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Objects $a$ and $b$ are isomorphic exactly when $C (-, a)$ and $C (-, b)$ are naturally isomorphic

Statement

Let $a$ and $b$ be objects of a locally small category $C$ . Then

$a ≅ b ⟺ C (-, a) ≅ C (-, b)$

by a natural isomorphism of presheaves.

Facts & Assumptions

Given: Objects $a, b$ of a locally small category $C$ .

[L1]

The Yoneda assignment induces a bijection $C (x, y) ≅ Nat (C (-, x), C (-, y))$ for every $x, y$ (The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective).

[F1]

A natural isomorphism $α : F \Rightarrow G$ has a natural transformation $β : G \Rightarrow F$ with $β \circ α = 1_{F}$ and $α \circ β = 1_{G}$ (Natural isomorphism).

[F2]

An isomorphism $f : a \to b$ has a morphism $f^{- 1} : b \to a$ with $f^{- 1} f = 1_{a}$ and $f f^{- 1} = 1_{b}$ (Isomorphism, groupoid, and connected category).

Proof

technique · direct

1.1

If $f : a \to b$ is an isomorphism, its Yoneda image $y (f)$ has inverse $y (f^{- 1})$ , since the postcomposition formulas give $y (f^{- 1}) y (f) = y (1_{a})$ and $y (f) y (f^{- 1}) = y (1_{b})$ ; hence the representable presheaves are naturally isomorphic.

givenL1F2

1.2

Conversely, let $α : C (-, a) \Rightarrow C (-, b)$ be a natural isomorphism with inverse $β$ . By the surjectivity in [L1], there are $f : a \to b$ and $g : b \to a$ with $y (f) = α$ and $y (g) = β$ .

L1F1choose

2.1

The inverse equations of [F1] give $y (g f) = y (g) y (f) = 1_{y (a)} = y (1_{a})$ and $y (f g) = y (1_{b})$ ; injectivity in [L1] therefore gives $g f = 1_{a}$ and $f g = 1_{b}$ , so $f$ is an isomorphism.

step 1.2L1F1F2

3.1

Step 1.1 proves the forward implication and steps 1.2--2.1 prove the reverse implication.

step 1.1step 2.1∎

Depends on

Used by

Non-isomorphic objects can have naturally isomorphic representable presheaves False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 12 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, Proposition 2.3.1 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Corollary 4.3.10 (standard reference, not scraped)

Objects aa and bb are isomorphic exactly when C(−,a)C(−,a) and C(−,b)C(−,b) are naturally isomorphic