## Calibrated toposes.(English)Zbl 1266.18007

This rather technical paper focuses on the description of the relationship between the slice categories $$\varepsilon/A$$, $$A$$ being an object of the elementary topos $$\varepsilon$$, and the corresponding topos of sheaves on $$A$$, hereafter called the petit topos on $$A$$. The idea is to consider $$\varepsilon$$, the gros topos, as a space of spaces, the petit toposes. Essentially, a petit topos on $$A$$ is the full subcategory of $$\varepsilon/A$$ whose objects we may think as fibrewise discrete morphisms with codomain $$A$$; a calibration is an attempt to axiomatise this notion.
Although many fundamental properties of calibrations are derived in the article, together with a few examples, the work is not yet conclusive: not all the objects of a gros topos are sufficiently spacelike to be represented up to homotopy equivalence by their petit toposes. An analysis on how to characterize those objects which have this property is left to a subsequent work.
In conclusion, this paper describes a very technical result which is still partial. Thus, the work is of interest mainly for specialists in the theory of toposes, both because of the required technical level, and because of the intermediate nature of the achievements.

### MSC:

 18B25 Topoi

### Keywords:

topos; calibration; locally connected morphism; local morphism
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