We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given `geometric' invariant as a property of its sites of definition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to specific invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equivalent to a presheaf topos), obtaining explicit site characterizations for them.
Keywords: Grothendieck topos, site characterizations, geometric logic
2010 MSC: 03G30, 18C10, 18B25
Theory and Applications of Categories, Vol. 26, 2012, No. 25, pp 710-728.
Published 2012-11-23.
http://www.tac.mta.ca/tac/volumes/26/25/26-25.dvi
http://www.tac.mta.ca/tac/volumes/26/25/26-25.ps
http://www.tac.mta.ca/tac/volumes/26/25/26-25.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/25/26-25.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/25/26-25.ps
Revised 2013-02-20. Original version at
http://www.tac.mta.ca/tac/volumes/26/25/26-25a.pdf