# Talk:Dual (category theory)

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Field:  Foundations, logic, and set theory

Actually the Pontryagin duality is between the category of locally compact abelian groups and its own opposite; and restricts to a duality between compact and discrete groups. So I prefer the earlier wording. We could do with having the PD article in place, naturally.

Charles Matthews 08:38, 9 Nov 2003 (UTC)

## Terminology: Dual vs. Opposite

Mirror question to one I asked in Talk:Equivalence_of_categories: isn't opposite category a better established usage than dual category? ---- Charles Stewart 11:24, 31 Aug 2004 (UTC)

My understanding is that these are different words. Any category has an opposite category by just 'reversing all arrows', though often this is pretty meaningless (e.g. I think the morphisms in the opposite category of groups don't correspond to maps of the objects in any reasonable sense). The dual category is an opposite category that happens to be equivalent to something (e.g. the morphisms in the opposite category of commutative rings give you maps between affine schemes in a nice way). Expert commentary is of course appreciated. —Preceding unsigned comment added by 67.188.117.167 (talk) 06:30, 12 November 2009 (UTC)

## Unclearness

I rated the article as unclear because it is not clear how to reverse a morphism. I'm learning category theory on Wikipedia, but I can read and understand all other articles about category theory.

I have now (after rereading Morphism and Concrete category) maybe realized where the confusion comes from, but still have doubts: how do I reverse a morphism (i.e. find its inverse function), if it is not injective? The question is probably the wrong one, when one realizes the dual of a category has not to be a concrete category: the category supplies a set of morphisms (and does not need to supply them through axioms they must obey), and a morphism is just characterized by having a domain and a codomain.

At this point, it is however unclear how Boolean algebras + Boolean isomorphisms are the opposite of Stone spaces + continuous functions. I think that after defining the opposite of the former category, one can show that it is "isomorphic" to the second category, right? I'm still using an algebraic language, not the one of categories; I should talk about Equivalence of categories, even because I'm reading that it's different from isomorphisms.

Plus, the fact that a partial order is a category is not obvious - reading the example, I thought that the article was making a parallel with reversing a poset. Maybe it's my fault, but the article looks still more unclear. At least, it deserves the {{technical}} template.--Blaisorblade (talk) 01:17, 21 June 2008 (UTC)

I've removed the unclearness tag. I reworte what it means to reverse morphism DesolateReality (talk) 10:05, 31 January 2009 (UTC)

## Dual always symmetric?

If S is dual to R, is R always dual to S? The page strongly suggests it, but doesn't explicitly say so.--greenrd (talk) 12:44, 4 August 2011 (UTC)

## a more introductory explanation of the concept of dual would be nice.

In general I find those tags on articles which say "this material may not be accessible enough" to be irritating. So it is with some chagrin that I find myself writing this:

"Dual" is a very interesting concept, which is (I hope) at least partially explainable without recourse to category theory.

It may well be that unless one understands the meaning in the context of category theory, one does not really understand the concept of the dual, but that's an awfully big mountain to climb for someone who's not a professional mathematician, or aiming to be one.

My path to this entry was via a link in the article Corecursion.

I, for one, would like to see a more introductory discussion of the concept, perhaps how it arose in the history of mathematics, and how it was generalized over time. Graph theory might be a good starting place.

Unfortunately, as one who came here trying to understand a bit more about the concept of duality, to flesh it out a bit more beyond my intuition, I'm not the right person to write such an article.

My intuition about the concept of the dual leads me to define it as a kind of transformation of one mathematical object into another, equivalent form, which may sometimes lead to new insights or easier approaches to solving problems than were apparent in the original form. But I am afraid I'm missing some aspect of the thing, because this seems like a description which does not merit a new name beyond "isomorphism". A "verbal walk" starting from these misapprehensions and correcting them would make me a lot happier. I don't see the path for that walk in the current article. Ngvrnd (talk) 13:28, 12 September 2012 (UTC)