It's a structure. Consider the following category C. Two objects x and y, with hom-categories C(x,x)=C(y,y)={0,1} C(y,x)={0} C(x,y)=M with composition defined so that each 1 is an identity morphism and each 0 a zero morphism, and with M an arbitrary set. Any commutative monoid structure on M makes C into a linear category. Steve. -----Original Message----- From: cat-dist@mta.ca on behalf of Michael Barr Sent: Fri 8/11/2006 6:14 AM To: Categories list 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.