Lemma 4.22.9. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with filtered index category $\mathcal{I}$. The following are equivalent

1. $M$ is an essentially constant ind-object,

2. there exists a cocone $(X, \{ M_ i \to X\} _ i)$ such that for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X)$ is bijective,

3. $X = \mathop{\mathrm{colim}}\nolimits _ i M_ i$ exists and for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X)$ is bijective, and

4. there exists an $i$ in $\mathcal{I}$ and a morphism $X \to M_ i$ such that for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \to \mathop{\mathrm{colim}}\nolimits _{j \geq i} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ j)$ is bijective.

In cases (2), (3), and (4) the value of the essentially constant system is $X$.

Proof. It is clear that (3) implies (2). Assume (2). Then $\text{id}_ X \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, X)$ comes from a morphism $X \to M_ i$ for some $i$, i.e., $X \to M_ i \to X$ is the identity. Then both maps

$\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X)$

are bijective for all $W$ where the first one is induced by the morphism $X \to M_ i$ we found above, and the composition is the identity. This means that the composition

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W, M_ i)$

is the identity too. Setting $W = M_ j$ and starting with $\text{id}_{M_ j}$ in the colimit, we see that $M_ j \to X \to M_ i \to M_ k$ is equal to $M_ j \to M_ k$ for some $k$ large enough. This proves (1) holds.

Assume (4). Let $k$ be an object of $\mathcal{I}$. Setting $W = M_ k$ we deduce there exists a unique morphism $M_ k \to X$ such that there exists a $j$ and morphisms $k \to j$ and $i \to j$ in $\mathcal{I}$ such that $M_ k \to X \to M_ i \to M_ j$ is equal to $M_ k \to M_ j$. The uniqueness guarantees that we obtain a cocone $(X, \{ M_ k \to X\} )$. In this way we see that (4) implies (2); some details omitted.

We omit the proof that (1) implies the other conditions. $\square$

Comment #8282 by Et on

It seems aa though in (2) => (1) you don't actually use the colimit aasumption, right? That is to say it is enough to assume we have a cocone X for which the map in (2) is bijective.

The reason I'm asking is that in lemma 13.14.6, you spend the last part of the proof proving (in a purely formal manner) the colimit assumption on the object denoted C in that lemma, and it seems as though that may be unecessary.

Comment #8915 by on

OK, I upgraded this lemma to deal with the use case you mentioned and I fixed the proof in derived categories as you suggested. It still isn't optimal perhaps because it could be formulated on the level of functors, but we can add that if we ever need it. Thanks! Changes are here.

There are also:

• 10 comment(s) on Section 4.22: Essentially constant systems

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).