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nLab > Latest Changes: convex space, homomorphism

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    • CommentRowNumber1.
    • CommentAuthorJohn Baez
    • CommentTimeDec 8th 2009
    I added some references to convex space and began a new entry on homomorphism.

    It would be great to see the article on convex spaces continue... it sort of trails off now. I've tried to enlist Tobias Fritz.
    • CommentRowNumber2.
    • CommentAuthorJohn Baez
    • CommentTimeDec 8th 2009
    I also added an entry conical space. I would love to see the relation between various generalized versions of linear algebra neatly explained: conical spaces, affine spaces, convex spaces and vector spaces for a start! But we don't even have the definition of Durov's generalized rings! Someone could say what it means for a finitary monad to be commutative...
    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeDec 8th 2009
    • (edited Dec 8th 2009)

    Thanks John. As far as commutativity, entry commutative algebraic theory might have escaped your attention. The entry needs more explanation though; it is in the equivalent language of theories rather than monads.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeAug 8th 2012

    I added extended conical spaces (to the previously existing article).

  5. replaced “monoid” by “semiring” in the introductory example in convex space.

  6. changed “commutative semiring” to “commutative nonunital ring”.

    But my main problem is this: The definition of convex space requires the averaging variables (let’s call it like that) to be from a “(semi)ring PP”. But in this case the term 1−p1-p is undefined. Requiring an additive inverse is not an option as it does not exist in the standard example P=[0,1]P = [0,1]. Rather the appropriate algebraic structure here is something with a multiplication and an inverse “1−p1-p”. My guess would be a Heyting algebra (ab=a∧ba b = a\wedge b and 1−a=a⇒01-a = a \Rightarrow 0).

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMar 18th 2017
    • (edited Mar 18th 2017)

    My own inclination would be to ask for an operation c p,qc_{p,q} for any pair of elements p,q∈Pp,q\in P such that p+q=1p+q=1. Then the symmetry axiom would be c p,q(x,y)=c q,p(y,x)c_{p,q}(x,y) = c_{q,p}(y,x) and the assocativity axiom would be c p,q(x,c r,s(y,z))=c p′,q′(c r′,s′(x,y),z)c_{p,q}(x,c_{r,s}(y,z)) = c_{p',q'}(c_{r',s'}(x,y),z) whenever (p,qr,qs)=(p′r′,p′s′,q′)(p,q r,q s)= (p'r', p's',q').

    Edit: I see the page already takes this approach in the “unbiased” case.

  8. Thanks, I put this remedy in the definition.

  9. I reworked the definition again because the interval was not subsumed by it (the interval is not closed under addition).

  10. Is there a categorical way to define/identify which convex spaces are the cancellative ones?

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMar 20th 2017

    My proposal of defining c p,qc_{p,q} whenever p+q=1p+q=1, rather than c pc_p for all pp, would eliminate the need to consider QQ.

    • CommentRowNumber12.
    • CommentAuthorkirksturtz
    • CommentTimeMar 21st 2017
    I updated the page with three additional references and further details about the category. The last paragraph
    ``Of particular importance are convex spaces parametrized by the interval P=[0,1] or
    the Boolean algebra P={0,1}.'' I find very odd. The Boolean algebra 2={0,1}, and more generally, ``discrete convex spaces'' I believe you are thinking about can be viewed within the context of parameterization over the unit interval. I am basing my ideas on the work of
    J.Isbell,M. Klum, and S.Schanuel, Affine part of algebraic theories - which shows that Cvx is just the affine part of the
    theory of K-modules, as detailed in Mengs thesis (which is referenced).
    • CommentRowNumber13.
    • CommentAuthorRodMcGuire
    • CommentTimeMar 21st 2017

    I changed the link to your Stutz paper to https://arxiv.org/abs/1703.03240. In the future please link to the abs page not the pdf one.

    I also cleaned up the page a little, making inline references link to the list at the bottom.

    [ I don’t know how to enclose such links in square brackets ]

  14. I adapted the second paragraph to the new version of Sturtz’s paper (he updated the link but did not changed what he had written into the text).

1 to 14 of 14

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