Yoneda reduction

…

The term *Yoneda reduction* is was coined by Todd Trimble in his (unpublished) thesis. It refers to a technique based on the Yoneda lemma for performing a number of end and coend calculations which arise in coherence theory and enriched category theory.

There are various formulations of the Yoneda lemma. One says that given a presheaf $F: C^{op} \to Set$, there is a canonical isomorphism

$F(c) \cong Nat(\hom_C(-, c), F)$

where “Nat” refers to the set of natural transformations between presheaves $C^{op} \to Set$; in other words, the hom

$Set^{C^{op}}(\hom_C(-, c), F)$

appropriate to the presheaf category.

There is an $V$-enriched category version, whenever $C$ is a category enriched in a complete, cocomplete, symmetric monoidal closed category $V$. Here “Nat” is constructed as an enriched end (an example of a weighted limit):

$V^{C^{op}}(C(-, c), F) = \int_d F(d)^{C(d, c)}$

and therefore the enriched Yoneda lemma gives an isomorphism

$F(c) \cong \int_d F(d)^{C(d, c)} \qquad (1)$

which is ($V$-)natural in $c$; we may therefore write

$F(-) \cong \int_d F(d)^{C(d, -)} \qquad (2)$

and this isomorphism is $V$-natural in $F$.

We pause to give an instance of the Yoneda lemma which is both familiar and which serves to inform much of the module-theoretic terminology in the discussion below. Let $V = Ab$; let $R$ be a ring (conceived as an $Ab$-enriched category with exactly one object $\bullet$). Then $Ab^{R^{op}}$ is the ($Ab$-enriched) category of right $R$-modules, or equivalently, left $R^{op}$-modules). The presheaf $\hom_R(-, \bullet)$ is just the underlying abelian group of $R$ seen as a right module over the ring $R$, also known as the *regular representation*.

The first formulation (1) of the Yoneda lemma would simply say that at the level of *abelian groups*, we have for any right $R$-module $M$

$M(\bullet) \cong RightMod_R(R, M)$

Further taking into account the “naturality” in the argument bullet, the formulation (2) says that actually we have an isomorphism at the level of *right $R$-modules*

$M \cong RightMod_R(R, M)$

where the module structure on the right side arises by considering the argument $R$ now as a *bimodule* over the (ring) $R$.

The (enriched) Yoneda lemma is nothing but a far-reaching extrapolation of this basic isomorphism: it says

$F \cong RightMod_C(\hom_C, F)$

where the $C$-presheaf or right $C$-module hom on the right is appropriately constructed as an enriched end, and $\hom_C: C^{op} \otimes C \to V$ is a treated as a $V$-enriched “bimodule” over $C$, and plays the role of the “regular representation” of $C$.

The analogy between presheaves and modules can be pursued considerably further. Again, we start with the perhaps more familiar context of rings and modules.

In the first place, given a ring $R$, there is a familiar monoidal category of $R$-bimodules (and bimodule morphisms). If $M, N$ are bimodules over $R$, with left $R$-actions denoted by $\lambda$’s and the right actions by $\rho$’s, their tensor product $M \otimes_R N$, defined by the coequalizer

$M \otimes R \otimes N \stackrel{\to}{\to} M \otimes N \to M \otimes_R N$

(where the two parallel arrows are $M \otimes \lambda$, $\rho \times N$) carries an evident $R$-bimodule structure. Each of the functors $M \otimes_R -$ and $- \otimes_R N$ admits a right adjoint expressed by natural isomorphisms of abelian groups

$Bimod(N, Left_R(M, Q)) \cong Bimod(M \otimes_R N, Q) \cong Bimod(M, Right_R(N, Q))$

where $Left_R(M, Q)$ denotes the abelian group of left $R$-module maps $M \to Q$, equipped with its natural $R$-bimodule structure; $Right(N, Q)$ is similar. Thus the monoidal category of $R$-bimodules is biclosed.

More generally, there is a bicategory whose objects or 0-cells are rings $R, S, \ldots$, and whose morphisms or 1-cells $R \to S$ are left $R$-, right $S$-bimodules. 2-cells are homomorphisms of bimodules. If $M: R \to S$ and $N: S \to T$ are bimodules, then their bimodule composite is $M \otimes_S N: R \to T$. This too is a biclosed bicategory, meaning that

This generalized module theory can be pursued much further.

(Lost a bunch of work, due to vagaries of computers. Sigh. Will return later.)

If $V$ is symmetric monoidal, then the monoid of endomorphisms on the $n$-fold tensor functor

$\bigotimes^n: V^{\otimes n} \to V$

is in bijection with the monoid of endomorphisms on the unit object $I$.

By fully and faithfully embedding $V$ (as a symmetric monoidal category) into $Set^{V^{op}}$, we may without loss of generality suppose $V$ is complete, cocomplete, symmetric monoidal closed.

The result is by induction on $n$: observe that a map

$x_1 \otimes \x_2 \otimes \ldots \otimes x_n \to x_1 \otimes x_2 \otimes \ldots \otimes x_n$

natural in all the arguments $x_i$, in particular in $x_n$, corresponds to a map dinatural in $x_n$:

$x_1 \otimes \ldots \otimes x_{n-1} \to x_1 \otimes \ldots \otimes x_{n-1} \otimes x_n)^{x_n}$

and hence to a map to the end

$x_1 \otimes \ldots \otimes x_{n-1} \to \int_{x_n} (x_1 \otimes \ldots \otimes x_{n-1} \otimes x_n)^{x_{n}^{I}}$

where the end exists and is isomorphic to

$x_1 \otimes \ldots \otimes x_{n-1} \otimes I \cong x_1 \otimes \ldots \otimes x_{n-1}$

by Yoneda reduction. This completes the induction.

(It’s been ages since I’ve thought about this. I need to think through the argument carefully again.)

Todd Trimble talks about Yoneda reduction on the $n$Café here.

Revised on March 30, 2011 07:32:44
by Urs Schreiber
(89.204.153.85)