From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/378 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: injectivity Date: Fri, 9 May 1997 13:11:56 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016929 25491 80.91.229.2 (29 Apr 2009 14:55:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:55:29 +0000 (UTC) To: categories Original-X-From: cat-dist Fri May 9 13:13:35 1997 Original-Received: by mailserv.mta.ca; id AA20359; Fri, 9 May 1997 13:11:56 -0300 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:378 Archived-At: Date: Fri, 9 May 1997 15:30:53 +0200 (MET DST) From: Marek Golasinski Dear Colleagues, Let $Vect_k$ be the category of vector spaces over a field $k$ and $I$ a small category. Consider the category $I-Vect_k$ of all covariant functors from $I$ to $Vect_k$. For two object $F,F'$ of the category $I-Vect_k$ consider their tensor product $F\otimes F'$ such that $(F\otimes F')(i)=F(i)\otimes F'(i)$ for all $i\in I$ and in the obvious way on the morphisms of $I$. 1) Is it true that this tensor product $F\otimes F'$ is injective provided that $F$ and $F'$ are injective? I am really intersted in its particular case. Namely, let $G$ be a finite group and $O(G)$ the finite associated category of canonical orbits. Objects of $O(G)$ are given by the finite $G$-sets $G/H$ for all subgroups $H\subsetq G$ and morphisms by eqivariant maps. 2) What about preserving the injectivity by the above defined tensor product in the functor category $O(G)-Vect_k$? If that is not true for $I=O(G)$ then I would greatly appreciate getting a counterexample. Many thanks in advance for your kind attention on the problem above. With my best regards, Marek Golasinski