Definition of Category of Locales - MathOverflow

Definition of Category of Locales

2009-12-29T17:58:49Z

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

Answer by Tom Leinster for Definition of Category of Locales

2009-12-29T20:20:36Z

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.

Answer by Paul Taylor for Definition of Category of Locales

2009-12-29T22:24:37Z

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.