Dear Michael, I have a trivial comment to the first paragraph of your message. You ask: "...Could you have two (semi)ring structures on the same set with the same associative multiplication?" Take any (semi)ring R with a multiplicative automorphism f that is not an additive automorphism, and transport the structure along f. For instance, both in the semiring of natural numbers and in the ring of integers, any non-identity permutation of (positive) prime numbers determines such an f. An example for students: take, say, f(2) = 3, f(3) = 2, and f(p) = p for all primes p different from 2 and 3. Then, denoting the new addition by *, we calculate (usung the fact that f coincides with its inverse) 1*1 = f(f(1)+f(1)) = f(2+2) = f(2x2) = f(2)xf(2) = 3x3 = 9. Best regards, George ----- Original Message ----- From: "Michael Barr" <mbarr@math.mcgill.ca> To: "Categories list" <categories@mta.ca> Sent: Thursday, August 10, 2006 10:14 PM Subject: categories: Linear--structure or property?
Bill Lawvere uses "linear" for a category enriched over commutative semigroups. Obviously, if the category has finite products, this is a property. What about in the absence of finite products (or sums)? Could you have two (semi)ring structures on the same set with the same associative multiplication?
Robin Houston's startling (to me, anyway) proof that a compact *-autonomous category with finite products is linear starts by proving that 0 = 1. Suppose the category has only binary products? Well, I have an example of one that is not linear: Lawvere's category that is the ordered set of real numbers has a compact *-autonomous structure. Tensor is + and internal hom is -. Product is inf and sum is sup, but there are no initial or terminal objects and the category is not linear.