Talk:Comma category

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So I just expanded this quite a bit - some of the previous phrasing has been incidentally squelched, but everything that was there before is still there now. I'm pretty sure that the new stuff is accurate, but I'm not too happy with the notation: there are far too many different symbols. It could also use a commutative diagram for the most general case. The material on adjoints is currently minimal; I've tried to say just enough to show "look, there's a connection!" without actually going into any details, but perhaps further details are needed. --AlexG 00:44, 19 Jul 2004 (UTC)

Definition of morphisms

I'm new to category theory, so I won't make changes myself here. But I want to suggest the following:

There's no definition of the identity morphism of each object in the comma category.
By the given definition of the morphisms, it is not specified but I suppose it must be obvious that the morphisms between two objects are taken uniquely. For example: if there are objects (a,b,f1) (a',b',f1') (a,b,f2) (a',b',f2') in the comma category, then there could be g:a->a' h:b->b' s.t. (g,h) is a morphism both in hom((a,b,f1), (a',b',f1')) and hom((a,b,f2), (a',b',f2')), but these are two separate morphism with the same notation, isn't it so?

--Itaj Sherman (talk) 23:28, 15 February 2008 (UTC) ’[reply]

The identity morphism for (a, b, f) is (id_a, id_b). You are right that the notation for morphisms is a bit ambiguous. Normally one thinks of the morphism as being given with its domain and codomain, but sometimes writing a map properly — (g, h) : (a, b, f) → (a’, b’, f’) — requires too much notation. I found a reference for the identity morphism in Adámek–Herrlich–Strecker, which does not comment on the ambiguity. It's possible that Mac Lane talks about it, but I don't have my copy handy at the moment. Michael Slone (talk) 01:29, 16 February 2008 (UTC)[reply]

Well, I think it should be at least noted that the meaning of the definition is to be exclusive per domain and codomain, whether the books state it or not, this is what they mean. It would be helpful for people who are new to category theory - I was confused and had to go through all the definitions again to understand what was meant there. --Itaj Sherman (talk) 00:00, 23 February 2008 (UTC)[reply]

There really shouldn't be any confusion here. Two morphisms with different domains (or codomains) are never equal. This isn't special to comma categories. I agree that the language used in the article is a little loose in this regard, but I think to state it formally would add to the confusion rather than subtract from it. -- Fropuff (talk) 05:53, 23 February 2008 (UTC)[reply]

"Common domain"

From the article:

The most general comma category construction involves two functors with the same domain. [...]

Suppose that

{\mathcal {A}}

,

{\mathcal {B}}

, and

{\mathcal {C}}

are categories, and

T

and

S

are functors

File:CommaCategory-04.png

[...]

The diagram defining morphisms is identical to the diagram which defines the components of a natural transformation (assuming the domains of the two functors agree).

It seems to me that it's the codomains that agree, not the domains. Hairy Dude (talk) 18:27, 5 July 2008 (UTC)[reply]