12.30 Essentially constant systems
In this section we discuss essentially constant systems with values in additive categories.
Lemma 12.30.1. Let $\mathcal{I}$ be a category, let $\mathcal{A}$ be a pre-additive Karoubian category, and let $M : \mathcal{I} \to \mathcal{A}$ be a diagram.
Assume $\mathcal{I}$ is filtered. The following are equivalent
$M$ is essentially constant,
$X = \mathop{\mathrm{colim}}\nolimits M$ exists and there exists a cofinal filtered subcategory $\mathcal{I}' \subset \mathcal{I}$ and for $i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')$ a direct sum decomposition $M_{i'} = X_{i'} \oplus Z_{i'}$ such that $X_{i'}$ maps isomorphically to $X$ and $Z_{i'}$ to zero in $M_{i''}$ for some $i' \to i''$ in $\mathcal{I}'$.
Assume $\mathcal{I}$ is cofiltered. The following are equivalent
$M$ is essentially constant,
$X = \mathop{\mathrm{lim}}\nolimits M$ exists and there exists an initial cofiltered subcategory $\mathcal{I}' \subset \mathcal{I}$ and for $i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')$ a direct sum decomposition $M_{i'} = X_{i'} \oplus Z_{i'}$ such that $X$ maps isomorphically to $X_{i'}$ and $M_{i''} \to Z_{i'}$ is zero for some $i'' \to i'$ in $\mathcal{I}'$.
Proof.
Assume (1)(a), i.e., $\mathcal{I}$ is filtered and $M$ is essentially constant. Let $X = \mathop{\mathrm{colim}}\nolimits M_ i$. Choose $i$ and $X \to M_ i$ as in Categories, Definition 4.22.1. Let $\mathcal{I}'$ be the full subcategory consisting of objects which are the target of a morphism with source $i$. Suppose $i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')$ and choose a morphism $i \to i'$. Then $X \to M_ i \to M_{i'}$ composed with $M_{i'} \to X$ is the identity on $X$. As $\mathcal{A}$ is Karoubian, we find a direct summand decomposition $M_{i'} = X_{i'} \oplus Z_{i'}$, where $Z_{i'} = \mathop{\mathrm{Ker}}(M_{i'} \to X)$ and $X_{i'}$ maps isomorphically to $X$. Pick $i \to k$ and $i' \to k$ such that $M_{i'} \to X \to M_ i \to M_ k$ equals $M_{i'} \to M_ k$ as in Categories, Definition 4.22.1. Then we see that $M_{i'} \to M_ k$ annihilates $Z_{i'}$. Thus (1)(b) holds.
Assume (1)(b), i.e., $\mathcal{I}$ is filtered and we have $\mathcal{I}' \subset \mathcal{I}$ and for $i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')$ a direct sum decomposition $M_{i'} = X_{i'} \oplus Z_{i'}$ as stated in the lemma. To see that $M$ is essentially constant we can replace $\mathcal{I}$ by $\mathcal{I}'$, see Categories, Lemma 4.22.11. Pick any $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and denote $X \to M_ i$ the inverse of the isomorphism $X_ i \to X$ followed by the inclusion map $X_ i \to M_ i$. If $j$ is a second object, then choose $j \to k$ such that $Z_ j \to M_ k$ is zero. Since $\mathcal{I}$ is filtered we may also assume there is a morphism $i \to k$ (after possibly increasing $k$). Then $M_ j \to X \to M_ i \to M_ k$ and $M_ j \to M_ k$ both annihilate $Z_ j$. Thus after postcomposing by a morphism $M_ k \to M_ l$ which annihilates the summand $Z_ k$, we find that $M_ j \to X \to M_ i \to M_ l$ and $M_ j \to M_ l$ are equal, i.e., $M$ is essentially constant.
The proof of (2) is dual.
$\square$
Lemma 12.30.2. Let $\mathcal{I}$ be a category. Let $\mathcal{A}$ be an additive, Karoubian category. Let $F : \mathcal{I} \to \mathcal{A}$ and $G : \mathcal{I} \to \mathcal{A}$ be functors. The following are equivalent
$\mathop{\mathrm{colim}}\nolimits _\mathcal {I} F \oplus G$ exists, and
$\mathop{\mathrm{colim}}\nolimits _\mathcal {I} F$ and $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} G$ exist.
In this case $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} F \oplus G = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} F \oplus \mathop{\mathrm{colim}}\nolimits _\mathcal {I} G$.
Proof.
Assume (1) holds. Set $W = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} F \oplus G$. Note that the projection onto $F$ defines natural transformation $F \oplus G \to F \oplus G$ which is idempotent. Hence we obtain an idempotent endomorphism $W \to W$ by Categories, Lemma 4.14.8. Since $\mathcal{A}$ is Karoubian we get a corresponding direct sum decomposition $W = X \oplus Y$, see Lemma 12.4.2. A straightforward argument (omitted) shows that $X = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} F$ and $Y = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} G$. Thus (2) holds. We omit the proof that (2) implies (1).
$\square$
Lemma 12.30.3. Let $\mathcal{I}$ be a filtered category. Let $\mathcal{A}$ be an additive, Karoubian category. Let $F : \mathcal{I} \to \mathcal{A}$ and $G : \mathcal{I} \to \mathcal{A}$ be functors. The following are equivalent
$F \oplus G : \mathcal{I} \to \mathcal{A}$ is essentially constant, and
$F$ and $G$ are essentially constant.
Proof.
Assume (1) holds. In particular $W = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} F \oplus G$ exists and hence by Lemma 12.30.2 we have $W = X \oplus Y$ with $X = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} F$ and $Y = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} G$. A straightforward argument (omitted) using for example the characterization of Categories, Lemma 4.22.9 shows that $F$ is essentially constant with value $X$ and $G$ is essentially constant with value $Y$. Thus (2) holds. The proof that (2) implies (1) is omitted.
$\square$
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