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Motivated by potential applications to theoretical computer science, inparticular those areas where the Curry-Howard correspondence plays an importantrole, as well as by the ongoing search in pure mathematics for feasibleapproaches to higher category theory, we undertake a detailed study of a newmathematical abstraction, the generalized category. It is a partially definedmonoid equipped with endomorphism maps defining sources and targets onarbitrary elements, possibly allowing a proximal behavior with respect tocomposition.  We first present a formal introduction to the theory of generalizedcategories. We describe functors, equivalences, natural transformations,adjoints, and limits in the generalized setting. Next we indicate how thetheory of monads extends to generalized categories. Next, we present a variantof the calculus of deductive systems developed by Lambek, and give ageneralization of the Curry-Howard-Lambek theorem giving an equivalence betweenthe category of typed lambda-calculi and the category of cartesian closedcategories and exponential-preserving morphisms that leverages the theory ofgeneralized categories. Next, we develop elementary topos theory in thegeneralized setting of ideal toposes, utilizing the formalism developed for theCurry-Howard-Lambek theorem. In particular, we prove that ideal toposes possessthe same Heyting algebra structure and squares of adjoints that ordinarytoposes do. Finally, we develop generalized sheaves, and show that suchcategories form ideal toposes. We extend Lawvere and Tierney's theorem relating$j$-sheaves and sheaves in the sense of Grothendieck to the generalizedsetting.

Towards Theory and Applications of Generalized Categories to Areas of Type Theory and Categorical Logic