119 2.11. Main Theorem. Exercise 2.11.1. Show that for any M ∈ M the object Hom(M, M) with the multiplication defined above is an algebra (in particular, define the unit morphism!). Theorem 2.11.2. Let M be a module category over C, and assume that M ∈ M satisfies two conditions: 1. The functor Hom(M, ) is right exact (note that it is automatically •left exact). 2. For any N ∈ M there exists X ∈ C and a surjection X ⊗M N.→Let A = Hom(M, M). Then the functor F := Hom(M, ) : M → •ModC (A) is an equivalence of module categories. Proof. We will proceed in steps: (1) The map F : Hom(N1, N2) HomA(F (N1), F (N2)) is an iso-→morphism for any N2 ∈ M and N1 of the form X ⊗ M, X ∈ C. Indeed, F (N1) = Hom(M, X ⊗ M) = X ⊗ A and the statement follows from the calculation: HomA(F (N1), F (N2)) = HomA(X ⊗ A, F (N2)) = Hom(X, F (N2)) = = Hom(X, Hom(M, N2)) = Hom(X ⊗ M, N2) = Hom(N1, N2). (2) The map F : Hom(N1, N2) HomA(F (N1), F (N2)) is an iso-→morphism for any N1, N2 ∈ M. By condition 2, there exist objects X, Y ∈ C and an exact sequence Y ⊗ M X ⊗ M N1 → 0.→ → Since F is exact, the sequence F (Y ⊗ M) F (X ⊗ M) F (N1) 0→ → → is exact. Since Hom is left exact, the rows in the commutative diagram 0 −→ ⏐⏐�Hom(N1, N2) F −→ ⏐⏐�Hom(X ⊗ M, N2) F −→ ⏐⏐�Hom(Y ⊗ M, N2) F 0 − Hom(F (N1), F (N2)) − Hom(F (X ⊗ M), F (N2)) − Hom(F (Y ⊗ M), F (N2))→ → → are exact. Since by step (1) the second and third vertical arrows are isomorphisms, so is the first one. (3) The functor F is surjective on isomorphism classes of objects of ModC (A). We know (see Exercise 2.9.15) that for any object L ∈ ModC(A) there exists an exact sequence f˜ Y ⊗ A −→ X ⊗ A → L → 0� 120 for some X, Y ∈ C. Let f ∈ Hom(Y ⊗ M, X ⊗ M) be the preimage of f˜ under the isomorphism Hom(Y ⊗M, X⊗M) ∼= HomA(Y ⊗A, X⊗A)= HomA(F (Y ⊗M), F (X⊗M)) ∼and let N ∈ M be the cokernel of f. It is clear that F (N) = L. We proved that F is an equivalence of categories and proved the Theorem. Remark 2.11.3. This Theorem is a special case of Barr-Beck Theorem in category theory, see [ML]. We leave it to the interested reader to ⏐⏐� deduce Theorem 2.11.2 from Barr-Beck Theorem. We have two situations where condition 1 of Theorem 2.11.2 is sat-isfied: 1. M is an arbitrary module category over C and M ∈ M is projec-tive. 2. M is an exact module category and M ∈ M is arbitrary. ⏐⏐�Exercise 2.11.4. Check that in both of these cases Hom(M, ) is exact •(Hint: in the first case first prove that Hom(M, N) is a projective object of C for any N ∈ M). Exercise 2.11.5. Show that in both of these cases condition 2 is equiv-alent to the fact that [M] generates Gr(M) as Z+−module over Gr(C). Thus we have proved Theorem 2.11.6. (i) Let M be a finite module category over C. Then there exists an algebra A ∈ C and a module equivalence M � ModC (A). (ii) Let M be an exact module category over C and let M ∈ M be an object such that [M] generates Gr(M) as Z+−module over Gr(C). Then there is a module equivalence M � ModC (A) where A = Hom(M, M). 2.12. Categories of module functors. Let M1, M2 be two module categories over a multitensor category C, and let (F, s), (G, t) be two module functors M1 → M2. Definition 2.12.1. A module functor morphism from (F, s) to (G, t) is a natural transformation a from F to G such that the following diagram commutes for any X ∈ C, M ∈ M: sF (X ⊗ M) −−−→ X ⊗ F (M) (2.12.1) a id⊗a tG(X ⊗ M) −−−→ X ⊗ G(M)121 It is easy to see that the module functors with module functor mor-phisms introduced above form a category called the category of mod-ule functors. This category is very difficult to manage (consider the case = Vec !) and we are going to consider a subcategory. LetCF unC (M1, M2) denote the full subcategory of the category of module functors consisting of right exact module functors (which are not nec-essarily left exact). First of all this category can be described in down to earth terms: Proposition 2.12.2. Assume that M1 � ModC (A) and M2 � ModC (B) for some algebras A, B ∈ C. The category F unC (M1, M2) is equiva-lent to the category of A − B−bimodules via the functor which sends a bimodule M to the functor • ⊗A M. Proof. The proof repeats the standard proof from ring theory in the categorical setting. � Thus we have the following Corollary 2.12.3. The category F unC(M1, M2) of right exact module functors from M1 to M2 is abelian. Proof. Exercise. � In a similar way one can show that the category of left exact module functors is abelian (using Hom over A instead of tensor product over A). We would like now to construct new tensor categories in the fol-lowing way: take a module category M and consider the category F unC (M, M) with composition of functors as a tensor product. Exercise 2.12.4. The category F unC (M, M) has a natural structure of monoidal category. But in general the category F unC (M, M) is not rigid (consider the case C = Vec!). Thus to get a good theory (and examples of new tensor categories), we restrict ourselves to the case of exact module categories. We will see that in this case we can say much more about the categories F unC (M, M) than in general. 2.13. Module functors between exact module categories. Let M1 and M2 be two exact module categories over C. Note that the category F unC(M1, M2) coincides with the category of the additive module functors from M1 to M2 by Proposition 2.7.8. Exercise 2.13.1. Any object of F unC (M1, M2) is of finite length.� �122 Lemma 2.13.2. Let M1, M2, M3 be exact module categories over C. The bifunctor of composition F unC(M2, M3) × F unC(M1, M2) →F unC (M1, M3) is biexact. Proof. This is an immediate consequence of Proposition 2.7.8. � Another immediate consequence of Proposition 2.7.8 is the following: Lemma 2.13.3. Let M1, M2 be exact module categories over C. Any functor F ∈ F unC (M1, M2) has both right and left adjoint. We also have the following immediate Corollary 2.13.4. Let M1, M2 be exact module categories over C. Any functor F ∈ F unC (M1, M2) maps projective objects to projectives. In view of Example 2.6.6 this Corollary is a