Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
In the article on essential geometric morphisms it is stated that a local geometric morphism is an essential geometric morphism such that f* has a right adjoint and f! is full and faithful. On the other hand, in the article on local geometric morphisms the definition given is a geometric morphism such that f* has a right adjoint and f! is fully faithful. Are these two definitions equivalent? In particular, does f* need to have a left adjoint f! in a local geometric morphism? Which functor is required to be fully faithful?
No, the existence of f! is not required. If it exists, then there are alternative conditions for f to be local.
Thanks for catching that. The paragraph talked about adding to “essentiality” the condition of “connected surjectivity” and “locality”, but I can see how that didn’t become very clear at all. (Also the notation suddenly changed from f to p etc., quite a mess.) I have edited now.
So that definition is for a local and essential geometric morphism. Thanks Urs.
1 to 4 of 4