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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 8th 2011

    I meant to ask this some time ago. Under the examples section of Tall-Wraith monoid, I don’t understand why in the case of commutative unital rings the notion of Tall-Wraith monoid is identified with biring. It seems to me that this example is just like the next example, except that we’re working over \mathbb{Z} instead of a field kk, and therefore the notions of Tall-Wraith monoid should be pretty much the same in these examples. In other words, in the first example it ought to be a plethory over the base ring \mathbb{Z}, not a biring – right?

    • CommentRowNumber2.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 8th 2011

    You’re right. A biring is a co-ring object in rings (in both cases, ring = commutative unital ring). In Tall and Wraith’s paper, what we call a Tall-Wraith monoid is there a biring triple.

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 8th 2011

    Corrected.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 9th 2011

    Thank you!

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 28th 2014

    In view of recent discussions which touched on Λ\Lambda-rings, I felt an urge to add a few more theoretical details to Tall-Wraith monoid, which I have now done.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 29th 2014

    Added still more details. At some point I need to check whether I got some things switched around, i.e., whether what I call RSR \odot S is what others call SRS \odot R, and the like.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2014
    • (edited Jul 29th 2014)

    Thanks, Todd!

    I am watching “from a distance”, being on the road and quasi-offline and distracted. But I really appreciate what you are writing here, very valuable. Thanks.

    In free minutes I keep trying to see what to make of the various observations that led to the discussion on Lambda-rings lately, which in turn led you to your latest additions here.

    Maybe the following might help me. Consider the endofunctor on Sh(CRing^op) which sends representables SpecR to the disjoint union over all prime ideals of the formal completions of R at these ideals, and which is defined by left Kan extension from this to all of Sh(CRing^op). Does this endofunctor have a chance to preserve limits, hence to have a left adjoint?

    (For any topology, and maybe for other flavors of ideals/valuations.)

    This question may seem rather off-topic here, but the relation to Lambda-rings is there, namely via their relation, in turn, to arithmetic jet spaces.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 29th 2014
    • (edited Jul 30th 2014)

    Something of the codensity monad or perhaps density comonad here?

    EDIT: Not even sure why I said that now.