For those who are into the sort of stuff described by Jesus, here's another puzzle.

Start with the adjacency matrix of a graph and take matrix powers in the usual sense. In other words, consider the entries to live in the semiring \$$(\mathbb{N},+,\cdot)\$$. Then it's well-known that the \$$n\$$-th power is the matrix whose \$$(i,j)\$$-entry counts *the number of paths* from the \$$i\$$th vertex to the \$$j\$$th vertex of length \$$n\$$. So is there a way to phrase this in terms of enriched category theory as well?

As a warm-up to this question, let me thus pose:

**Puzzle TF4:** We know that we can keep track of *whether it's possible* to get from one vertex to another using a \$$\mathbf{Bool}\$$-enriched category; we can also keep track of *how many steps it takes* using a \$$\mathbf{Cost}\$$-enriched category. Can we also keep track of *how many paths there are* using an enriched category? In which monoidal poset would we have to enrich? And can we do this in such a way that we count paths of length \$$n\$$ separately for each \$$n\$$?