2-Yoneda lemma

Short description: Result in category theory

In mathematics, especially category theory, the 2-Yoneda lemma is a generalization of the Yoneda lemma to 2-categories. Precisely, given a contravariant pseudofunctor $F$ on a category C, it says: for each object $x$ in C, the natural functor (evaluation at the identity)

$\underset{\_}{\mathrm{Hom}}(h_x,F)\to F(x)$

is an equivalence of categories, where $\underset{\_}{\mathrm{Hom}}(-,-)$ denotes (roughly) the category of natural transformations between pseudofunctors on C and $h_x=\mathrm{Hom}(-,x)$.

Under the Grothendieck construction, $h_x$ corresponds to the comma category $C\downarrow x$. So, the lemma is also frequently stated as:

$F(x)\simeq \underset{\_}{\mathrm{Hom}}(C\downarrow x,F),$

where $F$ is identified with the fibered category associated to $F$.

As an application of this lemma, the coherence theorem for bicategories holds.

Sketch of proof

First we define the functor in the opposite direction

$\mu :F(x)\to \underset{\_}{\mathrm{Hom}}(h_x,F)$

as follows. Given an object $\bar{x}$ in $F(x)$, define the natural transformation

$\mu (\bar{x}):h_x\to F,$

that is, $\mu (\bar{x}{)}_y:\mathrm{Hom}(y,x)\to F(y),$ by

$\mu (\bar{x}{)}_y(f)=(Ff)\bar{x}.$

(In the below, we shall often drop a subscript for a natural transformation.) Next, given a morphism $\phi :\bar{x}\to x^{\prime }$ in $F(x)$, for $f:y\to x$, we let $\mu (\phi )(f)$ be

$(Ff)\phi :(Ff)\bar{x}\to (Ff)x^{\prime }.$

Then $\mu (\phi ):\mu (\bar{x})\to \mu (x^{\prime })$ is a morphism (a 2-morphism to be precise or a modification in the terminology of Bénabou). The rest of the proof is then to show

1. The above $\mu$ is a functor,
2. $e\circ \mu \simeq \mathrm{id}$, where $e$ is the evaluation at the identity; i.e., $e(\lambda )=\lambda ({\mathrm{id}}_x),$ $e(\alpha :\lambda \to \rho )=\alpha ({\mathrm{id}}_x):\lambda ({\mathrm{id}}_x)\to \rho ({\mathrm{id}}_x),$
3. $\mu \circ e\simeq \mathrm{id}.$

Claim 1 is clear. As for Claim 2,

$e(\mu (\bar{x}))=\mu (\bar{x})({\mathrm{id}}_x)=F({\mathrm{id}}_x)\bar{x}\simeq {\mathrm{id}}_{F(x)}\bar{x}=\bar{x}$

where the isomorphism here comes from the fact that $F$ is a pseudofunctor. Similarly,$e(\mu (\phi ))\simeq \phi .$ For Claim 3, we have:

$\mu (e(\lambda ))(f)=(Ff\circ \lambda )({\mathrm{id}}_x)\simeq (\lambda \circ h_{x}f)({\mathrm{id}}_x)=\lambda (f).$

Similarly for a morphism $\alpha :\lambda \to \rho .$ $◻$

∞-Yoneda

Given an ∞-category C, let $\hat{C}=\underset{\_}{\mathrm{Hom}}(C^{op},\text{<mrow data-mjx-texclass="ORD">Kan</mrow>})$ be the ∞-category of presheaves on it with values in Kan = the ∞-category of Kan complexes. Then the ∞-version of the Yoneda embedding $C\hookrightarrow \hat{C}$ involves some (harmless) choice in the following way.

First, we have the hom-functor

$\mathrm{Hom}:C^{op}\times C\to \text{<mrow data-mjx-texclass="ORD">Kan</mrow>}$

that is characterized by a certain universal property (e.g., universal left fibration) and is unique up to a unique isomorphism in the homotopy category $\mathrm{ho}\underset{\_}{\mathrm{Hom}}(C\times C^{op},\text{<mrow data-mjx-texclass="ORD">Kan</mrow>}).$ Fix one such functor. Then we get the Yoneda embedding functor in the usual way:

$y:C\to \hat{C},a\mapsto \mathrm{Hom}(-,a),$

which turns out to be fully faithful (i.e., an equivalence on the Hom level). Moreover and more strongly, for each object $F$ in $\hat{C}$ and object $a$ in $C$, the evaluation $e$ at the identity (see below)

$\mathrm{Hom}(y(a),F)\to F(a)$

is invertible in the ∞-category of large Kan complexes (i.e., Kan complexes living in a universe larger than the given one). Here, the evaluation map $e$ refers to the composition

$\mathrm{Hom}(y(a),F)\stackrel{-(a)}{\to }\mathrm{Hom}(y(a)(a),F(a))=\mathrm{Hom}(\mathrm{Hom}(a,a),F(a))\to F(a)$

where the last map is the restriction to the identity ${\mathrm{id}}_a$.

The ∞-Yoneda lemma is closely related to the matter of straightening and unstraightening.

References

• Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory". http://homepage.sns.it/vistoli/descent.pdf. 
• Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, 64, Cambridge University Press, Cambridge-New York, ISBN 0-521-28702-2, http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf 
• Cisinski, Denis-Charles (2023) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1108473200. https://cisinski.app.uni-regensburg.de/CatLR.pdf. 

Further reading

• Street, Ross (1974). "Fibrations and Yoneda's lemma in a 2-category". Category Seminar. Lecture Notes in Mathematics. 420. pp. 104–133. doi:10.1007/BFb0063102. ISBN 978-3-540-06966-9. https://doi.org/10.1007/BFb0063102. 
• Street, Ross (1980). "Fibrations in bicategories". Cahiers de Topologie et Géométrie Différentielle Catégoriques 21 (2): 111–160. http://eudml.org/doc/91227. 
• Johnson, Niles (2008). "Morita Theory for Derived Categories: A Bicategorical Perspective". arXiv:0805.3673 [math.AT].
• Johnson, Niles; Yau, Donald (2021). 2-Dimensional Categories. Oxford University Press. doi:10.1093/oso/9780198871378.001.0001. ISBN 978-0-19-887137-8. 
• Lurie, Jacob (2009). "5.1.3 Yoneda’s Lemma". Higher Topos Theory. Princeton University Press. ISBN 978-0-691-14048-3. 
• Kudasov, Nikolai; Riehl, Emily; Weinberger, Jonathan (2024). "Formalizing the ∞-Categorical Yoneda Lemma". Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs. pp. 274–290. doi:10.1145/3636501.3636945. ISBN 979-8-4007-0488-8. 
• https://math.stackexchange.com/questions/1293920/yoneda-lemma-for-2-categories-lax-version
• "Yoneda lemma for bicategories in nLab". https://ncatlab.org/nlab/show/Yoneda+lemma+for+bicategories. 
• "94.5 The 2-Yoneda lemma—The Stacks project". https://stacks.math.columbia.edu/tag/04SS. 
• "Lemma 4.41.2 (2-Yoneda lemma)—The Stacks project". https://stacks.math.columbia.edu/tag/004B. 
• "2.2 The Theory of 2-Categories". https://kerodon.net/tag/007K.