We now assemble the objective. The construction has two levels: an abelian category becomes a module category, and the category of permitted functors between two such categories becomes a category of bimodules. Composition and identities agree through explicit natural isomorphisms.
10.1. Composition and identities
Theorem 10.1. The tensor dictionary preserves composition
Let \({}_S B_R\) and \({}_T C_S\) be bimodules. Their tensor product \(C\otimes_S B\) is a \(T\)-\(R\) bimodule, with
\[t(c\otimes b)=(tc)\otimes b,\qquad(c\otimes b)r=c\otimes(br).\]There are natural isomorphisms
\[T_C\circ T_B\cong T_{C\otimes_S B},\qquad T_{{}_R R_R}\cong1_{R\text{-Mod}}.\]They send \(c\otimes(b\otimes m)\) to \((c\otimes b)\otimes m\), and \(r\otimes m\) to \(rm\), respectively.
Proof.
Both outer actions respect the additive and \(S\)-balancing relations defining \(C\otimes_SB\), since the original actions commute. The left and right actions commute with each other and satisfy their module axioms on pure generators, hence on every tensor. This establishes the asserted bimodule.
The typed associativity and unit isomorphisms were constructed in Chapter 6. The balancing checks in this instance are
\[(cs\otimes b)\otimes m=(c\otimes sb)\otimes m,\qquad(c\otimes br)\otimes m=(c\otimes b)\otimes rm.\]They are exactly the two relations in \(C\otimes_S(B\otimes_RM)\), so reassociating defines a map in both directions. These maps are inverse on pure tensors, which generate. They are \(T\)-linear and natural in \(M\) because a module morphism replaces only \(m\) by its image. The unit inverse is \(m\mapsto1\otimes m\); the relation \(r\otimes m=1\otimes rm\) verifies the other composite. Naturality is the equation \(u(rm)=ru(m)\).
These comparisons are coherent: for four tensor factors, each route through repeated reassociation sends any pure tensor to the same rebracketing with the same four entries. Pure tensors generate every iterated tensor product, so the routes agree as maps. This verifies the associativity pentagon directly. Removing a middle regular-ring factor either by its action first or by reassociation first gives the same tensor by balancing; this verifies the unit triangle. No separate monoidal-category theorem is required.
□Proposition 10.2. Composition of transformations
For bimodule morphisms \(\beta:B\to B'\), \(\gamma:C\to C'\) with the same ring types, composition of their associated functor transformations is represented by
\[\gamma\otimes\beta:C\otimes_SB\to C'\otimes_SB',\qquad c\otimes b\longmapsto\gamma(c)\otimes\beta(b).\]Proof.
The map is additive and balanced, since
\[\gamma(cs)\otimes\beta(b)=\gamma(c)s\otimes\beta(b)=\gamma(c)\otimes s\beta(b)=\gamma(c)\otimes\beta(sb).\]It respects both outer actions because \(\gamma\) is \(T\)-linear and \(\beta\) is right \(R\)-linear. At a module \(M\), the transformation of the composite functors is the map obtained by first applying \(1_C\otimes(\beta\otimes1_M)\), then \(\gamma\otimes1_{B'\otimes_RM}\). It sends \(c\otimes(b\otimes m)\) to \(\gamma(c)\otimes(\beta(b)\otimes m)\). After reassociation this is exactly \((\gamma\otimes\beta)\otimes1_M\). Changing the order of the two operations has the same effect on generators, hence the same effect everywhere. Identities and iterated compositions are respected for the same reason.
□10.2. Returning to abstract abelian categories
Lemma 10.3. Equivalences preserve the hypotheses on functors
An equivalence between abelian categories is additive and preserves kernels, cokernels, and all existing set-indexed coproducts. In particular it is exact and carries small projective generators to small projective generators.
Alphabeta Math: exactness of equivalences · preservation of limits and colimits · preservation of small projective generators
Proof.
