Gabbay and Pitts observed that the Fraenkel–Mostowski model of set-theory supports useful notions of “name-abstraction” and “fresh-name”. In order to understand their work in a more general setting we introduce the notions of И-units and И-relations in a regular category D. A И-relation is given by a functor A # (-):D→D and we show that in the case that D is a topos then A # (-) has a right adjoint [A](-) that can be thought of as an object of abstractions. We also explore the existence of a right adjoint to [A](-) and relate it to the “name swapping” operations considered as fundamental by Gabbay and Pitts. We present many examples of categories where this notions occur and we relate the results here with Pitts' Nominal Logic.
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