Remark [efr-LPX2]

Since we describe properties of kernels in terms of their corresponding linear operator on function spaces, it would seem natural to take the function spaces as the basic object. Hence we might consider the category C^\infty -algebras with some relaxed notion of maps between them. The tricky part there is to find some reasonable class of maps so that the tensor product (coproduct) of C^\infty -algebras extends to these. (Since it is not the same as the tensor product of \mathbb {R}-algebras, linear maps do not automatically extend to the tensor product). In particular, when considering kernels *_1 \to X where *_1 is a "fat point of order 1"---that is, functions on 1_* have a value at the point and a derivative---it is not clear what sort of continuity condition the derivative operation on C^\infty (X) should satisfy, nor how to define this in a general way for all C^\infty -algebras. It would certainly be of interest to synthetic computational geometry to have such a Markov category, but we leave this for future work.