Timeline for How do I split a homotopy idempotent?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Jan 24, 2011 at 18:33 | comment | added | darij grinberg | Ah! (I know. Homological and -topical algebra seems to consist of definitions, theorems and abuses of notation.) | |
| Jan 24, 2011 at 10:49 | comment | added | Theo Buehler | @darij: $C \otimes I$ is the mapping cylinder over the identity $C \to C$, see Weibel Ch. 1.6 for an explicit description of the complex. The notation $- \otimes I$ is a standard abuse from homotopical algebra. | |
| Jan 24, 2011 at 9:10 | comment | added | darij grinberg | Also, it really doesn't work without countable sums? Can we prove it? | |
| Jan 24, 2011 at 9:09 | comment | added | darij grinberg | What is $\otimes$ if the category is not moonoidal? | |
| Jan 24, 2011 at 2:52 | history | edited | John Klein | CC BY-SA 2.5 |
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| Jan 24, 2011 at 2:48 | vote | accept | Eitan Chatav | ||
| Jan 24, 2011 at 2:39 | comment | added | John Klein | @Theo: Thanks for the reference and the observation. | |
| Jan 24, 2011 at 2:30 | comment | added | Theo Buehler | But don't you assume that there are countable coproducts (or co-powers) in your argument? In that case it's a general fact about such pre-additive categories proved with essentially the same 'mapping telescope argument', see Freyd, Splitting homotopy idempotents, in: Proceedings of the Conference on Categorical Algebra, La Jolla, CA, 1965, Springer, New York, 1966, pp. 173–176. | |
| Jan 24, 2011 at 2:15 | comment | added | John Klein | @Eitan: Good. Then my argument does apply. | |
| Jan 24, 2011 at 2:03 | comment | added | Eitan Chatav | Yes, I mean that: the homotopy category of a category of chain complexes. In general I just want the underlying category to be Ab-enriched and idempotent-splitting. The categories I'm looking at are not monoidal though. | |
| Jan 24, 2011 at 0:37 | comment | added | John Klein | Good point. Does Eitan's category have mapping cylinders and can one form colimits? I don't actually know what a "chain homotopy category" is. Maybe he means the homotopy category of a category of chain complexes. What are these complexes defined over? | |
| Jan 24, 2011 at 0:15 | comment | added | darij grinberg | $\otimes$? $I$? What do we know about our category? | |
| Jan 24, 2011 at 0:13 | history | answered | John Klein | CC BY-SA 2.5 |