The Stacks project

Definition 4.17.1. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say $\mathcal{I}$ is cofinal in $\mathcal{J}$ or that $H$ is cofinal if

  1. for all $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$ there exist an $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a morphism $y \to H(x)$, and

  2. given $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$, $x, x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and morphisms $y \to H(x)$ and $y \to H(x')$ there exist a sequence of morphisms

    \[ x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x' \]

    in $\mathcal{I}$ and morphisms $y \to H(x_ i)$ in $\mathcal{J}$ such that the diagrams

    \[ \xymatrix{ & y \ar[ld] \ar[d] \ar[rd] \\ H(x_{2k}) & H(x_{2k + 1}) \ar[l] \ar[r] & H(x_{2k + 2}) } \]

    commute for $k = 0, \ldots , n - 1$.

Comments (4)

Comment #6492 by George on

It seems like the arrows in the base of the diagram should be reversed; and likewise we should have morphisms in , or am I misunderstanding the notation? (I'm looking at p.217 of Mac Lane's "Categories for the Working Mathematician").

Comment #6493 by on

Although psychologically it might be better to change it the way you say (and I might do so the next time I go through the comments), mathematically speaking there is no difference. Namely, given consider the set of pairs where . Consider the relation on given by . Then what the condition (written either way) guarantees is that has exactly one equivalence class for the equivalence relation generated by . OK?

Comment #6495 by Laurent Moret-Bailly on

@#6492, #6493: Concretely, you can switch between the two versions by taking the first and last arrows to be identities.

Comment #6497 by on

@#6492 Yes, indeed! Should have said so.

There are also:

  • 3 comment(s) on Section 4.17: Cofinal and initial categories

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