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Chapter 4
The basic structure of
morphisms in Loc
4.1. Localic maps versus frame homomorphisms. First we should realize that
the one-one (resp. onto) localic maps correspond precisely to the onto (resp. one-one) frame homomorphisms
(as it always is with Galois adjoints: it follows from the identities f∗ff∗ = f∗ and ff∗f = f).
4.2. Localic maps versus continuous maps. It has been already said that localic maps (and frame homomorphisms, if we do not mind about contravariance) fairly well represent continuous maps. The reader may wonder how it can be: homomorphisms of algebras behave in a special way contrasting with the behav- ior of structure preserving maps in other contexts (like for instance continuous maps between spaces). Thus for instance if h is a one-one homomorphism, if f is a mapping, and if hf is a homomorphism then f is a homomorphism, and similarly if h is onto and fh is a homomorphism then f is a homomorphism again. Such statements hold for frame homomorphisms and hence, by 4.1, they hold for localic maps as well. Nothing of the kind holds for continuous maps between spaces, though. The explanation of this seeming paradox is that in the correspondence between localic and continuous maps the properties to be onto or one-one are not copied. Thus for instance, one-one localic maps correspond to embeddings, not to general one-one maps, as we will see shortly.
4.3. Special monomorphisms and epimorphisms. The reader knows that in everyday-life categories, monomorphisms model (roughly speaking) one-one maps, and that epimorphisms model (very roughly speaking) onto maps.
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