  Positive data languages are languages over an infinite alphabet closed under
possibly non-injective renamings of data values. Informally, they model
properties of data words expressible by assertions about equality, but not
inequality, of data values occurring in the word. We investigate the class of
positive data languages recognizable by nondeterministic orbit-finite nominal
automata, an abstract form of register automata introduced by Boja\'nczyk,
Klin, and Lasota. As our main contribution we provide a number of equivalent
characterizations of that class in terms of positive register automata, monadic
second-order logic with positive equality tests, and finitely presentable
nondeterministic automata in the categories of nominal renaming sets and of
presheaves over finite sets.
