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\\)?

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\\)?