Re: injectivity

[categories <cat-dist@mta.ca>]

Date: Tue, 13 May 1997 17:52:00 -0400
From: Michael Barr <barr@triples.math.mcgill.ca>

I have given some thought to this question.  I do not have a complete
answer, but no one else has posted anything, so I will give what I 
have.  First off, the functor category [I,Vect_k] is an AB5 category
with a projective generator and hence a module category.  In the particular
case that I is the orbits of a group, finite or not, it is just k[G]
modules.  Now if k is finite, then k[G] is semisimple, whence all modules
are injective, unless char(k) | #(G), the so-called modular case.  In that
case, I haven't worked out the details, but I think the tensor product
of finite-dimensional injectives is injective.  The argument uses duality
in k.  In fact, the category is self dual (a *-autonomous category).
On the other hand, I think it unlikely that this is true for infinite
dimensional spaces, but I do not have a counter-example.  There are 
categories, for instance Ab, in which the tensor product of injectives
is injective.  The reason for Ab is that every injective is a direct sum
of indecomposable injectives and the only non-zero tensor product of
indecomposable injectives is Q tensor Q = Q.

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Date: Fri, 9 May 1997 13:11:56 -0300 (ADT)
From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: injectivity 
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Date: Fri, 9 May 1997 15:30:53 +0200 (MET DST)
From: Marek Golasinski <marek@mat.uni.torun.pl>

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