Spectral maps are the morphisms of the category Spec. They relate different spectral spaces to each other. So far we have mostly analyzed properties of spectral spaces. Now it is time to study spectral maps more closely. Various questions need to be addressed.
In our category-theoretic context the characterization of monomorphisms and epimorphisms in Spec is a fundamental task. We present a complete answer in Section 5.2, showing that the monomorphisms are the injective spectral maps and the epimorphisms are the surjective spectral maps. The analysis of monomorphisms is continued in Section 5.4, where we consider spectral maps that are homeomorphisms onto the image. Similar more detailed questions about epimorphisms are studied in Chapter 6, particularly in Section 6.4. Other questions about spectral maps are concerned with topological properties. Closed maps and open maps are studied in Section 5.3, dominant and irreducible maps in Section 5.5.
It is an important problem in general topology to study extensions of continuous maps, cf. [GiJe60], [Kel75, p. 115, p. 242], [Eng89, p. 69]. We consider the extension problem for spectral maps and ask: given a spectral map defined on a spectral subspace of a spectral space, does there exist an extension to a spectral map defined on a larger subspace, or even on the entire ambient space? A first answer is presented in Section 5.6.
In Section 5.1 we describe images of spectral subspaces under spectral maps, which provides useful tools for the later sections.
By Stone duality every spectral map corresponds to a unique homomorphism of bounded distributive lattices, see Section 3.2. Therefore every property of a spectral map may be considered as the topological expression of an algebraic property of the corresponding lattice homomorphism, and conversely. Throughout we keep an eye on these connections and find several useful translations between algebraic and topological properties.
The notion of a spectral map was introduced in Section 1.2. Later we proved several alternative characterizations of spectral maps. As a reminder and a starting point we collect the equivalent conditions here.
Reminder Let f : X → Y be a map between spectral spaces. The following conditions are equivalent.