nLab Yoneda reduction

Idea

The term Yoneda reduction was coined by Todd Trimble in his (unpublished) thesis.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.

The module perspective on the Yoneda lemma

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 $C$.

Calculus of bimodules

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.)

Examples

Relative coherence theorem for symmetric monoidal categories

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 $I$.

Proof

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.)

Blog resources

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