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nLab > Latest Changes: tensor triangulated category

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2016
    • (edited Jun 29th 2016)

    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”?

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeJun 29th 2016

    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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2016

    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?

  4. I’m pretty sure it’s just a synonym for ’triangulated functor’, i.e. the usual notion of a functor between triangulated categories.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2016

    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?

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeJun 30th 2016

    Why not ask Paul Balmer? He is very approachable.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2020

    added pointer to

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 24th 2021

    Added a reference to

    • {#May01} Peter May, The Additivity of Traces in Triangulated Categories, Advances in Mathematics, Volume 163, Issue 1, 2001, Pages 34-73, doi

    which includes three more axioms.

    diff, v12, current

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