On 15/01/2012, at 9:22 AM, Fred E.J. Linton wrote:
On Sat, 14 Jan 2012 09:51:24 AM EST, George Janelidze <janelg@telkomsa.net
wrote:
... What about biproducts? ...
... [snip] ...
(b) if a category admits infinite biproducts, then it is indiscrete (=every object in it is zero) ...
Let me refute that by channeling the voice of Dana May Latch, the late Alex Heller's student-of-yore, who stumped me once, decades ago, by asking:
Another counterexample is the category Spn(E) of spans in (for example) a Grothendieck topos E. The objects are those of E and the morphisms are isomorphism classes of spans in E. This occurs in studying Mackey functors thanks to Harald Lindner. By not taking the isomorphism classes and looking at the bicategory Spn(E) we are presented with another terminological problem: the prefix "bi" in "biproduct" could mean "bicategorical product". For me this is much worse than conflicts with "linear logic", "linear order" and "k-linear category". It is sad that we cannot work out good names for even the most basic concepts. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]