Lemma 4.23.2 (0035)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.23: Exact functors
Lemma 4.23.2 (cite)

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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. $\square$

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