User winfried - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:11:18Z http://mathoverflow.net/feeds/user/21336 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17532/does-linearization-of-categories-reflect-isomorphism/88258#88258 Answer by Winfried for Does linearization of categories reflect isomorphism? Winfried 2012-02-12T03:26:53Z 2012-02-12T03:26:53Z <p>Hi Tilman. I believe I proved that (in your language) linearization reflects isomorphism. The following is a sketch. I will send you a more detailed version. The general case may be reduced to the case of prime fields $F_p$ and certain categories $C$ with fixed objects $x$ and $y$ and morphisms $f_1,\dots,f_m\colon x\to y$ and $g_1,\dots,g_n\colon y\to x$ subject to relations which correspond to the fact that $u=f_1+\dots+f_m$ and $u^{-1}=g_1+\dots+g_n$ are mutually inverse in the $F_p$-linearization. Apart from trivial cases, we may reindex these generators such that $f_1g_1 = 1_y$ and $g_nf_m=1_x$, while the other summands in the expansion of $uu^{-1}$ and $u^{-1}u$, respectively, fall into equivalence classes whose size is a multiple of $p$. It is then possible to derive a sequence of pairs $(i_1,j_1),(i_2,j_2),\dots,(i_k,j_k)$ such that $f_{i_r}g_{j_r} = f_{i_{r+1}}g_{j_{r+1}}$ for $r=2,3,\dots,k-1$ and $g_{j_r}f_{i_r}=g_{j_{r+1}}f_{i_{r+1}}$ for $r=1,3,\dots,k-2$. Then $f_{i_1}g_{j_2}f_{i_3}g_{j_4}\dots f_{i_k}$ and $g_{j_k}f_{i_{k-1}}g_{j_{k-2}}f_{i_{k-3}}\dots g_{j_1}$ are mutual inverses of $C$.</p> http://mathoverflow.net/questions/17532/does-linearization-of-categories-reflect-isomorphism/88258#88258 Comment by Winfried Winfried 2012-03-04T08:13:20Z 2012-03-04T08:13:20Z A new version of this note can be downloaded from <a href="http://www.idmp.uni-hannover.de/downloads_wd.html" rel="nofollow">idmp.uni-hannover.de/downloads_wd.html</a> - Winfried