Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
In articles by Balmer I see “tensor monoidal category” to be explained as a triangulated category equipped with a symmetric monoidal structure such that tensor product with any object “is an exact functor”, but I don’t see where he is specific about what “exact functor” is meant to mean. Maybe I am just not looking in the right article.
Clearly one wants it to mean “preserving exact triangles” in some evident sense. One place where this is made precise is in def. A.2.1 (p.106) of Hovey-Palmieri-Strickland’s “Axiomatic stable homotopy theory” (pdf).
However, these authors do not use the terminology “tensor triangulated” but say “symmetric monoidal compatible with the triangulation”. On the other hand, Balmer cites them as a reference for “tensor triangulated categories” (e.g. page 2 of his “The spectrum of prime ideals in tensor triangulated categories” ).
My question is: may I assume that “tensor triangulated category” is used synonymously with Hovey-Palmieri-Strickland’s “symmetric monoidal comaptible with the triangulation”?
I know that in algebraic K-theory they often refer to exact’ in the sense near to Quillen’s notion of exact category, so I expect your assumption is correct. The term is used in that way in wikipedia.
Thanks for the pointer. So Hovey-Palmieri-Strickland actually require more than is stated on that Wikipedia page.
Is there some authorative textbook definition on the convention used? I checked in Neeman’s book on triangulated categories, but I don’t see this discussed there?
I’m pretty sure it’s just a synonym for ’triangulated functor’, i.e. the usual notion of a functor between triangulated categories.
That must be it. Thanks. I have edited accordingly at tensor triangulated category.
Who is the one to first say “tensor triangulated category”? Is it Balmer 05?
Why not ask Paul Balmer? He is very approachable.
1 to 6 of 6