The Stacks project

4.20 Cofiltered limits

Limits are easier to compute or describe when they are over a cofiltered diagram. Here is the definition.

Definition 4.20.1. We say that a diagram $M : \mathcal{I} \to \mathcal{C}$ is codirected or cofiltered if the following conditions hold:

  1. the category $\mathcal{I}$ has at least one object,

  2. for every pair of objects $x, y$ of $\mathcal{I}$ there exist an object $z$ and morphisms $z \to x$, $z \to y$, and

  3. for every pair of objects $x, y$ of $\mathcal{I}$ and every pair of morphisms $a, b : x \to y$ of $\mathcal{I}$ there exists a morphism $c : w \to x$ of $\mathcal{I}$ such that $M(a \circ c) = M(b \circ c)$ as morphisms in $\mathcal{C}$.

We say that an index category $\mathcal{I}$ is codirected, or cofiltered if $\text{id} : \mathcal{I} \to \mathcal{I}$ is cofiltered (in other words you erase the $M$ in part (3) above).

We observe that any diagram with cofiltered index category is cofiltered, and this is how this situation usually occurs.

As an example of why cofiltered limits of sets are “easier” than general ones, we mention the fact that a cofiltered diagram of finite nonempty sets has nonempty limit (Lemma 4.21.7). This result does not hold for a general limit of finite nonempty sets.

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