Dear Marco, This has been considered by Brian Day. He spoke about it in a talk *-autonomous convolution in the Australian Category Seminar on 5 March 1999, You can also transform this via the log/exponential functions to an additive tensor product on the extended (positive and negative) reals. Regards, Steve Lack. -----Original Message----- From: cat-dist@mta.ca on behalf of Marco Grandis Sent: Tue 14/03/2006 12:45 AM To: categories@mta.ca Subject: categories: An autonomous category The Lawvere category of extended positive real numbers has also an autonomous structure, with a multiplicative tensor product (instead of the original additive one). Has this been considered somewhere? To be more explicit: The well-known article of Lawvere on "Metric spaces..." (Rend. Milano 1974, republished in TAC Reprints n. 1) introduced the category of extended positive real numbers, from 0 to oo (infinity included), with arrows x \geq y, equipped with a strict symmetric monoidal closed structure: the tensor product is the sum, the internal hom is truncated difference (with oo - oo = 0). Now, the same category can be equipped with a multiplicative tensor product, x.y. Provided we define 0.oo = oo (so that tensoring by any element preserves the initial object oo), this is again a strict symmetric monoidal closed structure, with hom(y, z) = z/y. Now, the 'undetermined forms' 0/0 and oo/oo are defined to be 0. The new multiplicative structure is even *-autonomous, with involution x* = 1/x (and 'nearly' compact). (Note that this choice of values of the undetermined forms comes from privileging the direction x \geq y, which is necessary if we want to view metric spaces, normed categories etc. as enriched categories). Marco Grandis