The Stacks project

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, and

  2. $X = \mathop{\mathrm{colim}}\nolimits _ i M_ i$ exists and for any $W$ in $\mathcal{C}$ the map

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

    is bijective.

Proof. Assume (2) holds. 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. The proof of (1) $\Rightarrow $ (2) is omitted. $\square$

Comments (2)

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

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05PY. Beware of the difference between the letter 'O' and the digit '0'.