Tries to reframe relational data in terms of functions over finite domain.
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$f = g \circ h$ . -
$g(x) = f(x, y)$ (this could just be a combination of$i(x) = (x, y)$ which can be seen as combing a morphism$j(1) = y$ and then using the isomorphim$x \leftrightarrow (x, 1)$ . -
$g(y) = f^{-1}(y)$ -
$g(x) = f(x, -)$ (function valued) -
$g(x) = \sum_y f(x, y) = \mathbb{E}(f(x, y) | ((x, y) \mapsto x))$ . -
$f(x, y) = g(x) \times g(y)$ -
$dup(x) = (x, x)$ . -
$g(a) = \mathbb{E}(f(b) | \phi)$ where$\phi(b) = a$ . So what we really care about is this sort of labelled$\sigma$ -algebra$\phi^{-1}$ . -
$h(i, k) = \mathbb{E}_j(\mathrm{prod} \circ (f \times g) \circ \mathrm{match}_{j=k}^{-1}(i, j, k, true))$ (matrix multiplication of$f(i, j)$ and$g(k, l)$ ) and$\mathrm{match}((i, j), (k, l)) = (i, j, l, j==k)$ -
$2^X \Leftrightarrow S \hookrightarrow X$ .