1. "forall" goes with "weakening" because it is adjoint to it on the right. 2. "exists" goes with "weakening" because it is adjoint to it on the left.
What's the definition of "weakening"? I've not seen this word used formally.
-> A to the subobject M x B >-> A x B, i.e., it is pullback along the
Weakening is the map Sub(A) -> Sub(A x B) which takes a subobject M projection pi1 : A x B -> A. The existential quantifier over B is the left adjoint to weakening, where Sub(A) and Sub(A x B) are viewed as posetal categories. The universal quantifier is the right adjoint. Working out the details in Set is instructive. As far as I know the terminology comes from logic. The rule of weakening says that a logical derivation may be "wakened" with an additional (but unneeded) hypothesis: Gamma |- A ---------------------- Gamma, H |- A With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]