Talk:Enriched category

rewrite, now merge?

Merged from Talk:Enriched functor, now a redirect to here.

I've just rewritten the article. No offense to Clotito (talk · contribs) intended, but the article we had before was completely incomprehensible. The article is nothing but a definition though (both before and after the rewrite) and I think it should probably be merged into enriched category. Any opinions? Is anyone watching this article? -lethe ^{talk +} 01:16, 30 April 2006 (UTC)

I mean, we also have enriched-category-theoretic definitions for natural transformations. Do we give an independent article to every enriched-category analogue of a category-theoretic concept? -lethe ^{talk +} 14:51, 30 April 2006 (UTC)

Why are all the images color-reversed?

Is there any particular reason? Is the problem in my browser? --Acepectif 04:47, 13 May 2007 (UTC)

Correspondance not identity

"For each object A in C, the identity arrow idA must be an arrow in M from I to the hom-object Hom(A,A)..." So, an arrow of one category is also an arrow of another one with diffent domain and codomain. Very counterintuitive. Define it as correspondence rather than identity. --84.228.240.181 (talk) 14:28, 29 September 2011 (UTC)

Enriched categories are not categories

At least, not in general. It's only when the hom-objects (of M) have some kind of canonical realization as sets (which happens to be the case for M=Set, M=2, and the other examples given thus far) that the corresponding enriched categories are indeed categories, otherwise they're not. An example of the latter would be Lawvere's metric spaces where M is the real numbers ordered by >=, i.e., the hom-object is merely a number, composition is the triangle inequality, there's no notion of individual morphism, and thus no way to view this as a traditional category. I may try to edit this in. Rfcrew (talk) 22:14, 16 October 2011 (UTC)

and fixed. Also expanded the intro to give a bit more motivation. Still bothered by the verbosity of the axioms. Rfcrew (talk) 01:51, 17 October 2011 (UTC)

Inconsistent notation

Now that I've merged, I see that I've introduced an unfortunate inconsistency in the notation. In the functors section, objects are denoted with miniscule a,b in contrast to the majuscule A,B in the rest of the article. Hom-objects are denoted C(a,b), instead of Hom(A,B) (which might prefer to be written Hom_C(A,B) in the functor section). Do you think this is bad enough to merit redoing the diagrams? Probably so. -lethe ^{talk +} 18:15, 1 May 2006 (UTC)

There's also confusion about the order of composition:

• If composition is Hom(B,C)⊗Hom(A,B)->Hom(A,C) then the top lines of the identity diagrams should be

I⊗Hom(A,B) -> Hom(B,B)⊗Hom(A,B) and Hom(A,B)⊗I -> Hom(A,B)⊗Hom(A,A) respectively.

• If composition is Hom(A,B)⊗Hom(B,C)->Hom(A,C) then the top lines of the identity diagrams should be

I⊗Hom(A,B) -> Hom(A,A)⊗Hom(A,B) and Hom(A,B)⊗I -> Hom(A,B)⊗Hom(B,B) respectively.

Right now, we're getting a mix of both worlds, so at the very least one of the diagrams needs to be redone. Also, there's also some question as to what "left" and "right" refer to. It would be better to just call these "the identity axioms" without getting bogged down in which is which.

And if it has to be rewritten anyway, I'd just as soon use Kelly's notation as in functor section, which is more concise anyway.

Rfcrew (talk) 18:03, 26 October 2011 (UTC)

Are you sure?

in a text: "0 ≥ d(a,a)". Inequality is usually in opposite direction. --Jj14 (talk) 22:47, 24 November 2015 (UTC)

... in the section Examples of enriched categories, near the end. — Preceding unsigned comment added by Jj14 (talk • contribs) 22:50, 24 November 2015 (UTC)

Not sure what you mean. The actual pseudoquasimetric-space axiom calls for equality, but that follows from R^+∞ (with the inverse ordering) being an ordered set with 0 terminal (so that 0 ≥ d implies d = 0). As for why it's "0 ≥ d(a,a)" rather than "d(a,a) ≤ 0", that's in order to emphasize the parallelism with the other examples. Rfcrew (talk) 22:48, 25 June 2016 (UTC)