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• 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.