Are there examples of functors from the category of a single group to the category a partially ordered set (some sort of representation of the group in a poset) ?
Functor from a group to a poset
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group-theory
category-theory
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8Since you have only one object in a group and partially ordered sets are the small categories with *at most* one morphism between any two objects, there isn't much interesting to say... – 2011-07-26
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0On the other hand, a functor from a group-as-a-category to the category of posets would indeed be a poset representation. Such a poset representation might be obtained by composing a $G$-set (thought of as a functor) with the covariant powerset functor. I'm not sure what you mean about viewing simplicial complexes as posets though... – 2011-07-26
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0I don't really understand what the second question has to do with the first. – 2011-07-26
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0Or maybe Qiaochu Y's rhetorical incomprehension is a (valid) "no" answer to the second question... The question(s) as posed do not seem grounded. – 2011-07-26
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1I thought a simplicial complex could be viewed as a poset ordered by inclusion. Thus if a functor could be found between a group-as-category and a poset-as-category, one could do the same between the group and the complex. I will edit my question and remove that second part, since it raises more issues than expected. – 2011-07-26
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2(I guess one problem with too-broad, or too-amorphous questions, that verge on "can one make sense of X", is that they always have a (pointless) "yes" answer, insofar as a person sufficiently mathematically literate can _contrive_ connections between _any_ two things... upon demand, in effect. But that proves little. In general, telling the context in which a seemingly broad question arises is very helpful, enabling people to give _useful_, rather than pointless-formal responses.) – 2011-07-26
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1@paul garrett: I agree, but currently, if I start giving my motivations, it will be way too long and too boring for MathSE. To make it short : - I'm studying (as a hobby) mathematical music theory, in particular transformational music theory - I have intuitions about some new stuff. - However I hold a PhD in chemistry and my math education stopped at the undergraduate level, so that my reasoning is floppy. Hence the multiple "amorphous" questions on MathSE... – 2011-07-27
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0Ah! I'll think about this as context. I think it wouldn't hurt to mention a bit of the context, to genuinely help people who're trying to give genuinely helpful answers. One can be forthright, as opposed to "coy", even while avoiding being long-winded...? :) – 2011-07-27