nLab Yoneda lemma for (infinity,1)-categories

Idea

The statement of the Yoneda lemma generalizes from categories to (∞,1)-categories.

Yoneda embedding

Definition

For $C$ an (∞,1)-category and $PSh(C)$ its (∞,1)-category of (∞,1)-presheaves, the $(\infty,1)$-Yoneda embedding is the (∞,1)-functor

$$y \colon C \to PSh(C)$$

given by $y(X) \colon U \mapsto C(U,X)$.

Seen under the $\infty$-Grothendieck construction this is Riehl & Verity 2018, Def. 6.2.3.

Properties

Yoneda lemma

Proposition

$(\infty,1)$-Yoneda embedding

Let $C$ be an (∞,1)-category and $PSh(C) \coloneqq Func(C^\op, \infty Grpd)$ be the corresponding (∞,1)-category of (∞,1)-presheaves. Then the canonical (∞,1)-functor

$$Y \colon C \to PSh(C)$$

is a full and faithful (∞,1)-functor.

For small $\infty$-categories this is HTT, prop. 5.1.3.1. For possibly large $\infty$-categories see Riehl & Verity 2018, Thm. 7.2.22 (which considers $\infty$-presheaves regarded under the $\infty$-Grothendieck construction) and Kerodon, Thm. 8.2.5.4.

Proposition

$(\infty,1)$-Yoneda theorem

For $C$ a small $(\infty,1)$-category and $F \colon C^{op} \to \infty Grpd$ an $(\infty,1)$-functor, the composite

$$C^{op} \to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to} \infty Grpd$$

is equivalent to $F$.

For small $\infty$-sites this is HTT, Lemma 5.5.2.1. For possibly large $\infty$-sites see Kerodon, Prop. 8.2.1.3.

Proof

For small $\infty$-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 $PSh_{(\infty,1)}(C)$ is modeled by the enriched functor category $[C^{op}, sSet]_{proj}$ with $C$ regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.

Naturality

Proposition

$PSh$ can be extended to a functor $PSh \colon (\infty,1)Cat \to (\infty,1)\widehat{Cat}$ so that the yoneda embedding $C \to PSh(C)$ is a natural transformation.

Here, $(\infty,1)\widehat{Cat}$ is the (∞,1)-category of large (∞,1)-categories.

This follows from (HTT, prop. 5.3.6.10), together with the identification of $PSh(C)$ with the category obtained by freely adjoining small colimits to $C$. This functor is locally left adjoint to the contravariant functor $C \mapsto Func(C^\op, \infty Grpd)$.

Preservation of limits

Proposition

The $(\infty,1)$-Yoneda embedding $y : C \to PSh(C)$ preserves all (∞,1)-limits that exist in $C$.

(HTT, prop. 5.1.3.2)

Local Yoneda embedding

Proposition

For $C$ an (∞,1)-site and $\mathcal{X}$ an (∞,1)-topos, (∞,1)-geometric morphisms $(f^* \dashv f_*) Sh(C) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{X}$ from the (∞,1)-sheaf (∞,1)-topos $Sh(C)$ to $\mathcal{X}$ correspond to the local (∞,1)-functors $f^* : C \to \mathcal{X}$, those that

• are left exact (∞,1)-functors;

• send covering families $\{U_i \to X\}$ in $\mathcal{G}$ to effective epimorphism

  $$\coprod_i f^*(U_i) \to f^*(X) \,.$$

More preseicely, the (∞,1)-functor

$$Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G})) \stackrel{L}{\to} Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G})) \stackrel{y}{\to} Func(\mathcal{X}, \mathcal{G})$$

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.

(HTT, prop. 6.2.3.20)

Related concepts

• Brown representability theorem

• Yoneda Lemma for higher categories

References

• 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 $(\infty,1)$-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 $(\infty,2)$-Naturality of Yoneda [arXiv:2405.08799]

Discussion in the context of an ∞-cosmos:

• Emily Riehl, Dominic Verity, Section 6 of: Fibrations and Yoneda’s lemma in an $\infty$-cosmos, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 (arXiv:1506.05500, doi:10.1016/j.jpaa.2016.07.003)

Discussion internal to any (∞,1)-topos:

• Louis Martini, Yoneda’s lemma for internal higher categories, [arXiv:2103.17141]

Formalization of the $(\infty,1)$-Yoneda lemma via simplicial homotopy type theory (in Rzk):

• Nikolai Kudasov, Emily Riehl, Jonathan Weinberger. Formalizing the $\infty$-categorical Yoneda lemma (2023) [arXiv:2309.08340]