Quotient of a category by a group action - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:13:41Z http://mathoverflow.net/feeds/question/22686 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22686/quotient-of-a-category-by-a-group-action Quotient of a category by a group action Mariano Suárez-Alvarez 2010-04-27T06:13:53Z 2010-04-27T18:46:29Z <p>Let a group $G$ act on a small category $C$. If $G$ acts freely on objects, there is a sensible construction of the quotient $C/G$ (this is briefly spelt out <a href="http://mathoverflow.net/questions/636/quotient-of-a-category-by-a-free-group-action/22682#22682" rel="nofollow">here</a>)</p> <p>What about the non-free case?</p> http://mathoverflow.net/questions/22686/quotient-of-a-category-by-a-group-action/22695#22695 Answer by Dai Tamaki for Quotient of a category by a group action Dai Tamaki 2010-04-27T07:47:00Z 2010-04-27T07:47:00Z <p>From the view point of representations of finite dimensional algebras, there is a construction called the "orbit category" or "skew category" construction. The construction appears, for example, <a href="http://arxiv.org/abs/math/0312214" rel="nofollow">a paper by Cibils and Marcos</a>, <a href="http://arxiv.org/abs/math/0503240" rel="nofollow">a paper by Keller</a>, and <a href="http://arxiv.org/abs/0807.4706" rel="nofollow">a paper by Asashiba</a>.</p> <p>If you browse these papers, you will notice that their construction is a version of the Grothendieck construction. Here we regard an action of a group $G$ on a category $C$ as a functor $$G \longrightarrow Cats.$$ </p> <p>So it's related to Reid's comment.</p> http://mathoverflow.net/questions/22686/quotient-of-a-category-by-a-group-action/22760#22760 Answer by Dan Ramras for Quotient of a category by a group action Dan Ramras 2010-04-27T18:46:29Z 2010-04-27T18:46:29Z <p>Here is a particular special case that seems to be useful to me:</p> <p>Let <code>$F: D \to G$</code>-Sets be a diagram of <code>$G$</code>-sets, indexed by some small category D. Then ignoring the G-actions, one can form the Grothendieck wreath product <code>$D\wr F$</code>. Objects are pairs (d, x) with <code>$x\in F(d)$</code>, and morphisms <code>$(d,x)\to (d', x')$</code> are arrows <code>$a:d\to d'$</code> such that <code>$F(a)(x) = x'$</code>. This category inherits an action of G from the actions on the sets in the diagram; <code>$g\cdot (d,x) = (d, gx)$</code> (on morphisms, $g$ looks like the identity: <code>$g\cdot (a:(d,x)\to(d',x')) = a:(d,gx)\to(d',gx')$</code>, which makes since $F(a)$ is $G$-equivariant. </p> <p>There is another diagram <code>$F/G : D\to$</code> Sets, which takes <code>$d\in D$</code> to <code>$F(D)/G$</code>, and there's a natural functor $D\wr F \to D\wr (F/G)$. I claim that this functor satisfies the universal property of the colimit, i.e. <code>$(D\wr F)/G = D\wr (F/G)$</code>. </p> <p>This special case has the interesting property that the nerve of <code>$D\wr (F/G)$</code> is precisely <code>$N_\cdot (D\wr F)/G$</code>. I don't think that will hold for arbitrary group actions on categories.</p>