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For a presheaf $P$ , $Nat (C (-, a), P) ≅ P (a)$ naturally in $a$ and $P$

Statement

Let $C$ be locally small, let $a$ be an object, and let $P : C^{o p} \to S e t$ be a presheaf. Evaluation at the identity gives a bijection

$E_{P}^{a} : Nat (C (-, a), P) \overset{≅}{\to} P (a), E_{P}^{a} (α) = α_{a} (1_{a}),$

whose inverse sends $x \in P (a)$ to the natural transformation with $c$ -component $f : c \to a \mapsto P (f) (x)$ . It is natural in both variables. In particular, for $h : a \to a^{'}$ and $β : C (-, a^{'}) \Rightarrow P$ ,

$E_{P}^{a} (β \circ C (-, h)) = P (h) (E_{P}^{a^{'}} (β)),$

and postcomposition by a natural transformation $P \Rightarrow Q$ corresponds to its component at $a$ .

Facts & Assumptions

Given: The locally small category $C$ , object $a$ , presheaf $P$ , and the morphisms and natural transformations in the statement.

[L1]

For a covariant functor $F : D \to S e t$ , evaluation at the identity is a bijection $Nat (D (d, -), F) \to F (d)$ whose inverse sends $x \in F (d)$ to the transformation with component $α_{c}^{x} (f) = F (f) (x)$ (Evaluation at the identity gives $Nat (C (a, -), F) ≅ F (a)$ and proves that the natural-transformation collection is a set), and this bijection is natural in $d$ and in $F$ (The Yoneda bijection $Nat (C (a, -), F) ≅ F (a)$ is natural in both $a$ and $F$ ).

[F1]

The opposite category has the same objects and satisfies $C^{o p} (a, c) = C (c, a)$ (Opposite category $C^{o p}$ ).

[L2]

A theorem derived from category axioms has a formal dual obtained by reversing morphisms and composition (Every theorem about categories has a formal dual obtained by reversing morphisms and composition).

[F2]

A presheaf is a functor $C^{o p} \to S e t$ (Presheaves, covariantly and contravariantly representable functors, and representations).

Proof

technique · direct

1.1

Apply [L1] to $D = C^{o p}$ , the object $a$ , and the covariant functor $P$ of [F2]; by [F1], its hom-functor $C^{o p} (a, -)$ is $C (-, a)$ , and the inverse formula becomes $f : c \to a \mapsto P (f) (x)$ .

L1F1F2

1.2

Under the same translation, a morphism $h : a \to a^{'}$ in $C$ is reversed in $C^{o p}$ , so naturality in the object becomes the displayed equation with $C (-, h)$ and $P (h)$ ; naturality in $P$ is unchanged.

L1F1L2

2.1

Steps 1.1 and 1.2 prove the bijection, its inverse formula, and both naturalities.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Exercise 2.2.i (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Theorem 4.2.1 (standard reference, not scraped)

For a presheaf PP, Nat⁡(C(−,a),P)≅P(a)Nat(C(−,a),P)≅P(a) naturally in aa and PP