2-Yoneda lemma

In mathematics, especially category theory, the 2-Yoneda lemma is a generalization of the Yoneda lemma to 2-categories. Precisely, given a contravariant pseudofunctor ${\displaystyle F}$ on a category C, it says: ^[1] for each object ${\displaystyle x}$ in C, the natural functor (evaluation at the identity)

${\displaystyle {\underline {\operatorname {Hom} }}(h_{x},F)\to F(x)}$

is an equivalence of categories, where ${\displaystyle {\underline {\operatorname {Hom} }}(-,-)}$ denotes (roughly) the category of natural transformations between pseudofunctors on C and ${\displaystyle h_{x}=\operatorname {Hom} (-,x)}$.

Under the Grothendieck construction, ${\displaystyle h_{x}}$ corresponds to the comma category ${\displaystyle C\downarrow x}$. So, the lemma is also frequently stated as: ^[2]

${\displaystyle F(x)\simeq {\underline {\operatorname {Hom} }}(C\downarrow x,F),}$

where ${\displaystyle F}$ is identified with the fibered category associated to ${\displaystyle F}$.

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

Sketch of proof

First we define the functor in the opposite direction

${\displaystyle \mu :F(x)\to {\underline {\operatorname {Hom} }}(h_{x},F)}$

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

${\displaystyle \mu ({\overline {x}}):h_{x}\to F,}$

that is, ${\displaystyle \mu ({\overline {x}})_{y}:\operatorname {Hom} (y,x)\to F(y),}$ by

${\displaystyle \mu ({\overline {x}})_{y}(f)=(Ff){\overline {x}}.}$

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

${\displaystyle (Ff)\varphi$ :(Ff){\overline {x}}\to (Ff)x'.}

Then ${\displaystyle \mu (\varphi ):\mu ({\overline {x}})\to \mu (x')}$ 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 ${\displaystyle \mu }$ is a functor,
2. ${\displaystyle e\circ \mu \simeq \operatorname {id} }$, where ${\displaystyle e}$ is the evaluation at the identity; i.e., ${\displaystyle e(\lambda )=\lambda (\operatorname {id} _{x}),}$${\displaystyle e(\alpha$ :\lambda \to \rho )=\alpha (\operatorname {id} _{x}):\lambda (\operatorname {id} _{x})\to \rho (\operatorname {id} _{x}),}
3. ${\displaystyle \mu \circ e\simeq \operatorname {id} .}$

Claim 1 is clear. As for Claim 2,

${\displaystyle e(\mu ({\overline {x}}))=\mu ({\overline {x}})(\operatorname {id} _{x})=F(\operatorname {id} _{x}){\overline {x}}\simeq \operatorname {id} _{F(x)}{\overline {x}}={\overline {x}}}$

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

${\displaystyle \mu (e(\lambda ))(f)=(Ff\circ \lambda )(\operatorname {id} _{x})\simeq (\lambda \circ h_{x}f)(\operatorname {id} _{x})=\lambda (f).}$

Similarly for a morphism ${\displaystyle \alpha$ :\lambda \to \rho .}${\displaystyle \square }$

∞-Yoneda

Given an ∞-category C, let ${\displaystyle {\widehat {C}}={\underline {\operatorname {Hom} }}(C^{op},{\textbf {Kan}})}$ be the ∞-category of presheaves on it with values in Kan = the ∞-category of Kan complexes. Then the ∞-version of the Yoneda embedding ${\displaystyle C\hookrightarrow {\widehat {C}}}$ involves some (harmless) choice in the following way.

First, we have the hom-functor

${\displaystyle \operatorname {Hom} :C^{op}\times C\to {\textbf {Kan}}}$

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 ${\displaystyle \operatorname {ho} {\underline {\operatorname {Hom} }}(C\times C^{op},{\textbf {Kan}}).}$ ^{[3] [4]} Fix one such functor. Then we get the Yoneda embedding functor in the usual way:

${\displaystyle y:C\to {\widehat {C}},\,a\mapsto \operatorname {Hom} (-,a),}$

which turns out to be fully faithful (i.e., an equivalence on the Hom level). ^[5] Moreover and more strongly, for each object ${\displaystyle F}$ in ${\displaystyle {\widehat {C}}}$ and object ${\displaystyle a}$ in ${\displaystyle C}$, the evaluation ${\displaystyle e}$ at the identity (see below)

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

is invertible in the ∞-category of large Kan complexes (i.e., Kan complexes living in a universe larger than the given one). ^[6] Here, the evaluation map ${\displaystyle e}$ refers to the composition

${\displaystyle \operatorname {Hom} (y(a),F){\overset {-(a)}{\to }}\operatorname {Hom} (y(a)(a),F(a))=\operatorname {Hom} (\operatorname {Hom} (a,a),F(a))\to F(a)}$

where the last map is the restriction to the identity ${\displaystyle \operatorname {id} _{a}}$. ^[7]

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

Notes

1. Kelly 1982 , § 2.4.
2. Vistoli 2008 , § 3.6.2.
3. Cisinski 2023 , § 5.8.1.
4. 8.3.3 Hom-Functors for ∞-Categories in Kerodon
5. Cisinski 2023 , Theorem 5.8.13. (i).
6. Cisinski 2023 , Theorem 5.8.13. (ii).
7. Cisinski 2023 , § 5.8.8.

References

• Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory" (PDF).
• Kelly, Gregory Maxwell (1982), Basic concepts of enriched category theory (PDF), London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, ISBN   0-521-28702-2, MR   0651714
• Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN   978-1108473200.

Further reading

• Street, Ross (1974). "Fibrations and Yoneda's lemma in a 2-category". Category Seminar. Lecture Notes in Mathematics. Vol. 420. pp. 104–133. doi:10.1007/BFb0063102. ISBN   978-3-540-06966-9.
• Street, Ross (1980). "Fibrations in bicategories". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 21 (2): 111–160.
• 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. arXiv: 2002.06055 . 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. arXiv: math/0608040 . ISBN   978-0-691-14048-3.
• E. Riehl and D. Verity, Fibrations and Yoneda’s lemma in an ∞-cosmos, J. Pure Appl. Algebra 221 (2017), no. 3, 499–564, arXiv:1506.05500
• https://math.stackexchange.com/questions/1293920/yoneda-lemma-for-2-categories-lax-version
• "Yoneda lemma for bicategories in nLab". ncatlab.org.
• "94.5 The 2-Yoneda lemma—The Stacks project". stacks.math.columbia.edu.
• "Lemma 4.41.2 (2-Yoneda lemma)—The Stacks project". stacks.math.columbia.edu.
• "2.2 The Theory of 2-Categories". Kerodon.