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
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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