Krzysztof Kapulkin

Univalent Categories and the Rezk Completion

When formalizing category theory in traditional set-theoretic foundations, a significant discrepancy between the foundational notion of “sameness” – equality – and its practical use arises: most category-theoretic concepts are invariant under weaker notions of sameness than equality, namely isomorphism in a category or equivalence of categories. We show that this discrepancy can be avoided when formalizing category theory in Univalent Foundations.

Homotopy limits in type theory

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

Quasicategories of Frames of Cofibration Categories

We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.

Homotopy-theoretic models of type theory

We introduce the notion of a logical model category, which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it while also having a purely intensional interpretation of the identity types. On the other hand, those conditions are easy to check and provide a wide class of models that are examined in the paper.

Locally cartesian closed quasi-categories from type theory

We prove that the quasi-categories arising from models of Martin-Lof type theory via simplicial localization are locally cartesian closed.

Threshold Properties of Prime Power Subgroups with Application to Secure Integer Comparisons

We present a semantically secure somewhat homomorphic public-key cryptosystem working in sub-groups of \(\mathbb {Z}_{n}^{*}\) of prime power order. Our scheme introduces a novel threshold homomorphic property, which we use to build a two-party protocol for secure integer comparison. In contrast to related work which encrypts and acts on each bit of the input separately, our protocol compares multiple input bits simultaneously within a single ciphertext. Compared to the related protocol of Damgard et al. [9, 10] we present results showing this approach to be both several times faster in computation and lower in communication complexity.