Towards Theory and Applications of Generalized Categories to Areas of Type Theory and Categorical Logic
Abstract
Motivated by potential applications to theoretical computer science, in particular those areas where the CurryHoward correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to higher category theory, we undertake a detailed study of a new mathematical abstraction, the generalized category. It is a partially defined monoid equipped with endomorphism maps defining sources and targets on arbitrary elements, possibly allowing a proximal behavior with respect to composition. We first present a formal introduction to the theory of generalized categories. We describe functors, equivalences, natural transformations, adjoints, and limits in the generalized setting. Next we indicate how the theory of monads extends to generalized categories. Next, we present a variant of the calculus of deductive systems developed by Lambek, and give a generalization of the CurryHowardLambek theorem giving an equivalence between the category of typed lambdacalculi and the category of cartesian closed categories and exponentialpreserving morphisms that leverages the theory of generalized categories. Next, we develop elementary topos theory in the generalized setting of ideal toposes, utilizing the formalism developed for the CurryHowardLambek theorem. In particular, we prove that ideal toposes possess the same Heyting algebra structure and squares of adjoints that ordinary toposes do. Finally, we develop generalized sheaves, and show that such categories form ideal toposes. We extend Lawvere and Tierney's theorem relating $j$sheaves and sheaves in the sense of Grothendieck to the generalized setting.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1803.00180
 Bibcode:
 2018arXiv180300180S
 Keywords:

 Mathematics  Category Theory;
 18C50;
 F.3.2;
 F.1.1;
 F.4.3
 EPrint:
 Author's PhD thesis. 131 pages