If $\cal M$ is both an abelian category and a symmetric monoidal closed
category, then it is natural to ask whether projective objects in $\cal M$
are flat, and whether the tensor product of two projective objects is
projective. In the most familiar such categories, the answer to these
questions is obviously yes. However, the category $\cal M_G$ of Mackey
functors for a compact Lie group $G$ is a category of this type in which
projective objects need not be so well-behaved. This category is of
interest since good equivariant cohomology theories are Mackey functor
valued. The tensor product on $\cal M_G$ is important in this context
because of the role it plays in the not yet fully understood universal
coefficient and K\"{u}nneth formulae. This role makes the relationship
between projective objects and the tensor product especially critical.
Unfortunately, if $G$ is, for example, $O(n)$, then projectives need not
be flat in $\cal M_G$ and the tensor product of projective objects need
not be projective. This misbe haviorcomplicates the search for full
strength equivariant universal coefficient and K\"{u}nneth formulae.
The primary purpose of this article is to investigate these questions
about the interaction of the tensor product with projective objects in
symmetric monoidal abelian categories. Our focus is on functor categories
whose monoidal structures arise in a fashion described by Day.
Conditions are given under which such a structure interacts appropriately
with projective objects. Further, examples are given to show that, when
these conditions aren't met, this interaction can be quite bad. These
examples were not fabricated to illustrate the abstract possibility of
misbehavior. Rather, they are drawn from the literature. In particular,
$\cal M_G$ is badly behaved not only for the groups $O(n)$, but also for the
groups $SO(n)$, $U(n)$, $SU(n)$, $Sp(n)$, and $ Spin(n)$. Similar
misbehavior occurs in two categories of global Mackey functors which are
widely used in the study of classifying spaces of finite groups. Given
the extent of the homological misbehavior in Mackey functor categories
described here, it is reasonable to expect that similar problems occur in
other functor categories carrying symmetric monoidal closed structures
provided by Day's machinery.
