User dylan rupel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:09:22Z http://mathoverflow.net/feeds/user/4115 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60241/why-dont-ideals-and-quotients-work-well-for-categories/60467#60467 Answer by Dylan Rupel for Why don't ideals and quotients work well for categories? Dylan Rupel 2011-04-03T18:20:28Z 2011-04-03T18:20:28Z <p>I don't think I saw anyone give this construction above and if so I apologize for the repetition.</p> <p>One could consider the orbit category by an endofunctor. The example I have in mind is the cluster category $\mathcal{C}$ important in the categorification of cluster algebras. One starts with the category of finite-dimensional representations $mod_\mathbb{C}~Q$ of a Dynkin quiver $Q$ over the field $\mathbb{C}$ of complex numbers. Then the cluster category is the orbit category of the bounded derived category $D^b(mod_\mathbb{C}~Q)$ by the endofunctor $F=\tau^{-1}[1]$ where $\tau$ is the Auslander-Reiten translation of $mod_\mathbb{C}~Q$ and $[1]$ is the shift functor. The paper at <a href="http://de.arxiv.org/pdf/math.RT/0402054.pdf" rel="nofollow">http://de.arxiv.org/pdf/math.RT/0402054.pdf</a> was one of the first on cluster categories.</p> <p>I guess to answer your main question: the objects in these categories are very different. For example $D^b(mod_\mathbb{C}~Q)$ has infinitely many isomorphism classes of indecomposable representations, for example the shifts of indecomposable objects in $mod_\mathbb{C}~Q$. However $\mathcal{C}$ has only finitely many indecomposable objects, namely the indecomposable representations of $Q$ considered as complexes concentrated in degree 0 and the shifts $P[1]$ of the projective representations of $Q$. </p> <p>I suppose this construction has a slightly different flavor than the quotient of a ring by an ideal.</p> http://mathoverflow.net/questions/23643/books-about-history-of-recent-mathematics/46697#46697 Answer by Dylan Rupel for Books about history of recent mathematics Dylan Rupel 2010-11-20T01:22:22Z 2010-11-20T01:22:22Z <p>I enjoyed the book "Remarkable Mathematicians: From Euler to von Neumann" by Ioan James. It gives a good account of what the mathematicians were doing (in their personal lives and professional) and how their interactions shaped mathematics. It is fairly light on the mathematical content but an enjoyable read nonetheless. </p>