories whose objects are not sets. (Perhaps categorically better: many categories are not directly presented as having objects which are sets.) to repeat: The concept "category" is larger in extension than the concept "category whose objects are sets and whose morphisms are maps of sets". ad representations of categories: http://en.wikipedia.org/wiki/Yoneda_Lemma [page was last modified on 1 April 2012 at 05:17]

I guess that this made me think of idA as idA(x) = x for each x of A. Later when I saw other (more general) definitions I did not read carefully to realize the difference.

Thanks a lot for making this clear to me.

-- Manfred

I will let stand my restatement of what you already know ;) oo--JS.