Let $\mathcal{C}$ be a $k$-linear abelian category. Are the following two definitions of enough projectives equivalent?

(i) For every object $A$ there exists a projective object $P$ s.t. $P\twoheadrightarrow A$

(ii) Every simple object $U_i$ has a projective cover $p_i: P_i\twoheadrightarrow U_i$

The first one is taken from wikipedia, the second from the book Tensor Categories by Etingof et al.

I think this cannot be the same, for the superficial reason that nothing in (ii) makes a reference to arbitrary (generally indecomposable) objects of $\mathcal{C}$. However, I might be very wrong since I cannot seem to prove it.

Any pointers?

  • $\begingroup$ What's your definition of simple object? $\endgroup$ – Randall Apr 11 '18 at 15:31
  • $\begingroup$ One whose only subobjects are $0$ and itself. $\endgroup$ – Jo Be Apr 11 '18 at 15:32

They do not, in general. I bet Etingof is assuming we're in a semisimple category, as is usually the case, if not always, in that book. If every object is a sum of simples, then it should be clear that these conditions are equivalent, since a sum of projectives is projective and a sum of epis is epi. The conditions are not equivalent in general, since abelian categories need not have any simple objects. See Jeremy Rickard's example here.

  • $\begingroup$ To be fair, I lied a bit (by omission). We are actually assuming local finiteness (Hom-finite and every object has finite length) and finitely many iso classes of simple objects. As you might have noticed, this is precisely the definition of a finite abelian cat over $k$, which is generally not semisimple. $\endgroup$ – Jo Be Apr 11 '18 at 15:50

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