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Tag 08WF

Chapter 34: Descent > Section 34.4: Descent for universally injective morphisms

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 \begin{equation} \tag{34.4.1.1} \xymatrix{ A \ar[r]^f & B \ar@<1ex>[r]^{g_1} \ar@<-1ex>[r]_{g_2} & C } \end{equation} 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 \begin{equation} \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. \end{equation}

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.

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