Lemma 4.19.6 (09WS)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.19: Filtered colimits
Lemma 4.19.6 (cite)

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Lemma 4.19.6. Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every solid diagram

$\xymatrix{ x \ar[d] \ar[r] & y \ar@{..>}[d] \\ z \ar@{..>}[r] & w }$

in $\mathcal{I}$ there exist an object $w$ and dotted arrows making the diagram commute. Then $\mathcal{I}$ is either empty or a nonempty disjoint union of connected categories having the same property.

Proof. If $\mathcal{I}$ is the empty category, then the lemma is true. Otherwise, we define a relation on objects of $\mathcal{I}$ by saying that $x \sim y$ if there exist a $z$ and morphisms $x \to z$ and $y \to z$ . This is an equivalence relation by the assumption of the lemma. Hence $\mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ is a disjoint union of equivalence classes. Let $\mathcal{I}_ j$ be the full subcategories corresponding to these equivalence classes. Then $\mathcal{I} = \coprod \mathcal{I}_ j$ with $\mathcal{I}_ j$ nonempty as desired. $\square$

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