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A left adjoint exists exactly when chosen initial objects are supplied in every comma category

Statement

Let $G : D \to C$ be a functor. A left adjoint to $G$ is supplied exactly by choosing, for every $c \in C$ , an initial object $(F c, η_{c})$ of the comma category $(c ↓ G)$ . These choices determine the action of $F$ on morphisms and the adjunction uniquely.

Dually, a right adjoint to $F : C \to D$ is supplied exactly by choosing a terminal object in every comma category $(F ↓ d)$ .

Facts & Assumptions

Given: A functor $G : D \to C$ .

[F1]

The comma category $(c ↓ G)$ has objects $(d, f : c \to G d)$ , and a morphism from $(d, f)$ to $(d^{'}, f^{'})$ is a morphism $h : d \to d^{'}$ with $G (h) \circ f = f^{'}$ (Comma category, slice category, and coslice category).

[L1]

If $F ⊣ G$ , then $(F c, η_{c})$ is initial in $(c ↓ G)$ for every $c$ ; dually, counit components are terminal in $(F ↓ d)$ (Unit components are initial in comma categories, and counit components are terminal).

[L2]

Unit-counit data with the triangle identities and a natural family of universal arrows $(F c, η_{c})$ from each $c$ to $G$ carry the same adjunction data, and neither description requires local smallness (The unit-counit, hom-set, unit-universal, and counit-universal encodings of an adjunction are equivalent).

Proof

technique · direct

1.1

If a left adjoint $F$ is supplied, [L1] gives the required chosen initial object $(F c, η_{c})$ for every $c$ .

1.2

Conversely, suppose such initial objects are supplied. For $a : c \to c^{'}$ , initiality gives a unique $F (a) : F c \to F c^{'}$ satisfying $G (F (a)) \circ η_{c} = η_{c^{'}} \circ a$ .

F1choose

2.1

The identity $1_{F c}$ satisfies the defining equation for $F (1_{c})$ , so uniqueness gives $F (1_{c}) = 1_{F c}$ .

step 1.2F1

2.2

If $a : c \to c^{'}$ and $b : c^{'} \to c^{''}$ , then $F (b) F (a)$ satisfies the defining equation for $F (b a)$ ; uniqueness gives $F (b a) = F (b) F (a)$ . Thus $F$ is a functor and $η$ is natural.

step 1.2F1

3.1

For every $f : c \to G d$ , initiality supplies a unique $\hat{f} : F c \to d$ with $G (\hat{f}) η_{c} = f$ . Steps 2.1 and 2.2 make $F$ a functor and $η$ natural, so $(F c, η_{c})_{c}$ is a natural family of universal arrows from each $c$ to $G$ ; by [L2] that data is an adjunction, so $F ⊣ G$ .

step 1.2step 2.1step 2.2F1L2

4.1

Any functor action compatible with the chosen initial objects must satisfy the equation in step 1.2 and is therefore equal to this one. Passing to opposite categories proves the terminal-object criterion for right adjoints.

step 1.2step 3.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 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, 2nd ed., Lemma 4.7.1 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Corollary 2.3.7 (standard reference, not scraped)