Kerodon

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Definition 6.3.3.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $W$ is localizing if the following conditions are satisfied:

$(1)$

Every isomorphism of $\operatorname{\mathcal{C}}$ is contained in $W$.

$(2)$

The collection of morphisms $W$ satisfies the two-out-of-three property. That is, for every $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]

of $\operatorname{\mathcal{C}}$, if any two of the morphisms $u$, $v$, and $w$ belong to $W$, then so does the third.

$(3)$

For every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $w: Y \rightarrow Z$ which belongs to $W$, where the object $Z$ is $W$-local.

We say that $W$ is colocalizing if it satisfies conditions $(1)$ and $(2)$ together with the following dual version of $(3)$:

$(3')$

For every object $Y \in \operatorname{\mathcal{C}}$, there exists a morphism $w: X \rightarrow Y$ which belongs to $W$, where $X$ is $W$-colocal.