In this post I want to review some basic general properties of representable functors. It seems these `God-given’ machines play important roles in all sorts of different ways throughout algebraic geometry (and presumably other categorically founded areas of mathematics).

Firstly, recall that if we have a category {\mathcal{C}}, each object {U} has two associated representable functors, one covariant, the other contravariant: namely {h^A: B \mapsto \text{Hom}(A,B)} and {h_A: B \mapsto \text{Hom}(B,A)}, with arrows corresponding to post and pre composition by themselves respectively. A general set-valued functor on {\mathcal{C}} is called representable if it is isomorphic to such a functor (we say it is represented by {A}).

The most fundamental result about these functors is that they actually embed the category {\mathcal{C}} into the category Hom({\mathcal{C}}, Set). This is part of a story that culminates in the classical Yoneda Lemma, which says more: that the representable functors detect all the information contained in an arbitrary functor. Note that the sets of homomorphisms between two functors are just the sets of natural transformations.

Theorem 1 (Yoneda Lemma) Let {F: \mathcal{C} \rightarrow Set} be a covariant functor. Then for all objects {X} of {\mathcal{C}}, there is an isomorphism natural in {X}:

\displaystyle F(X) \cong \text{Hom}(h^X, F).

Let {G: \mathcal{C} \rightarrow Set} be a contravariant functor. Then for all objects {X} of {\mathcal{C}}, there is an isomorphism natural in {X}:

\displaystyle G(X) \cong \text{Hom}(h_X, G).

 

Note that setting {G=h_Y}, we get our embedding property:

\displaystyle \text{Hom}(X,Y) \cong \text{Hom}(h_X, h_Y).

In other words, the functor from {\mathcal{C}} into its category of set-valued presheaves is fully faithful.

The next property I want to discuss was pointed out to me recently as `almost tautological’, but seems to be responsible for much of what gives these functors their character.

Theorem 2 (Preservation of Limits) The functors {h^X} preserve all limits which already exist, and the {h_X} send existing colimits to limits.

Proof: Suppose {(L, p_D)} is a limit for a functor {G: \mathcal{D} \rightarrow \mathcal{C}}. Consider any cone {(A \rightarrow \text{Hom}(X,GD))_{D \in \mathcal{D}}} over {h^X \circ G}. Then any {a \in A} maps to a cone {(X \rightarrow GD)_{D \in \mathcal{D}}}, which corresponds simply to a map {f_a: X \rightarrow L} which together with the {p_D} factors our cone, since {L} is a limit in {\mathcal{C}}. Hence, we have a map {A \rightarrow \text{Hom}(X,L) = h^X(L)}. Since {A} was arbitrary, we deduce that {h^X(L)} is the limit of {h^X \circ G}, as was required. The contravariant case follows by considering it as covariant {\mathcal{C}^{op} \rightarrow \text{Set}}. \Box

Note that this is a very general statement. We now know that in particular the representable functors respect products, pullbacks, and equalisers.

Corollary 3 In an abelian category, the representable functors are left exact.

Proof: They preserve equalisers (or send coequalisers to equalisers). \Box

On a slightly deeper level, recall the adjoint functor theorem, which basically says that any suitably nice functor which preserves limits admits a left adjoint. In other words, we might hope that together with the functor which `forgets everything about {Y} except its relationships with {X}‘ we also have a functor which associates to each set {A} a universal object {Y_A} in {\mathcal{C}} which is `best’ at being described by something like {A}. I am not really sure how to push this idea further in general, but it can of course be remarked that in the category of {R}-modules (where the representable functors actually land back in the category {R}-modules: {h^X: R-Mod \rightarrow R-Mod \rightarrow Set}) we have such a left adjoint:

Proposition 4 The tensor product by {X}, {Y \mapsto X \otimes_R Y} is a left adjoint to the representable functor {h^X} (considered as taking values in {R}-modules).

It would be nice if in more general situations representable functors actually possessed concretely understandable left adjoints, especially if they shared properties with the tensor product. That they preserve limits certainly give us some hope that they might.

One final fact of a more geometrical flavour about representable functors. Let {X} be a topological space with an open cover {{U_i}}, and {Y} be some other topological space. If we are given continuous maps {f_i: U_i \rightarrow Y} with the property that {f_i = f_j} on {U_i \cap U_j} for all {i,j}, then since continuity is a local property, we can write down a unique function {f: X \rightarrow Y} which glues all the {f_i} together. Specifically, we just set {f(x) = f_i(x)} for some {i} where {x \in U_i}. This is well-defined because of the `agreement on intersections’ property, and it’s unique because {f} has to be a gluing.

From this discussion we conclude

Proposition 5 The representable presheaves (contravariant set-valued functors) {h_Y} on the category of topological spaces are in fact sheaves on the category of topological spaces (technically speaking, with the open cover topology).

In fact, it seems to be generally true that such results hold. In algebraic geometry however, a much less trivial phenomenon occurs. A discussion similar to the one above is enough to convince one that {h_Y} is a sheaf in the category of schemes with respect to the Zariski open cover topology. However, the Zariski topology is infamously deficient as a topology: open sets tend to be large and unwieldy, so one cannot take `small neighbourhoods’, one of the basic intuitions of classical topology and analysis, so algebraic geometers use the more general concept of `Grothendieck topologies’. These allow us to construct a much finer `topology’, in particular giving much more stringent sheaf conditions than those associated with merely the Zariski topology (having more sophisticated sheaf structures gives more information about the space – indeed, étale cohomology – a technique for working out algebraic topology type invariants from algebraic spaces, relies on this idea).

With these more stringent sheaf conditions, many structures which were sheaves in the Zariski topology are ruled out. However, Grothendieck proved that representable functors survive (we can still glue morphisms of schemes even when defined over very fine covers).

Theorem 6 (Grothendieck) The representable presheaves on the category of schemes are sheaves in the fpqc topology. In particular, they are sheaves in the étale topology.

Here is not the time or place to prove it, or indeed to define the terms above, but I hope this post gives a flavour of why representable functors are often rather special and why it’s likely I shall probably be mentioning them again before too long.

Advertisements