Groupoid structure on G/H? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:34:59Z http://mathoverflow.net/feeds/question/17792 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17792/groupoid-structure-on-g-h Groupoid structure on G/H? JunwuTu 2010-03-11T01:13:37Z 2012-01-25T10:52:05Z <p>Let $G$ be a group and let $H$ be a subgroup. If $H$ is normal in $G$, then $G/H$ has a group structure. But in general, can there be a groupoid structure on $G/H$(left cosets or right cosets) that generalizes the normal case?</p> http://mathoverflow.net/questions/17792/groupoid-structure-on-g-h/17798#17798 Answer by José Figueroa-O'Farrill for Groupoid structure on G/H? José Figueroa-O'Farrill 2010-03-11T01:42:34Z 2010-03-11T01:42:34Z <p>Any set on which a group acts can be given a natural groupoid structure, called the <em>action groupoid</em>. Since $G/H$ admits a left $G$-action, the action groupoid construction applies. You can find the definition in the <a href="http://en.wikipedia.org/wiki/Groupoid" rel="nofollow">wikipedia page</a> <em>de rigueur</em>, under <strong>Group actions</strong>.</p> http://mathoverflow.net/questions/17792/groupoid-structure-on-g-h/17799#17799 Answer by Theo Johnson-Freyd for Groupoid structure on G/H? Theo Johnson-Freyd 2010-03-11T01:43:59Z 2010-03-11T01:43:59Z <p>One answer to your question is that there is always the notion of an "action groupoid", although this does not reproduce the <em>group</em> structure on $G/H$ when $H$ is normal.</p> <p>Let $G$ be a group acting on a set $X$. (There are generalizations when both $X,G$ are groupoids.) Then the <strong>action groupoid</strong> $X//G$ is the groupoid with objects $X$, and morphisms $X\times G$. More precisely, if $x,y \in X$, then $\hom(x,y) = \{ g\in G \text{ s.t. } gx = y\}$. The groupoid axioms are essentially obvious.</p> <p>For example, let $X = G/H$ be the set of left $H$-cosets. Then the action groupoid is very simple: it is connected (any object is isomorphic to any other), with $\text{aut}(e) = H$, where $e = eH$ is the trivial left $H$-coset. And $\hom(e,g) = gH$, where $g = gH$ is a coset. So as a discrete groupoid, this action groupoid is equivalent to $\{\text{pt}\}//H$, also sometimes called the "classifying groupoid" $\mathcal B H$, because the geometric realization of the nerve of this groupoid is the usual classifying space of $H$.</p> <p>In short-hand, we have the following equation: $$(G/H) //G \cong 1//H$$ where $\cong$ denotes equivalence of groupoids. (Actually, since I'm talking about left actions, I should probably write $G \backslash \backslash X$ for the action groupoid, and so the equation really should be $G \backslash \backslash (G/H) = H \backslash \backslash 1$, but typing "/" is much faster than typing "\backslash", so I won't use the better notation.)</p> <p>But you are probably asking a different question. Recall that when $H$ is normal, then $G/H$ has a group structure, which is to say there is a groupoid with one object and whose morphisms are elements of $G/H$. Of course, as you know, if $H$ is not normal, then $G/H$ does not have a natural group structure, because in general $g_1Hg_2H$ is not a left $H$-coset.</p> <p>You can try to do the following. Any set is naturally a groupoid with only trivial morphisms, and then the set $G/H$ is equivalent to the groupoid $G//H$, where $H$ acts on the set $G$ by translation = right multiplication. (This is because $H$ acts on $G$ freely.) But $G$ is actually more than a set: it is a group. So let's think about it as a "groupal groupoid" or "2-group", i.e. a 2-groupoid with only one object; in this case, it will also only have identity 2-morphisms.</p> <p>Then I guess you should try to form the "action 2-group" or something, by adding 2-morphisms for the translations by $H$. But I think that if you do, you no longer have a groupal groupoid: I think that if $H$ is not normal, then the group multiplication is not a functor from the action groupoid $G//H$.</p> <p>The other only thing I can think of is to define $K = \text{Norm}_GH$, the normalizer, and then $K/H$ is a group that embeds in $G/H$, so let the objects of your groupoid be cosets of $K$ and the morphisms given by $K/H$?</p> <p>So, long story short: in the way that I think you are hoping, no, $G/H$ is not naturally a groupoid.</p> http://mathoverflow.net/questions/17792/groupoid-structure-on-g-h/17808#17808 Answer by Mariano Suárez-Alvarez for Groupoid structure on G/H? Mariano Suárez-Alvarez 2010-03-11T03:02:06Z 2010-03-11T03:38:49Z <p>When $G$ is finite, for arbitrary $H$ you can consider the quotient $G/H$ as an <em>association scheme</em> in the sense of [Zieschang, Paul-Hermann. An algebraic approach to association schemes. Lecture Notes in Mathematics, 1628. Springer-Verlag, Berlin, 1996. xii+189 pp. ISBN: 3-540-61400-1 <a href="http://www.ams.org/mathscinet-getitem?mr=MR1439253" rel="nofollow">MR1439253</a>]. </p> <p>The notion of association scheme generalizes (at least...) that of group, that of <a href="http://en.wikipedia.org/wiki/Distance-regular_graph" rel="nofollow">distance regular graph</a>, and that of (some types of?) <a href="http://en.wikipedia.org/wiki/Bruhat-Tits_building" rel="nofollow">building</a>. When $H$ is normal, then the association scheme $G/H$ is the same thing as the association scheme associated to the group $G/H$, so in a sense, for $H$ non-normal considering the association scheme $G/H$ is a quite natural.</p> <p>(In the infinite case, you can probably do more or less the same... I don't recall having seen infinite association schemes, though)</p> <p>(Of course, this does not answer your question... but as Theo pointed in an earlier answer, I do not think $G/H$ is a groupoid in any sensible way in general, so the bit of structure I am mentioning might be a useful consolation prize!)</p> http://mathoverflow.net/questions/17792/groupoid-structure-on-g-h/86616#86616 Answer by Ronnie Brown for Groupoid structure on G/H? Ronnie Brown 2012-01-25T10:52:05Z 2012-01-25T10:52:05Z <p>Another way of putting this is that there is an equivalences of categories between actions of a group (or groupoid) $G$ and covering morphisms of the group (or groupoid) $G$. See for example </p> <p>Higgins, P.J. Notes on categories and groupoids, Mathematical Studies, Volume~32. Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195.</p> <p>This links the notion of covering morphisms of groupoids nicely with the theory of covering spaces. There one finds that for "nice" spaces $X$ the fundamental groupoid $\pi_1$ gives an equivalence of categories from covering maps of $X$ to covering morphisms of $\pi_1 X$. Thus a map of spaces is modelled by a morphism of groupoids, and this is convenient. For an exposition in these terms, see my book "Topology and groupoids", or earlier editions (1968, 1988). </p>