# The Stacks Project

## Tag 08WF

#### 34.4.1. 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 $$\tag{34.4.1.1} \xymatrix{ A \ar[r]^f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} & C }$$ 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 34.4.2. A split equalizer is a diagram (34.4.1.1) 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 $$\tag{34.4.2.1} h \circ f = 1_A, \quad f \circ h = i \circ g_1, \quad i \circ g_2 = 1_B.$$

The point is that the equalities among arrows force (34.4.1.1) 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.

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 824–877 (see updates for more information).

\subsection{Category-theoretic preliminaries}
\label{subsection-category-prelims}

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

\medskip\noindent
For two morphisms $g_1, g_2: B \to C$, recall that an {\it 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

\label{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 {\it coequalizer}.
See Categories, Sections \ref{categories-section-equalizers} and
\ref{categories-section-coequalizers}.

\medskip\noindent
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.

\begin{definition}
\label{definition-split-equalizer}
A {\it split equalizer} is a diagram (\ref{equation-equalizer}) 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

\label{equation-split-equalizer-conditions}
h \circ f = 1_A, \quad f \circ h = i \circ g_1, \quad i \circ g_2 = 1_B.

\end{definition}

\noindent
The point is that the equalities among arrows force (\ref{equation-equalizer})
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 {\it split coequalizer}, whose definition is apparent.

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