Claude OCR'd topos anneles

Chapter IV

Spectrum of an n-topos with a ring

    Definition of the spectrum of an n-topos with a ring

The goal of this paragraph is to show that for every n-topos with a ring (X, A), one can universally associate a pair formed of an n-topos with local rings (III 2.3) (X, A) and a morphism of n-topoi with rings

π: (X, A) → (X, A).

Such a pair, as a solution to a universal 2-problem, is defined with uniqueness explicated in (I 2.5); that is to say, ((X, A), π) is defined up to an equivalence in the 2-category Top an((X, A)), the equivalence between two solutions itself being defined by a unique 2-isomorphism. One can further express this by the following theorem:

Theorem (1.4). The 2-functor of inclusion
i : Top an loc → Top an

admits a 2-adjoint to the right (I 1.10).
We will denote
(1.1.1)        Spec : Top an → Top an loc
the 2-functor adjoint to the right of i and
(1.1.2)        π : i o Spec → idTop an

the morphism of 2-functors that expresses the adjunction of i and Spec.

We will now describe the 2-functor Spec. Let (X, A) be an n-topos and E = (S, J), where S is an n-category and J is an n-topology on S, a standard n-site of definition of X (I 1.3.2). One defines X as the n-topos associated to the n-site E = (S, J) following:

(1.2) The category S
The objects of S are the pairs (U, s) formed of an object U of S and an element s of A(U). (Recall (II 1.3) that if k : S → X ...)