Dear all, I like very much Grothendieck's characterisation of the category of continuous G-sets for G a profinite group (from SGA 4, covered in a nice paper by Dubuc and de le Vega). Recently I was reading 'Topos Theory' for the first time (usually I have just looked at the Elephant), and I noticed that a Galois category is defined as a Boolean pretopos equipped with a functor to FinSet, satisfying some properties. This appears to me to be stronger than Grothendieck's axioms, and I'm wondering why Johnstone made the change (note that he explicitly credits Grothendieck for the definition). Was there another presentation of the ideas that lead to this change? The Johnstone definition is packaged very tightly (which is nice), but the way Dubuc and de la Vega present this definition, as well as variants appropriate for representable fibre functor and pro-groups, the way the axioms are altered for each case makes it very clear what is going on. Taking a pretopos and showing it is a topos and constructing the profinite group seems slightly less impressive than taking some axioms that don't quite look strong enough to get to a topos, and ending up with the same result. Any thoughts? Best regards, David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]