Funny identity matrix

Posted in Math by Mike Stay on 2013 June 11

Given any category C with finite products and coproducts such that the products distribute over the coproducts, you can make a compact closed bicategory Mat(C) of (natural numbers, matrices of elements of Ob(C), matrices of elements of Mor(C)) and define matrix composition by \displaystyle (M \circ N)(i,k) = \sum_j M(i,j) \times N(j,k).

Take the partial order whose objects are nonnegative integers and whose morphisms m \to n mean m | n; the product is gcd and the coproduct is lcm. In this category, the terminal object is 0 and the initial object is 1, so the identity matrix looks like

\left( \begin{array}{cccc}0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \ldots & 0 \end{array} \right)

and matrix multiplication is \displaystyle (M \circ N)(i,k) = \mathop{\mbox{lcm}}_j \mbox{gcd}(M(i,j), N(j,k)).