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Technical Report HW-MACS-TR-0077


TitleAn Observation on Support and Freshness in Nominal Sets
AuthorsMurdoch J. Gabbay
Date2010-05-20
Abstract

There are two natural ways of excluding an atom a in nominal techniques: we can either consider the sets X such that a is fresh for X, or we can consider the sets X such that a is fresh for every x ∈ X.

The statements of 'being fresh for all the elements of that set' and 'being fresh for a set' are not the same: it is not the case that ∀x ∈ X.a#x if and only if a#X. Both notions encode natural notions of 'fresh a'. In this paper, it is proved that these notions lead naturally to two categories that are isomorphic, so that in a suitable generalised sense they are the same.

The result is mathematically attractive and has an interesting reading: it is equivalent to add a fresh atom to the underlying universe, and to add a symbol to the meta-language referencing a fresh atom. Or to put it slightly differently: we prove the intuitively appealing but non-obvious fact that a fresh atom in the object-level is categorically isomorphic to a fresh atom in the meta-level.

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