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 , each object has two associated representable functors, one covariant, the other contravariant: namely and , with arrows corresponding to post and pre composition by themselves respectively. A general set-valued functor on is called representable if it is isomorphic to such a functor (we say it is *represented* by ).

The most fundamental result about these functors is that they actually *embed* the category into the category Hom(, 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 be a covariant functor. Then for all objects of , there is an isomorphism natural in :

Let be a contravariant functor. Then for all objects of , there is an isomorphism natural in :

Note that setting , we get our embedding property:

In other words, the functor from 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 preserve all limits which already exist, and the send existing colimits to limits.

*Proof:* Suppose is a limit for a functor . Consider any cone over . Then any maps to a cone , which corresponds simply to a map which together with the factors our cone, since is a limit in . Hence, we have a map . Since was arbitrary, we deduce that is the limit of , as was required. The contravariant case follows by considering it as covariant .

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

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

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

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 except its relationships with ‘ we also have a functor which associates to each set a universal object in which is `best’ at being described by something like . 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 -modules (where the representable functors actually land back in the category -modules: ) we have such a left adjoint:

Proposition 4The tensor product by , is a left adjoint to the representable functor (considered as taking values in -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 be a topological space with an open cover , and be some other topological space. If we are given continuous maps with the property that on for all , then since continuity is a local property, we can write down a unique function which glues all the together. Specifically, we just set for some where . This is well-defined because of the `agreement on intersections’ property, and it’s unique because has to be a gluing.

From this discussion we conclude

Proposition 5The representable presheaves (contravariant set-valued functors) on the category of topological spaces are in factsheaveson 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 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.

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August 3, 2011 at 11:36 am

Zhen LinYou should probably mention that a covariant functor into Set is representable if and only if it has a left adjoint. In some sense, this follows from the fact that the identity functor on Set is representable and that every set is canonically a coproduct of copies of 1. A more general statement of this result is given by Mac Lane [1998; Chapter IV, Corollary 1.2]. I imagine something similar is true for Ab-enriched categories, but I haven’t worked it out.

August 3, 2011 at 11:39 am

Zhen LinAh wait, I see the problem in extending the claim to abelian categories. Nevermind.