Definition 4.26.1 (0FWX)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.26: Categorically compact objects
Definition 4.26.1 (cite)

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Definition 4.26.1. Let $\mathcal{C}$ be a big ¹ category. An object $X$ of $\mathcal{C}$ is called a categorically compact if we have

$\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, \mathop{\mathrm{colim}}\nolimits _ i M_ i) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, M_ i)$

for every filtered diagram $M : \mathcal{I} \to \mathcal{C}$ such that $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ exists.

[1] See Remark 4.2.2.

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