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Relative topos theory via stacks
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abstract
We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site $({\mathcal{C}}, J)$ and that of ${\mathcal{C}}$-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a 'petit' topos associated with a 'gros' topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.
Forward citations
Cited by 5 Pith papers
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Generation of Grothendieck topologies, provability and operations on subtoposes
A systematic study of subtoposes yields explicit formulas for generating Grothendieck topologies and new preservation theorems for pullbacks of subtoposes.
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Sheaves on a bicategory
The paper proves a general adjunction between complete B-categories and 2-presheaves on Map(B), recovering sheaf theory over quantaloids and enrichment over monoidal categories as special cases.
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Local fibrations and morphisms of relative toposes
Local fibrations, defined using Grothendieck topologies, characterize the site-level functors that give morphisms of relative toposes and support a weak indexed Diaconescu theorem.
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Stackification via adjunction
A 2-adjunction between indexed categories and 2-toposes is claimed to compute stackification as the composite Γ∘Λ.
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Sites and Grothendieck toposes: an introduction
An expository book excerpt: it restates standard category theory and sheaf background as preparation for a planned introduction to Grothendieck toposes, with no new mathematical results.
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