35.4.1 Category-theoretic preliminaries

We start by recalling a few basic notions from category theory which will simplify the exposition. In this subsection, fix an ambient category.

For two morphisms $g_1, g_2: B \to C$, recall that an equalizer of $g_1$ and $g_2$ is a morphism $f: A \to B$ which satisfies $g_1 \circ f = g_2 \circ f$ and is universal for this property. This second statement means that any commutative diagram

$\xymatrix{A' \ar[rd]^ e \ar@/^1.5pc/[rrd] \ar@{-->}[d] & & \\ A \ar[r]^ f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} & C }$

without the dashed arrow can be uniquely completed. We also say in this situation that the diagram

35.4.1.1
$$\label{descent-equation-equalizer} \xymatrix{ A \ar[r]^ f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} & C }$$

is an equalizer. Reversing arrows gives the definition of a coequalizer. See Categories, Sections 4.10 and 4.11.

Since it involves a universal property, the property of being an equalizer is typically not stable under applying a covariant functor. Just as for monomorphisms and epimorphisms, one can get around this in some cases by exhibiting splittings.

Definition 35.4.2. A split equalizer is a diagram (35.4.1.1) with $g_1 \circ f = g_2 \circ f$ for which there exist auxiliary morphisms $h : B \to A$ and $i : C \to B$ such that

35.4.2.1
$$\label{descent-equation-split-equalizer-conditions} h \circ f = 1_ A, \quad f \circ h = i \circ g_1, \quad i \circ g_2 = 1_ B.$$

The point is that the equalities among arrows force (35.4.1.1) to be an equalizer: the map $e$ factors uniquely through $f$ by writing $e = f \circ (h \circ e)$. Consequently, applying a covariant functor to a split equalizer gives a split equalizer; applying a contravariant functor gives a split coequalizer, whose definition is apparent.

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