Martin Orr's Blog

Naturality in the Yoneda lemma for groups

Posted by Martin Orr on Saturday, 16 May 2009 at 19:54

In my last post on the Yoneda lemma for groups, I ignored the naturality part of the lemma. I want to work in detail what this means once - it is a lot of fiddly composing of morphisms and I probably won't do it again (at least in public). If you're not in the mood for following such details, then there is little point in reading this, although you could skip to the last paragraph.

First consider "natural in x". To explain what it means for the Yoneda map to be natural, we have to view both sides of the map themselves as functors C -> Set. For each object x the Yoneda lemma gives a function A_x : \mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(x, -), T) \to Tx. This family of functions is natural in x iff these A_x are the components of a natural transformation between the two functors.

When interpreting each side of the Yoneda map as a functor, the expression which appears in the definition of A_x tells us what the functor does to objects but we still have to specify what it does to morphisms in C. For the RHS, this is clear: the object map is x \mapsto Tx, so the morphism map should be f \mapsto Tf.

For the LHS, it is more complicated (but there is only one thing to do). The object map takes x \mapsto \mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(x, -), T). For a morphism f : x \to y, first we form

f^* : \mathop{\mathrm{Hom}}(y, z) \to \mathop{\mathrm{Hom}}(x, z) "precompose with f" for each z in C

then

f^{**} : \mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(x, -), T) \to \mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(y, -), T) "precompose each component with f^*".

(Note that we reversed direction, then back again.)

Let's get concrete and apply this to groups. Let T be a functor C \to \mathrm{Set} corresponding to G-set X. A morphism * \to * is just an element g of G, and then Tg is the function X \to X induced by g.

Recall that \mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(*, -), T) is the set of G-homomorphisms G \to X, which we shall write \mathop{\mathrm{Hom}}_G(G, X). Now for g in G, g^* : G \to G is "multiply on the right by g" and g^{**} sends u : G \to X to the map h \mapsto u(hg). Letting g act by g^{**} makes \mathop{\mathrm{Hom}}_G(G, X) itself into a G-set. (Which we should expect, since we are interpreting \mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(*, -), T) as a functor C \to \mathrm{Set}.)

The naturality says that the following diagram commutes:

\usepackage{amscd} \begin{CD} \mathop{\mathrm{Hom}}_G(G, X) @>A_*>> X \\ @Vg^{**}VV @VgVV \\ \mathop{\mathrm{Hom}}_G(G, X) @>A_*>> X \end{CD}

which is true since A_*(g^{**}(u)) = g^{**}(u)(1) = u(1g) = u(g) = g.u(1) = g.A_*(u).

To interpret the phrase "natural in T" in the Yoneda lemma, we have to fix x and consider both sides as functors of T. The domain category is now D = { functors C \to \mathrm{Set} }; morphisms in this category are natural transformations. The LHS of the Yoneda map is now a covariant Hom-functor from this category D, and the RHS is the functor "evaluate at x".

In the case where C is a group category, D is just the category of G-sets and G-homomorphisms. The LHS is the functor D \to Set : \mathop{\mathrm{Hom}}_G(G, -) and the RHS is the forgetful functor D \to \mathrm{Set} "take the underlying set". Naturality here says that the following diagram commutes for each G-homomorphism u : X \to Y:

\usepackage{amscd} \begin{CD} \mathop{\mathrm{Hom}}_G(G, X) @>A_*>> X \\ @Vu \circ -VV @VuVV \\ \mathop{\mathrm{Hom}}_G(G, Y) @>A_*>> Y \end{CD}

In other words, the function \mathop{\mathrm{Hom}}_G(G, X) \to \mathop{\mathrm{Hom}}_G(G, Y) "postcomposition with u" corresponds under the Yoneda bijection with the function u itself on the underlying set. To check this: for any v in \mathop{\mathrm{Hom}}_G(G, X), A_*(uv) = (uv)(1) = u(v(1)) = u(A_*(v)).

To summarise, the "natural in x" tells us that, for a fixed G-set X, the Yoneda map is itself a G-homomorphism Hom_G(G, X) \to X. And "natural in T" says that the Yoneda map behaves with respect to G-homomorphisms between G-sets. (The latter is probably what you would have expected "the Yoneda map for groups is natural" to mean.)

Tags categories, groups, maths, yoneda

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