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Adjointness in Foundations,
Dialectica 23 (1969), 281-296,
Reprints in Theory and Applications of Categories, No. 16 (2006), pp 1-16 (revised 2006-10-30)
(TAC)
on foundations of mathematics in categorical logic and the role of adjoints in the formalization of quantifiers (base change). The main point is that the logical operations on propositions
existential quantifier$\dashv$ context extension $\dashv$ universal quantifier
constitute an adjoint triple. In type theory this is lifted from operations on propositions to operations on types, where it becomes
dependent sum$\dashv$ context extension $\dashv$ dependent product.
In topos theory and geometry this adjoint triple is often known as base change.
William Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14. (pdf)
Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1981)
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