From non-unitary wheeled PROPs to smooth amplitudes and generalised convolutions

Abstract We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products Permutations) without unit. TRAPs are equipped with horizontal concatenation partial trace maps. Continuous morphisms on an infinite-dimensional topological space smooth kernels (respectively, smoothing operators) closed manifold form but not PROP. build free objects in category graphs show that completed to unitary (or PROP). further it vertical concatenation, linear homomorphisms vector space, amounts usual composition. The gives rise generalised convolutions. Graphs whose vertices decorated by TRAP. From their universal properties we amplitudes associated graph.