Abstract and References Vickers

Abstract: Grothendieck toposes are generalized spaces. Specifically, a classifying topos S[T] can be thought of as the space of models of the geometric theory T, but it relies on having a base elementary topos S. How do you choose S, especially if you don't like Set?
 I shall look at a 2-category GTop of Grothendieck toposes, fibred over a 2-category of elementary toposes, and relate it to my recent work on sketches for arithmetic universes. They do not depend on any particular base, only on the existence of an nno, and classifying toposes can be constructed in a fibred way that allows for variation of the base.

 

References

S. Vickers "Arithmetic universes and classifying toposes"
online at http://www.cs.bham.ac.uk/~sjv/papersfull.php#AUClTop

 Mitchell Buckley, Fibred 2-categories and bicategories, Journal
of Pure and Applied Algebra 218 (2014), 1034{1074.
 
Nick Gurski, Coherence in three-dimensional category theory, Cambridge
Tracts in Mathematics, no. 201, Cambridge University Press, 2013.
 
P.T. Johnstone, Sketches of an elephant: A topos theory compendium, vol.
1, Oxford Logic Guides, no. 44, Oxford University Press, 2002.
   -- Section B4.2 sets out the approach we use for classifying toposes.

Vickers, Topical categories of domains, Mathematical Structures in
Computer Science 9 (1999), 569--616.
   -- The Introduction sets out explicitly the of programme using
classifying toposes as spaces, while being vague about base toposes. The
Conclusion speculates about using arithmetic universes.

Vickers, Sketches for arithmetic universes, See arXiv:1608.01559, 2016.
   -- presents the underlying formal system for arithmetic universes.

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