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(infinity,1)-geometric morphism

Idea

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

Definition

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

• an (∞,1)-functor $f_{*}:H\to K$

  (called the direct image of the geometric morphism)

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

  (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 $(\mathrm{\infty },1)$-geometric morphisms between them is (∞,1)Toposes.

Further details

For the moment see the discussion and the further links at geometric morphism

Related concepts

• geometric morphism

• $(\mathrm{\infty },1)$-geometric morphism

References

section 6.3.1 in

• Jacob Lurie, Higher Topos Theory