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The statement of the Yoneda lemma generalizes from categories to (∞,1)-categories.
For an (∞,1)-category and its (∞,1)-category of (∞,1)-presheaves, the -Yoneda embedding is the (∞,1)-functor
given by .
Seen under the -Grothendieck construction this is Riehl & Verity 2018, Def. 6.2.3.
-Yoneda embedding
Let be an (∞,1)-category and be the corresponding (∞,1)-category of (∞,1)-presheaves. Then the canonical (∞,1)-functor
For small -categories this is HTT, prop. 5.1.3.1. For possibly large -categories see Riehl & Verity 2018, Thm. 7.2.22 (which considers -presheaves regarded under the -Grothendieck construction) and Kerodon, Thm. 8.2.5.4.
-Yoneda theorem
For a small -category and an -functor, the composite
is equivalent to .
For small -sites this is HTT, Lemma 5.5.2.1. For possibly large -sites see Kerodon, Prop. 8.2.1.3.
For small -sites, the statement may be obtained as a consequence of the sSet-enriched Yoneda lemma by using the fact that the (∞,1)-category of (∞,1)-presheaves is modeled by the enriched functor category with regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.
can be extended to a functor so that the yoneda embedding is a natural transformation.
Here, is the (∞,1)-category of large (∞,1)-categories.
This follows from (HTT, prop. 5.3.6.10), together with the identification of with the category obtained by freely adjoining small colimits to . This functor is locally left adjoint to the contravariant functor .
The -Yoneda embedding preserves all (∞,1)-limits that exist in .
For an (∞,1)-site and an (∞,1)-topos, (∞,1)-geometric morphisms from the (∞,1)-sheaf (∞,1)-topos to correspond to the local (∞,1)-functors , those that
are left exact (∞,1)-functors;
send covering families in to effective epimorphism
More preseicely, the (∞,1)-functor
given by precomposition of inverse image functors by ∞-stackification and by the (∞,1)-Yoneda embedding is a full and faithful (∞,1)-functor and its essential image is spanned by these local morphisms.
Jacob Lurie, Prop. 5.1.3.1 and Lemma 5.5.2.1 in: Higher Topos Theory (2009)
Emily Riehl, Dominic Verity, Def. 6.2.3 and Thm. 7.2.22 in: The comprehension construction, Higher Structures 2 1 (2018) (arXiv:1706.10023, hs:39)
MathOverflow, The Yoneda Lemma for -categories? (MO:9737/381)
Kerodon, Part 2, Chapter 8: The Yoneda Embedding kerodon:03JA, esp. Prop. 8.2.1.3 and Thm. 8.2.5.4
Shay Ben-Moshe, Uniqueness and -Naturality of Yoneda [arXiv:2405.08799]
Discussion in the context of an ∞-cosmos:
Discussion internal to any (∞,1)-topos:
Formalization of the -Yoneda lemma via simplicial homotopy type theory (in Rzk):
Last revised on May 15, 2024 at 06:58:32. See the history of this page for a list of all contributions to it.