Categories where projective objects are flat

In many categories arising in the theory of algebras or modules, projective objects are flat. For example each projective module over a ring with identity is flat. However, there are categories where this principle is violated: Lewis has shown that the category of Mackey functors for the orthogonal group $O(n)$ has non-flat projectives.

Q1: Is there a classification of symmetric monoidal abelian categories $\mathcal{A}$ where all projectives are flat ? BTW: Do such categories have a particular name ?

If a classification is too strong to establish, I would be interested in

Q2: Are there criteria that ensure that projectives are flat ?

An example for Q2 is, when $\mathcal{A}$ has direct sums and the unit is a progenerator of $\mathcal{A}$ (this is just the categorification of the module case mentioned above). Note that this criteria doesn't work for instance in the category of $RG$-modules ($R$ commutative ring, $G$ a group) where the tensor product is given by $M \otimes_R N$ with diagonal $G$-operation.

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For me, a flat object is by definition a filtered colimit of projective objets. All projective objects are flat in this sense. What do you mean by flat? –  Fernando Muro Feb 13 '13 at 23:36
An object $F$ is flat if the tensor functor $- \otimes F: \mathcal{A} \to \mathcal{A}, X \mapsto X \otimes F$ is exact. –  Ralph Feb 13 '13 at 23:48
So what happens for $\mathcal{A}=\mathrm{Rep}(G)$? –  Martin Brandenburg Feb 14 '13 at 0:04
Projectives are flat in Rep(G), but in the proof we use (1) projectives in RG-Mod are direct summands of direct sums of RG , and (2) RG is a direct sum of R in R-Mod. So the categorification is a bit more complicated than in the case of R-Mod described in the question. –  Ralph Feb 14 '13 at 0:18

As for the second question, an obvious sufficient condition is that $\mathcal{A}$ is AB4, the tensor product commutes with direct sums, and the existence of a generating family of flat objects.

Namely, if $\{F_i\}_{i \in I}$ is such a generating family and $P$ is a projective object of $\mathcal{A}$, then ${\bigoplus_{i \in I}} F_i^{\oplus \hom(F_i,P)} \to P$ is an epimorphism, which shows that $P$ is a direct summand of flat objects, therefore also flat.

Following the terminology of algebraic geometry, a generating family of invertible objects, or more generally dualizable objects (which are in particular flat) might be called an ample family. An abelian $\otimes$-category which has such an ample family might be called divisorial. When $X$ is a divisorial scheme, then $\mathrm{Qcoh}(X)$ is an example (but there are few projective objects). Other examples arise from $\otimes$-analogues of non-commutative projective schemes à la Artin-Zhang.

When $\mathcal{A}$ is as above and $G$ is a group, then $\mathrm{Rep}_{\mathcal{A}}(G) := [G,\mathcal{A}]$ has the same properties: If $\{F_i\}$ is a generating family of flat objects of $\mathcal{A}$, then $\{F_i[G]\}$ is a generating family of flat objects of $\mathrm{Rep}_{\mathcal{A}}(G)$. Here, $F \mapsto F[G]$ is the functor which is left adjoint to the forgetful functor, constructed via $F[G] = \bigoplus_{g \in G} F$ with the obvious $G$-action.

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Nice generalization of my example above where the generating family is $\lbrace 1\rbrace$. I think the example of group representations could be further generalized in the following way: Let $A$ be a monoidal abelian cat., $B$ an abelian cat., $F: A \to B$ a functor, $A' \subseteq A$ the image of $F$ and $G:A'\to B$ a functor. $B$ is a monoidal category with tensor product $\otimes_B := G \circ \otimes_A \circ (F \times F)$. Then: If $B$ has a generating family $P_i$ such that $F(P_i)$ is flat in $A$, then projectives in $B$ are flat (of course some more technical conditions are needed, like –  Ralph Feb 14 '13 at 22:57
... existence of direct sums, $F$ must commute with direct sums, etc.). –  Ralph Feb 14 '13 at 22:57
Of course we need $F: B \to A$. –  Ralph Feb 15 '13 at 0:34

This is the sort of question which would interest my advisor, Mark Hovey. In fact, he and Keir Lockridge wrote a paper called Semisimple Ring Spectra which addresses questions like this in triangulated categories (which cannot be abelian but make more sense to work with if you do homotopy theory). In this paper he has two results which answer your question in that context. I'll paraphrase them and highlight the bits that matter, but first some definitions.

A triangulated category come with an endofunctor $\Sigma$ which acts like suspension in topological spaces or shift in chain complexes (there are axioms for this). With this endofunctor you can define homotopy classes of maps and do cohomology theory. Call such a category $T$ a weak stable homotopy theory if every cohomology theory is representable. Suppose you have such a theory and it comes with a compact generator $S$ (i.e. the smallest localizing subcategory containing $S$ is $T$). Call such a theory a stable homotopy theory if there is also an axiom saying that $S$ plays nicely with the smash product. Call it a Brown category if all homology theories and natural transformations between them are representable. Assume below these hypotheses hold on $T$.

Proposition 2.1: The generator $S$ is semisimple iff projective objects and realizable objects coincide.

Remark 2.3: The generator $S$ is von Neumann regular iff flat objects and realizable objects coincide.

For the definitions of semisimple and von Neumann regular, see that paper. These are just the versions for triangulated categories of standard properties of rings. So semisimple implies von Neumann regular and there are classifications for when the converse holds. Furthermore, you can find a classification of semisimple objects in such theories $T$ as Theorem 4.1, which is a purely algebraic classification based on $\pi_*(S)$. So this gives some sense of a classification of theories where projectives and flats are the same. I don't know if anything like this has been done in the abelian setting. I'd be very interested to know!

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A pdf of the article can be found here: citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.148.9732. It came out in 2009 –  David White Feb 14 '13 at 19:58