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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 infinity integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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 stack string-theory subobject 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.
    • CommentAuthorzskoda
    • CommentTimeDec 7th 2009

    New entry Eckmann-Hilton duality. Discussion welcome.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2009
    Were there not some things deleted from the last entry worth saving? Unless they were wrong, it doesn't cost us to have as many views on a subject as possible.
    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeDec 7th 2009
    • (edited Dec 7th 2009)

    I have looked for old version and found mistakenly there is none. So I created the entry without knowing of the old one. Now I can not do anything as it is looked by Toby.

    In the old version there are of course some worthy points but "arrow reversal" is not the point. If such it would be easy, Eckmann and Hilton themselves talked about quote "heuristic duality".

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeDec 7th 2009

    I'm done; do what you need to do to fix it.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2009
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> Perhaps that came from<br/><br/><blockquote><br/><br/>Any notion (definition, theorem, etc.) in a category which can be expressed purely category-theoretically admits a formal dual in the opposite or dual category , which can then be re-interpreted as a notion in the original category ; this latter notion is the (Eckmann–Hilton) dual of the original notion.<br/><br/></blockquote><br/><br/><a href="http://eom.springer.de/E/e120020.htm">here</a>. </div>
    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeDec 7th 2009
    • (edited Dec 7th 2009)

    I have put most of the previous entry back inside. To David: Eckmann-Hilton duality in homotopy theory is saying much more than just about purely categorically expressed notions: it indeed has to do also with various constructions involving paths, homotopies (in classical sense) etc. Thus it has to do with functors. On the other hand, if you consider the representable functors, then one should use enriched Yoneda lemma to get, from the formulas for the functors, to get effective arrow reversal. See the way I expanded the formulation of Fuks' theorem in the new version of the entry.

    Now it would be interesting to see weather the fact that Fuks duality (which is well defined functor on endofunctors) is not everywhere idempotent has something to do with the phenomenona like that pushfoward of fibration along cofibration is not a fibration again.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeDec 7th 2009

    I created mapping cylinder required at Eckmann-Hilton duality. This helps supporting the entry Hurewicz cofibration as well. I have chosen to use the direct cylinder rather than the upside-down for the default. In other words, (x,0) is identified with f(x).

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeDec 8th 2009
    • (edited Dec 8th 2009)

    I have added one more theorem with a proof (check the proof!) to mapping cylinder. New entry homotopy inverse previously required by deformation retract, which is itself changed a bit.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeDec 8th 2009

    I created basic problems of algebraic topology about the 4 basic problems listed usually in intro chapters of textbooks on algebaric topology: lifting, section, extension and retraction probolems and their interrelations.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2009

    Concerning the entry mapping cylinder:

    in some sentences it said "mapping cone" in that entry. It seemed to me that this were typos, and I changed it to "mapping cylinder". I also added a pointer to the page on mapping cones.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeDec 8th 2009

    Thank Urs, it seems they were typoses. At some point I will check more carefully all statements in this and related entries. Too busy with preparing the lecture for tomorrow.

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)