TITLE: Definition of Category of Locales
QUESTION [5 upvotes]: 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

REPLY [6 votes]: 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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