nLab cotopological localization

Idea

Let $\mathcal{X}$ be an $\infty$-topos and let $\mathcal{Y} \subseteq \mathcal{X}$ be an accessible left exact localization of $\mathcal{X}$. We say that $\mathcal{Y}$ is a cotopological localization of $\mathcal{X}$ if the left adjoint $L : \mathcal{X} \to \mathcal{Y}$ to the inclusion of $\mathcal{Y}$ in $\mathcal{X}$ satisfies either of the following equivalent conditions:

1. For every monomorphism $u$ in $\mathcal{X}$, if $L^{\ast}u$ is an equivalence in $\mathcal{Y}$, then $u$ is an equivalence in $\mathcal{X}$.

2. For every morphism $u \in \mathcal{X}$, if $L^{\ast}u$ is an equivalence in $\mathcal{Y}$, then $f$ is ∞-connective.

The hypercompletion $\mathcal{X}^{\wedge}$ of an ∞-topos $\mathcal{X}$ can be characterized as the maximal cotopological localization of $\mathcal{X}$ (that is, the cotopological localization which is obtained by inverting as many morphisms as possible).

Every localization can be obtained by combining topological and cotopological localizations.

Examples

In the context of the Goodwillie calculus, the whole Goodwillie-Taylor tower consists of cotopological localizations of the classifying ∞-topos for pointed ∞-connective objects.

Related concepts

• Whitehead theorem in $(\infty,1)$-toposes

References

• 6.5.2.17 of Higher Topos Theory