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In the wikipedia entry for 'frames and locales', pains are taken to distinguish between the category of locales - defined to be the opposite of the category of frames - and the category whose objects are the complete Heyting algebras but whose arrows are the adjoints of the frame arrows. The two are clearly isomorphic as categories. How far are they from being identical, though? I became acquainted with locales through Borceux's excellent handbook and he defines arrow in locales to be the adjoints. So this is a little worrisome. Ok, I know this isn't precise... Let me put it this way: What are concrete examples of how these two differently defined categories actually differ?

Thank you in advance

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up vote 7 down vote accepted

Well, as you say, these two categories are isomorphic, so it's going to be hard to say how they differ! They only differ in the names the maps are given.

Maybe it would help to recap the definitions. I'll take them from p.39-40 of Peter Johnstone's book Stone Spaces.

A frame is a complete lattice $A$ satisfying the infinite distributive law $$ a \wedge \bigvee S = \bigvee \{ a \wedge s | s \in S \} $$ ($a \in A, S \subseteq A$). A homomorphism of frames is a function preserving finite meets and arbitrary joins. This defines the category Frm of frames.

Note (as you did) that every homomorphism of frames has a right adjoint.

The category Loc of locales is the opposite of the category of frames. Morphisms in Loc are called continuous maps.

Then Johnstone says: "We adopt the convention that if $f: A \to B$ is a continuous map of locales, we shall write $f^*: B \to A$ for the corresponding frame homomorphism, and $f_*: A \to B$ for the right adjoint of $f^*$."

So in Johnstone's convention (which is the one I know), the elements of Loc$(A, B)$ are identified with frame homomorphisms $B \to A$. In Borceux's convention, the elements of Loc$(A, B)$ are identified with order-preserving maps $A \to B$ that are right adjoint to frame homomorphisms.

I guess the other thing to say is that when you're dealing with ordered sets, adjoints are genuinely unique (not just unique up to isomorphism). So taking the right adjoint of a frame homomorphism is a bijective process.

I don't know what else to say. It's really just a matter of naming.

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do these notions come from logic or analysis? – john mangual Dec 30 '09 at 2:28
They come from logic (or more precisely, category theory) and its interaction with topology. The basic observation is that for many things that you do with topological spaces, the open sets matter more than the points. For instance, the notions of compactness and connectedness of a space refer only to the open sets, not the points. The same goes for the definition of sheaf. – Tom Leinster Dec 30 '09 at 3:24

The article on Heyting algebras and frames is one of many that are truly awful in Wikipedia.

Frames and complete Heyting algebras are completely different things. They are algebras for different theories and (so) their homomophisms are different.

Johnstone's convention, which Tom has described and to which there are now few dissenters, allows one to use the algebraic machinery to speak in topological language, but without mentioning points. For example, in Johnstone's book you will find definitions of locally compact locales and of open (continous) maps.

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Thanks to both Paul and Tom. I have a follow-up: If a category $D$ is defined to be the opposite of category $C$, then properties of $D$ can be discovered only up to categorical isomorphism, right? So, if Loc is defined to be the opposite of Frm then Loc then we are only able to conclude categorical properties of Loc - and, actually, that's all I think I need. However, it seems that Borceux's definition fixes one category among many possible opposites. Is this right? – John Iskra Jan 4 '10 at 16:40
@unknown(google): I'd say that the opposite of a category is well-defined on the nose, not just up to isomorphism. The objects and morphisms are the same as in the original category; the composition gets twisted, and the domain and codomain operations get interchanged. On the other hand, this "on the nose" definability should never be of any use --- you should care about the opposite category only up to isomorphism (and probably only up to equivalence). – Andreas Blass Feb 3 '11 at 11:01

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