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