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We introduce the concept of TRAP (Traces and Permutations), which can roughlybe viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs areequipped with a horizontal concatenation and partial trace maps. Continuousmorphisms on an infinite dimensional topological space and smooth kernels(resp. smoothing operators) on a closed manifold form a TRAP but not a wheeledPROP. We build the free objects in the category of TRAPs as TRAPs of graphs andshow that a TRAP can be completed to a unitary TRAP (or wheeled PROP). Wefurther show that it can be equipped with a vertical concatenation, which onthe TRAP of linear homomorphisms of a vector space, amounts to the usualcomposition. The vertical concatenation in the TRAP of smooth kernels givesrise to generalised convolutions. Graphs whose vertices are decorated by smoothkernels (resp. smoothing operators) on a closed manifold form a TRAP. Fromtheir universal properties we build smooth amplitudes associated with thegraph.

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