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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 8th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeDec 8th 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexYou'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*.

   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.

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeDec 8th 2011
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexCorrected.

   Corrected.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 9th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThank you!

   Thank you!

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 28th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIn 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 29th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexAdded 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 29th 2014
   • (edited Jul 29th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, 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]].

   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.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 29th 2014
   • (edited Jul 30th 2014)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSomething of the [[codensity monad]] or perhaps density comonad here? EDIT: Not even sure why I said that now.

   Something of the codensity monad or perhaps density comonad here?

   EDIT: Not even sure why I said that now.