nLab Yoneda lemma for bicategories

Contents

Context

Yoneda lemma

2-category theory

Statement
Implications
Proof

Explicit proof
High-technology proof

Related entries
References

Statement

The Yoneda lemma for bicategories is a version of the Yoneda lemma that applies to bicategories, the most common algebraic sort of weak 2-category. It says that for any bicategory $C$ , any object $x\in C$ , and any pseudofunctor $F\colon C^{op}\to Cat$ , there is an equivalence of categories

[C^{op},Cat](C(-,x), F) \simeq F(x)

which is pseudonatural in $x$ and $F$ , and which is given by evaluation at $1_x$ , i.e. $\alpha\colon C(-,x)\to F$ maps to $\alpha_x(1_x)$ .

For bicategories $A$ and $B$ , $[A,B]$ denotes the bicategory of pseudofunctors, pseudonatural transformations, and modifications from $A$ to $B$ . Note that it is a strict 2-category as soon as $B$ is.

Implications

In particular, the Yoneda lemma for bicategories implies that there is a Yoneda embedding for bicategories $C\to [C^{op},Cat]$ which is 2-fully-faithful?, i.e. an equivalence on hom-categories. Therefore, $C$ is equivalent to a sub-bicategory of $[C^{op},Cat]$ . Since Cat is a strict 2-category, it follows that $C$ is equivalent to a strict 2-category, which is one form of the coherence theorem for bicategories. (Conversely, another form of the coherence theorem can be used to prove the Yoneda lemma; see below.)

Proof

A detailed proof of the bicategorical Yoneda lemma is given in (Johnson & Yau 20, Chap. 8).

Explicit proof

An explicit proof, involving many diagrams, has been written up by Igor Baković and can be found here.

High-technology proof

We will take it for granted that $[C^{op},Cat]$ is a well-defined bicategory; this is a basic fact having nothing to do with the Yoneda lemma. We also take it as given that “evaluation at $1_x$ ” functor

[C^{op},Cat](C(-,x), F) \to F(x)

is well-defined and pseudonatural in $F$ and $x$ ; our goal is to prove that it is an equivalence. (Granted, these basic facts require a fair amount of verification as well.)

We will use part of the coherence theorem for pseudoalgebras?, which says that for a suitably well-behaved strict 2-monad $T$ , the inclusion $T$ - $Alg_{strict} \hookrightarrow Ps$ - $T$ - $Alg$ of the 2-category of strict $T$ -algebras and strict $T$ -morphisms into the 2-category of pseudo $T$ -algebras and pseudo $T$ -morphisms has a left adjoint, usually written as $(-)'$ . Moreover, for any pseudo $T$ -algebra $A$ , the unit $A\to A'$ is an equivalence in $Ps$ - $T$ - $Alg$ .

First, there is a 2-monad $T$ such that strict $T$ -algebras are strict 2-categories, strict $T$ -morphisms are strict 2-functors, pseudo $T$ -algebras are unbiased bicategories, and pseudo $T$ -morphisms are pseudofunctors. By Mac Lane’s coherence theorem for bicategories, any ordinary bicategory can equally well be considered as an unbiased one. Thus, since $Cat$ is a strict 2-category, for any bicategory $C$ there is a strict 2-category $C'$ such that pseudofunctors $C\to Cat$ are in bijection with strict 2-functors $C'\to Cat$ .

Now note that a pseudonatural transformation between two pseudofunctors (resp. strict 2-functors) $C\to D$ is the same as a single pseudofunctor (resp. strict 2-functor) $C\to Cyl(D)$ , where $Cyl(D)$ is the bicategory whose objects are the 1-cells of $D$ , whose 1-cells are squares in $D$ commuting up to isomorphism, and whose 2-cells are “cylinders” in $D$ . Likewise, a modification between two such transformations is the same as a single functor (of whichever sort) $C\to 2Cyl(D)$ , where the objects of $2Cyl(D)$ are the 2-cells of $D$ , and so on. Therefore, $C'$ classifies not only pseudofunctors out of $C$ , but transformations and modifications between them; thus we have an isomorphism

[C^{op},Cat] \cong [(C')^{op},Cat]_{strict,pseudo}

where $[A,B]_{strict,pseudo}$ denotes the 2-category of strict 2-functors, pseudonatural transformations, and modifications between two strict 2-categories. Thus we can equally well analyze the functor

[(C')^{op},Cat]_{strict,pseudo}(\overline{C(-,x)}, \overline{F}) \to \overline{F}(x) = F(x)

given by evaluation at $1_x$ . Here $\overline{C(-,x)}$ and $\overline{F}$ denote the strict 2-functors $(C')^{op}\to Cat$ corresponding to the pseudofunctors $C(-,x)$ and $F$ under the $(-)'$ adjunction. However, we also have a strict 2-functor $C'(-,x)$ , and the equivalence $C\simeq C'$ induces an equivalence $C'(-,x)\simeq \overline{C(-,x)}$ . Therefore, it suffices to analyze the functor

[(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x).

Now for any $A$ and $B$ , we have an inclusion functor $[A,B]_{strict,strict} \to [A,B]_{strict,pseudo}$ where $[A,B]_{strict,strict}$ denotes the 2-category of strict 2-functors, strict 2-natural transformations, and modifications. This functor is bijective on objects and locally fully faithful. Moreover, the composite

[(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x).

is an isomorphism, by the enriched Yoneda lemma, in the special case of $Cat$ -enrichment. Since

[(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F})

is fully faithful, if we can show that it is essentially surjective, then the 2-out-of-3 property for equivalences of categories will imply that the desired functor is an equivalence.

Here we at last descend to something concrete. Given $\alpha\colon C'(-,x)\to \overline{F}$ , we have an obvious choice for a strict transformation for it to be equivalent to, namely $\beta$ whose components $\beta_y\colon C'(y,x)\to \overline{F}(y)$ is given by $f \mapsto \overline{F}(f)(a)$ where $a = \alpha_x(1_x)\in \overline{F}(x)$ . Since $\alpha$ is pseudonatural, for any $f\colon y\to x$ in $C'$ we have an isomorphism

\alpha_y(f) = \alpha_y(f\circ 1_x) \cong \overline{F}(f)(\alpha_x(1_x)) = \overline{F}(f)(a) = \beta_y(f).

We then simply verify that these isomorphisms are the components of an (invertible) modification $\alpha\cong \beta$ . This completes the proof.

Related entries

Yoneda lemma for tricategories

References

Review:

Angelo Vistoli, §3.6.2 in: Grothendieck topologies, fibered categories and descent theory, in: Fundamental algebraic geometry – Grothendieck's FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) 1-104 [ISBN:978-0-8218-4245-4, math.AG/0412512]