Let be a topos, regarded as a base topos.
Definition 1.1. An -indexed topos is an -indexed category such that
for each object the fiber is a topos;
for each morphism in the corresponding transition functor is a logical morphism.
An -indexed geometric morphism is an -indexed adjunction between -indexed toposes, such that is left exact.
This yields a 2-category of -indexed toposes.
This appears at (Johnstone, p. 369).
Proposition 3.1. Write Topos for the slice 2-category of toposes over . This is a full sub-2-category of the 2-category of-indexed toposes:
This appears as (Johnstone, prop. 3.1.3).
Section B3.1 of
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