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The Yoneda embedding is its own pointwise left Kan extension

Statement

Let $C$ be small, and let $y : C \to [C^{o p}, S e t]$ be the Yoneda embedding. Then the identity functor on $[C^{o p}, S e t]$ , together with the identity natural transformation on $y$ , is a pointwise left Kan extension of $y$ along $y$ .

Equivalently, for every presheaf $P$ , the comma-category colimit computing ${Lan}_{y} y$ at $P$ is just $P$ itself.

Facts & Assumptions

Given: A small category $C$ , its Yoneda embedding $y$ , and a presheaf $P$ on $C$ .

[L1]

The density theorem expresses $P$ as the colimit of the diagram $\int P \to [C^{o p}, S e t]$ sending $(c, x)$ to $y (c)$ (Density theorem for a small category).

[F1]

Evaluation at the identity gives a natural bijection between morphisms $y (c) \Rightarrow P$ and elements $x \in P (c)$ ; under this bijection the comma category $(y ↓ P)$ is the category of elements of $P$ (For a presheaf $P$ , $Nat (C (-, a), P) ≅ P (a)$ naturally in $a$ and $P$ , The category of elements of a covariant functor or a presheaf, The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding).

[F2]

A pointwise left Kan extension at $P$ is computed by the colimit over $(y ↓ P)$ (Pointwise Kan extensions by the comma-category formula).

[L2]

If comma-category colimits and their universal cocones are supplied at every target object, they assemble uniquely into a left Kan extension functor, with unit given by the identity-indexed legs (Comma-category limit and colimit formulae compute Kan extensions).

Proof

technique · direct

1.1

By [F1], the comma category $(y ↓ P)$ is canonically the category of elements used in [L1], and under that identification its canonical diagram sends an object $(c, x)$ to the representable presheaf $y (c)$ .

F1L1

2.1

The colimit given by [L1] is therefore exactly the comma-category colimit required by [F2] to compute the pointwise left Kan extension of $y$ along itself at $P$ . That colimit object is $P$ .

F2step 1.1L1

3.1

For a natural transformation $θ : P \Rightarrow Q$ , the uniquely forced arrow between the two canonical density colimits is $θ$ itself, because its composites with all Yoneda legs are the legs indexed by the elements $θ_{c} (x)$ . Thus the assembled arrow maps are those of the identity functor, and the identity-indexed unit legs are identities.

L1step 2.1

4.1

Since these canonical colimits are supplied for every presheaf, [L2] assembles them into a left Kan extension; steps 2.1 and 3.1 identify it with the identity functor and the identity transformation on $y$ . By [F2] it is pointwise.

L2F2step 2.1step 3.1∎

Depends on

Used by

Dense subcategory Definition
The presheaf category on a small category is the free cocompletion Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Theorem 6.5.10 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Corollary 5.4.4 (standard reference, not scraped)