Given two topological spaces X,YX,Y and a continuous map f:XYf:X\to Y, or equivalently the functor f 1:Open(X)Open(Y)f^{-1}:\mathrm{Open}(X)\to\mathrm{Open}(Y), forming the morphism of sites Open(Y)Open(X)\mathrm{Open}(Y)\to\mathrm{Open}(X), one can introduce two kinds of “morphisms” between sheaves on XX and on YY: morphisms and comorphisms of sheaves over different base spaces. Depending on a motivation one or another is more natural.

  1. Morphisms. Recall that for a fixed base space, say XX, the category of etale spaces over XX is equivalent to the category of sheaves over XX. Etale spaces make a subcategory of the slice category Bun X=Top/X\mathrm{Bun}_X=\mathrm{Top}/X of spaces (“bundles”) over XX. Denote by p F:E(F)Xp_F:E(F)\to X the etale map corresponding to a sheaf FF. Thus for different bases we can simply define morphisms of sheaves FGF\to G in etale space language, as continuous maps between the etale spaces ξ:E(F)E(G)\xi: E(F)\to E(G) over ff, i.e. p Gξ=fp Fp_G\circ\xi=f\circ p_F. These maps restrict to set theoretic maps on the level of stalks.

  2. 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 MNM\to N amounts to a set theoretic map (ff above) of underlying sets together with a compatible family of maps of algebras k V:C (V)C (f 1V)k_V :C^\infty(V)\to C^\infty(f^{-1} V) where VNV\subset N runs through all open subsets of NN. Here of course k V:ggf f 1Vk_V : g\mapsto g\circ f|_{f^{-1}V}.


Let FF be a RR-valued presheaf over a category CC and GG a RR-valued presheaf over a category DD. A comorphism of RR-valued presheaves FGF\to G over a functor f 1:DCf^{-1}:D\to C (thought of as a morphism ff of sites in the opposite direction) is a morphism kk of RR-valued presheaves over CC: k:Gf *F=Ff 1k : G\to f_* F = F\circ f^{-1}. Instead of saying that kk is a co(homo)morphism over ff (or ff-comorphism), one often says that comorphism of presheaves is a pair (f,k)(f,k). 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 (f,f )(f,f^\sharp).

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 f:XYf:X\to Y and sheaves FF over XX and GG over YY, a comorphism (f,f )(f,f^\sharp) is a collection of maps f x :G f(x)F xf^\sharp_x : G_{f(x)}\to F_{x} for all xXx\in X such that for any section sG(V)s\in G(V) where V openYV^{\mathrm{open}}\subset Y the map xf x (s(f(x)))x\mapsto f^\sharp_x(s(f(x))) is a section of FF over f 1(V)f^{-1}(V). We can view all f x f^{\sharp}_x where xf 1(y)x\in f^{-1}(y) for a fixed yy as a single multi-valued map of stalks G y xf 1(y)F xG_y\to \coprod_{x\in f^{-1}(y)} F_x satisfying the continuity constraint mentioned.


  1. Notice that by adjunction, a morphism Gf *FG\to f_* F of sheaves of sets over YY corresponds to a unique morphism f *GFf^* G\to F of sheaves of sets over XX. However, in the case of ringed sites, a morphism Gf *FG\to f_* F is required to be morphism of sheaves of rings, whereas the adjunction does not guarantee that the corresponding morphism f *GFf^* G\to F of sheaves of sets over XX is a morphism of sheaves of rings over XX.

  2. In the case of sheaves of sets, the GG-component of the unit of adjunction η G:Gf *f *G\eta_G :G\to f_* f^* G is clearly a comorphism Gf *GG\to f^* G over ff. Given any comorphism (f,k):GF(f,k):G\to F (i.e. k:Gf *Fk:G\to f_* F) we can decompose it as a composition

    Gη Gf *Gkη G 1F G\stackrel{\eta_G}\to f^* G\stackrel{k\circ\eta_G^{-1}}\to F

    where the first map is a (canonical) comorphism over ff and the second map is a morphism of sheaves over XX. 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 ff-comorphism kk of sheaves of sets GFG\to F into a compositon of an ff-comorphism of sheaves of sets Gf *GG\to f^*G and a morphism of sheaves of sets over XX. There is also a unique decomposition of kk as Gf *FF, G\to f_* F\to F, where Gf *FG\to f_*F is a morphism and f *FFf_* F\to F is a comorphism over ff, namely

    Gi 1kf *FiF, G\stackrel{i^{-1}\circ k}\to f_* F\stackrel{i}\to F,

    where ii is the ff-comorphism which is given by the identity f *Ff *Ff_*F\to f_* F. In fact, this can be seen as an intermediate step in the adjunction f *f *f^*\dashv f_*:

    Hom X(f *G,F)cohom f(G,F)Hom Y(G,f *F). \mathrm{Hom}_X(f^*G,F)\cong \mathrm{cohom}_f(G,F)\cong\mathrm{Hom}_Y(G,f_* F).
Revised on May 23, 2009 15:15:27 by Toby Bartels (