Operads as double functors

Abstract

It is shown how double categories provide a direct abstract approach to coloured operads; namely, product-preserving normal lax functors from (Pb C)^op (the opposite of the double category of pullback squares in C) to Cat (the double category of functors and profunctors) can be seen as generalized operads, the standard ones arising when C = Set_f. In this context, generalized symmetric monoidal categories are considered, in particular those arising from indexed categories with sums or products.