Timeline for Is an "infinite compositions of arrows" meaningful?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2010 at 11:26 | vote | accept | Matthew | ||
| Apr 30, 2010 at 1:03 | comment | added | Peter LeFanu Lumsdaine | @dan: yes; as unknown said, "transfinite composition" refers to specific arrows coming from a very special class of limits/colimits. As Reid said, we restrict not just the index category (to ordinals), but also the functor itself: it has to be colimit-preserving. | |
| Apr 29, 2010 at 12:06 | comment | added | user2734 | But Dan, the definition only refers to one single arrow out of a whole colimiting cone. Also, only some special index categories (ordinals as pre-orders) are considered. So, I think that it is not possible to say that it is the same thing as a colimit. | |
| Apr 29, 2010 at 11:39 | comment | added | Matthew | I haven't learned about limits and colimits in categories yet. Perhaps that will shed some light on the subject. | |
| Apr 29, 2010 at 8:04 | comment | added | Dan Petersen | I've never heard the term transfinite composition, but it seems to be another word for the perfectly ordinary direct/inverse limit, right? If so, it certainly shows up absolutely everywhere in algebra, not just in categorical homotopy theory... consider the p-adic numbers or taking stalks of sheaves. | |
| Apr 29, 2010 at 6:49 | comment | added | Reid Barton | Yes. If $\alpha = n$ is a finite ordinal, then for the colimit we may take the last object $X_{n-1}$ and the structural map of the colimit cone is the composition of the maps $X_0 \to X_1 \to \cdots \to X_{n-1}$. | |
| Apr 29, 2010 at 6:36 | comment | added | user2734 | I apologize for the silly question, but in your first example of a colimit of a functor $\omega\to C$, is the transfinite composition just the 0-th arrow of the colimiting cone? Does this specialize in some sense to the usual composition if $\omega$ is replaced by some finite $n$? And if not, why is it called transfinite composition ? | |
| Apr 29, 2010 at 6:08 | history | answered | Reid Barton | CC BY-SA 2.5 |