In algebraic geometry, étale topos of scheme $X$ is sheaves on étale site. Étale cohomology $H^i_{\text{ét}}(X, \mathcal{F})$ computed in this topos provides arithmetic analog of singular cohomology.

Grothendieck's vision: study schemes through their toposes.

The Cahiers topos and smooth toposes provide models where:

• Infinitesimals exist as objects
• All functions are smooth
• Classical logic fails, enabling nilpotent infinitesimals

This gives synthetic approach to differential geometry without explicit limits.

Forcing in set theory can be formulated via Boolean-valued models: presheaf topos on Boolean algebra. Generic extensions correspond to geometric morphisms.

This categorical perspective clarifies forcing and provides intuitionistic generalization.

Toposes provide semantics for:

• Intuitionistic type theory
• Higher-order constructive logic
• Linear logic (via linear toposes)

Internal language of topos is powerful programming language with dependent types.

Recent work uses toposes of presheaves on quantum contexts to provide topos-theoretic semantics for quantum mechanics. The subobject classifier encodes quantum logic.

This is active research area connecting quantum physics and topos theory.

Higher toposes (∞-toposes) provide semantics for homotopy type theory (HoTT). Objects are homotopy types, morphisms are homotopy classes of maps.

This unifies homotopy theory, type theory, and logic in topos-theoretic framework.