Definition 4.17.1 (04E6)—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

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
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 August 11, 2021 at 19:50

It seems like the arrows in the base of the diagram should be reversed; and likewise we should have morphisms $x=x_0\to x_1\leftarrow x_2\to x_3\leftarrow\cdots\leftarrow x_{2n}=x'$ in $\mathcal{I}$ , or am I misunderstanding the notation? (I'm looking at p.217 of Mac Lane's "Categories for the Working Mathematician").

Comment #6493 by Johan on August 12, 2021 at 09:29

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 $y$ consider the set $S$ of pairs $(x, b)$ where $b : y \to H(x)$ . Consider the relation $R$ on $S$ given by $(x, b) \sim (x', b') \Leftrightarrow \exists a : x \to x' \mid b' = H(a) \circ b$ . Then what the condition (written either way) guarantees is that $S$ has exactly one equivalence class for the equivalence relation generated by $R$ . OK?

Comment #6495 by Laurent Moret-Bailly on August 12, 2021 at 09:38

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

Comment #6497 by Johan on August 12, 2021 at 10:17

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

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

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