Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeApr 25th 2012

    I have split off ordinal sum from the entry on joins as it is needed in other entries as well. I have not revised the entry just doing a cut and paste, so it needs more work!

  1. I added a note here on Lawvere’s definition of the ordinal sum of categories, from “Ordinal sums and equational doctrines”.

  2. Previous edit said there were isomorphisms from [i][j][i] \oplus [j] to [j][i][j] \oplus [i], this was false and has been corrected.


    diff, v9, current

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeJan 3rd 2019
    • (edited Jan 3rd 2019)

    @ Anonymous #3 ???? There are no natural isomorphisms but for finite ordinals, [i][j][i]\oplus [j] is the same ordinal as [j][i][j]\oplus [i] as both are [i+j+1][i+j+1], so is that what you ment?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJan 3rd 2019

    Further clarified that addition of finite ordinals is symmetric, but not infinite ones. Actually this page needs some more serious work in clarifying the finite/infinite distinction; it reads kind of as if whoever wrote it thought that “ordinal” meant “finite ordinal”. (Not that I think anyone actually thought that, I’m just saying the wording is confusing.)

    diff, v11, current

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 3rd 2019

    Wait a minute: anonymous must have been saying that there is no symmetry isomorphism (even for finite ordinals) for the ordinal sum monoidal product that is natural with respect to ordinal maps. That’s a correct statement! The current page suggests that (Δ a,+)(\Delta_a, +) carries symmetric monoidal structure, but that’s wrong.

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeJan 3rd 2019
    • (edited Jan 3rd 2019)

    I have tried to correct the entry a bit. There is probably a lot more that should be done however as it still is largely centred on finite ordinals and the application of ordinal sum in that context.

    diff, v12, current

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 3rd 2019

    Thanks Todd; that’s probably a better guess as to what Anonymous had in mind.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 2nd 2020

    Added a redirect for ordinal sum of categories.

    diff, v15, current

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)