Constructing Variants of the Category of Partial Equivalence Relations Yuning Feng Abstract: The thesis aims at providing a categorical model of modified realizability, understood as an interpretation of extensional Heyting arithmetic in all finite types. Two variants of the category of partial equivalence relations are studied, namely, PER~ and PER~~. The former is shown to be not regular, thus not suitable for our aim. The latter is regular and has enough projectives; it is a good candidate. Both variants are results of alternating iteration of the cover construction and the co-cover construction. The cover construction resembles but differs from the regular completion.