Abstract. Every directed graph can be factored into primes over the direct (or tensor) product. For some digraphs this factorization is unique: any two prime factorings differ only in the ordering of the factors. Other digraphs have many different prime factorizations.

We will examine some cases in which factorization is—or is not—unique. I will describe a new result which greatly enlarges the class of digraphs which are known to have unique prime factorization. I will paint a rough, global picture of the proof. The tone is visual. Nothing will be proved, but many pictures will be drawn.