Given two topological spaces and a continuous map , or equivalently the functor , forming the morphism of sites , one can introduce two kinds of “morphisms” between sheaves on and on : morphisms and comorphisms of sheaves over different base spaces. Depending on a motivation one or another is more natural.
Morphisms. Recall that for a fixed base space, say , the category of etale spaces over is equivalent to the category of sheaves over . Etale spaces make a subcategory of the slice category of spaces (“bundles”) over . Denote by the etale map corresponding to a sheaf . Thus for different bases we can simply define morphisms of sheaves in etale space language, as continuous maps between the etale spaces over , i.e. . These maps restrict to set theoretic maps on the level of stalks.
Comorphisms. The idea of comorphism is the idea of cospaces, rather than spaces; namely we like to replace a space, like in noncommutative geometry, by an object in a dually behaved category (say commutative rings; however it does not need to be strictly the dual category). For example, consider the ring of smooth functions on a manifold. A smooth morphism of manifolds amounts to a set theoretic map ( above) of underlying sets together with a compatible family of maps of algebras where runs through all open subsets of . Here of course .
Let be a -valued presheaf over a category and a -valued presheaf over a category . A comorphism of -valued presheaves over a functor (thought of as a morphism of sites in the opposite direction) is a morphism of -valued presheaves over : . Instead of saying that is a co(homo)morphism over (or -comorphism), one often says that comorphism of presheaves is a pair . If the presheaf is the structure sheaf of a ringed site one calls the comorphism a morphism of ringed sites; the usual notation is then .
In the case of a topological space, and sheaves of sets (or algebras of some kind), there is another description in terms of stalks, namely comorphisms may be described as certain multi-valued maps between stalks. More precisely, given a continuous map and sheaves over and over , a comorphism is a collection of maps for all such that for any section where the map is a section of over . We can view all where for a fixed as a single multi-valued map of stalks satisfying the continuity constraint mentioned.
Notice that by adjunction, a morphism of sheaves of sets over corresponds to a unique morphism of sheaves of sets over . However, in the case of ringed sites, a morphism is required to be morphism of sheaves of rings, whereas the adjunction does not guarantee that the corresponding morphism of sheaves of sets over is a morphism of sheaves of rings over .
In the case of sheaves of sets, the -component of the unit of adjunction is clearly a comorphism over . Given any comorphism (i.e. ) we can decompose it as a composition
where the first map is a (canonical) comorphism over and the second map is a morphism of sheaves over . Both maps may be viewed as multi-valued maps of etale spaces with the corresponding continuity conditions. In fact this is the unique decomposition of a given -comorphism of sheaves of sets into a compositon of an -comorphism of sheaves of sets and a morphism of sheaves of sets over . There is also a unique decomposition of as where is a morphism and is a comorphism over , namely
where is the -comorphism which is given by the identity . In fact, this can be seen as an intermediate step in the adjunction :