Timeline for Is an "infinite compositions of arrows" meaningful?
Current License: CC BY-SA 2.5
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| Apr 29, 2010 at 11:36 | comment | added | Matthew | @Pete: This makes the question even more striking in my view. For example, let's say we have the monoid {0,1} with operation xor and try to count its arrows. Should it have two arrows corresponding to the two elements of the monoid? What about the infinite composition ..0101.. (or the source-target-friendly 1..0101..0), which is neither equal to 0 nor to 1? I think this is the essence of my original question. | |
| Apr 29, 2010 at 6:10 | comment | added | Qfwfq | @Qiaochu: this problem of the target being not always well defined could perhaps be solved by requiring the infinite set of arrows one wants to "compose" to be linearly ordered (w.r.t. "source precedes target" order, let's say) and having a minimum (the source of the "infinite composite") and a maximum (the target of the "infinite composite"). | |
| Apr 29, 2010 at 6:04 | comment | added | Pete L. Clark | @Qiaochu: It's okay -- or rather not problematic for the reason you give -- in any category with a single object (i.e., a monoid). Seriously, that's a reasonable special case. But I certainly do take your point: we need to either very much restrict the categories in question, or cook up some extra structure that keeps track of convergence of objects (or do something else equally drastic). | |
| Apr 29, 2010 at 6:02 | comment | added | Steve D | Presumably, that example wouldn't be "convergent" in this "topology". | |
| Apr 29, 2010 at 5:51 | comment | added | Qiaochu Yuan | The example I gave is problematic in any category; the infinite composition ...fgfg doesn't have a well-defined codomain (maybe I should say target). | |
| Apr 29, 2010 at 5:46 | history | answered | Pete L. Clark | CC BY-SA 2.5 |