The group ring of a small category

Dear John Baez, see my abstract in the American Math Society Notices for 1963 Meeting Number 601, paper number 37, page 280. /A longer exposition including applications to network design, cohomology etc. was written then and I can make you a copy if you are interested. Even in 30 years not all the conclusions have been drawn. My colleague S.D. Schack also independently discovered the construction and has used it extensively in deformation theory. An important variant which applies to free categories and others was discovered by Pierre Leroux and used to construct the Moebius function of a category. Greetings, F.W. Lawvere +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++