A Batanin–Markl operadic category is unary if each map has a unique fibre. We explain how unary operadic categories comprise a full reflective subcategory of simplicial sets, and how discrete decomposition spaces (a.k.a. discrete 2-Segal spaces) are in their turn a reflective full subcategory of unary operadic categories. We also explain (after work of Andrianopoulos and Lack) the relation to monoidales in Span.
This is joint work with Joachim Kock and Mark Weber.