## 4.23 Exact functors

In this section we define exact functors.

Definition 4.23.1. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor.

Suppose all finite limits exist in $\mathcal{A}$. We say $F$ is *left exact* if it commutes with all finite limits.

Suppose all finite colimits exist in $\mathcal{A}$. We say $F$ is *right exact* if it commutes with all finite colimits.

We say $F$ is *exact* if it is both left and right exact.

Lemma 4.23.2. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. Suppose all finite limits exist in $\mathcal{A}$, see Lemma 4.18.4. The following are equivalent:

$F$ is left exact,

$F$ commutes with finite products and equalizers, and

$F$ transforms a final object of $\mathcal{A}$ into a final object of $\mathcal{B}$, and commutes with fibre products.

**Proof.**
Lemma 4.14.11 shows that (2) implies (1). Suppose (3) holds. The fibre product over the final object is the product. If $a, b : A \to B$ are morphisms of $\mathcal{A}$, then the equalizer of $a, b$ is

\[ (A \times _{a, B, b} A)\times _{(\text{pr}_1, \text{pr}_2), A \times A, \Delta } A. \]

Thus (3) implies (2). Finally (1) implies (3) because the empty limit is a final object, and fibre products are limits.
$\square$

Lemma 4.23.3. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. Suppose all finite colimits exist in $\mathcal{A}$, see Lemma 4.18.7. The following are equivalent:

$F$ is right exact,

$F$ commutes with finite coproducts and coequalizers, and

$F$ transforms an initial object of $\mathcal{A}$ into an initial object of $\mathcal{B}$, and commutes with pushouts.

**Proof.**
Dual to Lemma 4.23.2.
$\square$

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