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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 $k$, 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?
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.
Corrected.
Thank you!
In view of recent discussions which touched on $\Lambda$-rings, I felt an urge to add a few more theoretical details to Tall-Wraith monoid, which I have now done.
Added still more details. At some point I need to check whether I got some things switched around, i.e., whether what I call $R \odot S$ is what others call $S \odot R$, and the like.
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.
Something of the codensity monad or perhaps density comonad here?
EDIT: Not even sure why I said that now.
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