Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γ_{f} ⊂ X × Y of f, which (as a set) is defined as

Γ_{f} = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections p_{X}, p_{Y} from X × Y to the factors X, Y. Restricted to the subvariety Γ_{f}, p_{X} is an isomorphism (since f is a morphism). The fibres of p_{Y} restricted to Γ_{f} are just the fibres of f; so for example f is proper iff p_{Y} | Γ_{f} is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f_{∗} : H(X) → H(Y)

This map can be expressed in terms of the graph Γ_{f} and the projection maps as

f_{∗}(α) = p_{Y∗} (p_{X}^{∗}(α) ∪ [Γ_{f}]) —– (2)

where [Γ_{f}] ∈ H (X × Y) is the fundamental class of the subvariety [Γ_{f}]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that p_{X} be an isomorphism with some weaker assumption, such as p_{X} |Γ_{f}, p_{Y} | Γ_{f} finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ_{∗} : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf O_{Γ} on X × Y. Associated to the projection maps p_{X}, p_{Y}, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ D^{b}(X × Y) with support which is proper over both factors. A derived correspondence defines a functor Φ_{F} by

Φ_{F} : D^{b}(X) → D^{b}(Y)

(−) ↦ Rp_{Y∗}(Lp_{X}^{∗}(−) ⊗^{L} F)

where (−) could refer to both objects and morphisms in D^{b}(X). F is sometimes called the kernel of the functor Φ_{F}.

The functor Φ_{F} is exact, as it is defined as a composite of exact functors. Since the projection p_{X} is flat, the derived pullback Lp_{X}^{∗} is the same as ordinary pullback p_{X}^{∗}. Given derived correspondences E ∈ D^{b}(X × Y), F ∈ D^{b}(Y × Z), we obtain functors Φ_{E }: D^{b}(X) → D^{b}(Y), Φ_{F }: D^{b}(Y) → D^{b}(Z), which can then be composed to get a functor

Φ_{F} ◦ Φ_{E }: D^{b}(X) → D^{b}(Z)

which is a two-sided identity with respect to composition of kernels.