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We study the path category of an inverse semigroup admitting unique maximal idempotents and give an abstract characterization of the inverse semigroups arising from zigzag maps on a left cancellative category. As applications we show that every inverse semigroup is Morita equivalent to an inverse semigroup of zigzag maps and hence the class of Cuntz–Krieger $$C^*$$-algebras of singly aligned categories include the tight $$C^*$$-algebras of all countable inverse semigroups, up to Morita equivalence.

On zigzag maps and the path category of an inverse semigroup