2-Yoneda lemma

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 on a category C, it says:Kelly 1982 § 2.4. for each object in C, the natural functor (evaluation at the identity) is an equivalence of categories, where denotes (roughly) the category of natural transformations between pseudofunctors on C and . Under the Grothendieck construction, corresponds to the comma category . So, the lemma is also frequently stated as:Vistoli 2008 § 3.6.2. where is identified with the fibered category associated to . As an application of this lemma, the coherence theorem for bicategories holds. Sketch of proof First we define the functor in the opposite direction as follows. Given an object in , define the natural transformation that is, by (In the below, we shall often drop a subscript for a natural transformation.) Next, given a morphism in , for , we let be Then 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 The above is a functor, , where is the evaluation at the identity; i.e., Claim 1 is clear. As for Claim 2, where the isomorphism here comes from the fact that is a pseudofunctor. Similarly, For Claim 3, we have: Similarly for a morphism ∞-Yoneda Given an ∞-category C, let be the ∞-category of presheaves on it with values in Kan = the ∞-category of Kan complexes. Then the ∞-version of the Yoneda embedding involves some (harmless) choice in the following way. First, we have the hom-functor 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 Cisinski 2023 § 5.8.1.8.3.3 Hom-Functors for ∞-Categories in Kerodon Fix one such functor. Then we get the Yoneda embedding functor in the usual way: which turns out to be fully faithful (i.e., an equivalence on the Hom level).Cisinski 2023 Theorem 5.8.13. (i). Moreover and more strongly, for each object in and object in , the evaluation at the identity (see below) is invertible in the ∞-category of large Kan complexes (i.e., Kan complexes living in a universe larger than the given one).Cisinski 2023 Theorem 5.8.13. (ii). Here, the evaluation map refers to the composition where the last map is the restriction to the identity .Cisinski 2023 § 5.8.8. The ∞-Yoneda lemma is closely related to the matter of straightening and unstraightening. Notes References Angelo Vistoli Angelo Vistoli http://homepage.sns.it/vistoli/descent.pdf Notes on Grothendieck topologies, fibered categories and descent theory September 2, 2008 Max Kelly Kelly Gregory Maxwell 0-521-28702-2 651714 Cambridge University Press, Cambridge-New York London Mathematical Society Lecture Note Series Basic concepts of enriched category theory 64 1982 http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf Cisinski Denis-Charles Denis-Charles Cisinski https://cisinski.app.uni-regensburg.de/CatLR.pdf Higher Categories and Homotopical Algebra 2023 Cambridge University Press 978-1108473200 en Further reading https://doi.org/10.1007/BFb0063102 10.1007/BFb0063102 Fibrations and Yoneda's lemma in a 2-category Category Seminar Lecture Notes in Mathematics 1974 Street Ross Ross Street 420 104–133 978-3-540-06966-9 http://eudml.org/doc/91227 Fibrations in bicategories Cahiers de Topologie et Géométrie Différentielle Catégoriques 1980 21 2 111–160 Street Ross 0805.3673 Johnson Niles Morita Theory for Derived Categories: A Bicategorical Perspective 2008 math.AT 2002.06055 10.1093/oso/9780198871378.001.0001 2-Dimensional Categories 2021 Johnson Niles Yau Donald 978-0-19-887137-8 Oxford University Press Lurie Jacob Jacob Lurie 2009 Higher Topos Theory 5.1.3 Yoneda’s Lemma math/0608040 Princeton University Press 978-0-691-14048-3 Kudasov Nikolai Riehl Emily Weinberger Jonathan Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs Formalizing the ∞-Categorical Yoneda Lemma 2024 274–290 10.1145/3636501.3636945 979-8-4007-0488-8 2309.08340 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 ncatlab.org 94.5 The 2-Yoneda lemma—The Stacks project https://stacks.math.columbia.edu/tag/04SS stacks.math.columbia.edu Lemma 4.41.2 (2-Yoneda lemma)—The Stacks project https://stacks.math.columbia.edu/tag/004B stacks.math.columbia.edu 8.3.3 Hom-Functors for ∞-Categories https://kerodon.net/tag/03MR Kerodon 8.3.3 Hom-Functors for ∞-Categories in Kerodon Kelly Shane Derived Algebraic Geometry, UTokyo Autumn Semester 2023-2024 Lecture 9: Yoneda https://www.ms.u-tokyo.ac.jp/~kelly/Course2023-24DAG/2023DAG09.pdf Category:Category theory