nLab (infinity,1)-geometric morphism

Idea

By categorification of the notion of geometric morphism, an $(\infty,1)$-geometric morphism is a pair of adjoint (∞,1)-functors between (∞,1)-toposes where the leftadjoint is left-exact.

Definition

For $\mathbf{H}$ and $\mathbf{K}$ two (∞,1)-toposes, a $(\infty,1)$-geometric morphism $f : \mathbf{H} \to \mathbf{K}$ is

• an (∞,1)-functor $f_* : \mathbf{H} \to \mathbf{K}$

  (called the direct image of the geometric morphism)

• which has a left adjoint (∞,1)-functor $\mathbf{H} \leftarrow \mathbf{K} : f^*$;

  (called the inverse image of the geometric morphism)

• such that $f^*$ is a left exact (∞,1)-functor.

The (non-full) sub-(∞,1)-category of (∞,1)Cat on (∞,1)-toposes and $(\infty,1)$-geometric morphisms between them is (∞,1)Toposes.

Examples

• terminal geometric morphism

• locally $\infty$-connected $\infty$-topos

• cohesive $\infty$-topos

Related concepts

• geometric morphism

• terminal geometric morphism

References

section 6.3.1 in

• Jacob Lurie, Higher Topos Theory