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Tag 0105

Chapter 12: Homological Algebra > Section 12.3: Preadditive and additive categories

Lemma 12.3.7. Let $\mathcal{A}$, $\mathcal{B}$ be preadditive categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor. Then $F$ transforms direct sums to direct sums and zero to zero.

Proof. Suppose $F$ is additive. A direct sum $z$ of $x$ and $y$ is characterized by having morphisms $i : x \to z$, $j : y \to z$, $p : z \to x$ and $q : z \to y$ such that $p \circ i = \text{id}_x$, $q \circ j = \text{id}_y$, $p \circ j = 0$, $q \circ i = 0$ and $i \circ p + j \circ q = \text{id}_z$, according to Remark 12.3.6. Clearly $F(x), F(y), F(z)$ and the morphisms $F(i), F(j), F(p), F(q)$ satisfy exactly the same relations (by additivity) and we see that $F(z)$ is a direct sum of $F(x)$ and $F(y)$. Hence, $F$ transforms direct sums to direct sums.

To see that $F$ transforms zero to zero, use the characterization (3) of the zero object in Lemma 12.3.2. $\square$

    The code snippet corresponding to this tag is a part of the file homology.tex and is located in lines 162–167 (see updates for more information).

    \begin{lemma}
    \label{lemma-additive-additive}
    Let $\mathcal{A}$, $\mathcal{B}$ be preadditive categories.
    Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor.
    Then $F$ transforms direct sums to direct sums and zero to zero.
    \end{lemma}
    
    \begin{proof}
    Suppose $F$ is additive. A direct sum $z$
    of $x$ and $y$ is characterized by having morphisms
    $i : x \to z$, $j : y \to z$, $p : z \to x$ and
    $q : z \to y$ such that $p \circ i = \text{id}_x$,
    $q \circ j = \text{id}_y$, $p \circ j = 0$, $q \circ i = 0$
    and $i \circ p + j \circ q = \text{id}_z$, according
    to Remark \ref{remark-direct-sum}. Clearly $F(x), F(y), F(z)$
    and the morphisms $F(i), F(j), F(p), F(q)$ satisfy exactly the
    same relations (by additivity) and we see that $F(z)$ is
    a direct sum of $F(x)$ and $F(y)$.
    Hence, $F$ transforms direct sums to direct sums.
    
    \medskip\noindent
    To see that $F$ transforms zero to zero, use the
    characterization (3) of the zero object in
    Lemma \ref{lemma-preadditive-zero}.
    \end{proof}

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