This course provides an introduction to the theory of Grothendieck toposes from a meta-mathematical point of view. It presents the main classical approaches to the subject (namely, toposes as generalized spaces, toposes as mathematical universes and toposes as classifiers of models of first-order geometric theories) in light of the more recent perspective of toposes as unifying ‘bridges’ relating different mathematical contexts with each other and allowing to study mathematical theories from multiple points of view.
https://www.oliviacaramello.com/Talks/CourseGrothendieckToposesEdinburgh.pdf
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