Timeline for How canonical is cofibrant replacement?
Current License: CC BY-SA 2.5
6 events
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| Feb 8, 2011 at 2:30 | comment | added | Tyler Lawson | Actually, your side commentary is quite interesting. There are some monads in topology that require cofibrant input to be homotopically sensible, but typically will not produce cofibrant output. Having a more canonical version of cofibrant or fibrant replacement would be quite helpful. | |
| Feb 7, 2011 at 21:43 | comment | added | Richard Garner | The point I really wanted to make, had my head been screwed on correctly, was that for an algebraic w.f.s., one obtains a natural transformation G --> G o G (by lifting) which is a section of the two projections G o G --> G, and moreover coassociative in the obvious sense. It is this transformation, which, for an ordinary w.f.s., would fail to be natural (or coassociative). | |
| Feb 7, 2011 at 6:00 | comment | added | Richard Garner | @Harry - one can always exponentiate a category equipped with an algebraic WFS by a small category, and again obtain a category equipped with an algebraic WFS. Whether or not this is a useful thing to do is another question. The WFS one obtains in this way is neither the injective nor the projective one; and even if the original one was cofibrantly generated, the induced WFS will typically not be, at least not in the classical sense. | |
| Feb 7, 2011 at 5:56 | comment | added | Richard Garner | Sorry, what I wrote above seems to be complete rubbish. There is a perfectly good natural transformation G o F --> G under no assumptions at all, using only the functoriality of G. | |
| Feb 7, 2011 at 5:31 | comment | added | Harry Gindi | Dear Richard, are these algebraic notions compatible with exponentiation by small categories (That is, is this natural/algebraic WFS theory is an alternative to the theory of combinatoriality introduced by Jeff Smith)? | |
| Feb 7, 2011 at 5:22 | history | answered | Richard Garner | CC BY-SA 2.5 |