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.

On Mon, May 28, 2012 at 06:50:34PM +0200, Manfred Lotz wrote:

On Mon, 28 May 2012 10:57:11 -0400 Brent Yorgey wrote:

...

On Mon, May 28, 2012 at 04:14:40PM +0200, Manfred Lotz wrote:

...

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.

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.

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