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Simon Henry
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This is a very broad question, we have a huge numbers of such characterization.

But part C of "Sketches of an elephant" contains most of those I know. Basically all the notion introduced have such a "site characterization", i.e. a properties of a topos or a morphism is characterized by the existence of a Site description having some properties.

And I would like to add that Part C is in my opinion the better written and easiest to read part of the elephant.

From memory, you can find their at the very least conditions for:

  • Atomic toposes, as sheaves for a topology where all non-empty sieve are covering (called an atomic topology).
  • Proper geometric morphism from a 'finite subcovering property'
  • Coherent toposes by combining the fact that every covering is finitely generated with existence of finite limits in the site.

and a lots of other examples (open geometric morphisms, locally connected morphisms and so one...) but I do not remember all of them.

Simon Henry
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