Let \(F:\mathcal A\to\mathcal D\) have quasi-inverse \(G\) and natural isomorphisms \(GF\cong1_{\mathcal A}\), \(FG\cong1_{\mathcal D}\). The first comparison shows \(F\) is faithful, and the second shows \(G\) is faithful, by applying the comparison to any equality of image morphisms. To see fullness of \(F\), write \(\zeta:GF\Rightarrow1\) and take \(t:F(X)\to F(Y)\). Set \(u=\zeta_YG(t)\zeta_X^{-1}\). Naturality of \(\zeta\) at \(u\) gives \(GF(u)=G(t)\), and faithfulness of \(G\) gives \(F(u)=t\). Thus \(F\) is fully faithful; the same argument applies to \(G\). Both are essentially surjective because of the comparisons.
Suppose \(L\) is a colimit of a diagram \(D\) in \(\mathcal A\). A cocone from \(FD\) to any object of \(\mathcal D\) may be transported along an isomorphism to a target of the form \(F(X)\), by essential surjectivity. Full faithfulness identifies that cocone with a unique cocone from \(D\) to \(X\), including all its commutation equations. The colimit property gives a unique map \(L\to X\), hence a unique map \(F(L)\to F(X)\). This proves \(F(L)\) is the required colimit. Reversing arrows proves limit preservation. This includes zero objects, coproducts, products, kernels, and cokernels.
Addition in an additive category has the categorical expression \(f+g=\nabla_Y(f\oplus g)\Delta_X\): the diagonal \(\Delta_X:X\to X\oplus X\) has both projections \(1_X\), and the codiagonal \(\nabla_Y:Y\oplus Y\to Y\) has both restrictions \(1_Y\). Preserving zero objects and biproduct diagrams preserves these defining maps and block maps, so it preserves this formula. Hence \(F\) is additive.
For \(P\) a generator, test equality in \(\mathcal D\) after replacing its source by an isomorphic \(F(X)\); full faithfulness identifies the tests from \(F(P)\) with those from \(P\). Thus the generator property transfers. For projectivity, an epimorphism in \(\mathcal D\) can be transported to the image of an epimorphism in \(\mathcal A\); preservation and reflection of the cancellation property follow from full faithfulness and essential surjectivity. Lift a map from \(P\) there, then transport the lift back. For smallness, the canonical Hom-to-coproduct map for \(F(P)\) identifies through Hom bijections and coproduct preservation with that for \(P\), after replacing each family member by an isomorphic image. Set choice suffices to select these replacements for a set-indexed family. All three properties therefore transfer.
□Theorem 10.4. The complete dictionary
Let \(\mathcal A,\mathcal D\) be locally small abelian categories with all set-indexed coproducts and small projective generators \(P,Q\). Supply the chosen constructions of Chapter 8. Put
\[R=\operatorname{End}_{\mathcal A}(P)^{\mathrm{op}},\qquad S=\operatorname{End}_{\mathcal D}(Q)^{\mathrm{op}}.\]The category of additive, right-exact, coproduct-preserving functors \(\mathcal A\to\mathcal D\), with natural transformations, is equivalent to \(\mathscr B(S,R)\), in the size convention of Chapter 9. Explicitly, if \(H_{\mathcal A},L_{\mathcal A}\) and \(H_{\mathcal D},L_{\mathcal D}\) are the equivalences of Chapter 8, then a functor \(F\) corresponds to
\[B_F=H_{\mathcal D}\bigl(F(L_{\mathcal A}(R))\bigr),\]with its right \(R\)-action obtained by applying \(H_{\mathcal D}F L_{\mathcal A}\) to \(\rho_r\). A bimodule \(B\) corresponds to
\[L_{\mathcal D}\circ(B\otimes_R-)\circ H_{\mathcal A}.\]Proof.
Transport a functor by \(W(F)=H_{\mathcal D}F L_{\mathcal A}\), and transport a module functor by \(U(T)=L_{\mathcal D}T H_{\mathcal A}\). On transformations take the same composites: \(W(\eta)_M=H_{\mathcal D}(\eta_{L_{\mathcal A}(M)})\) and \(U(\theta)_X=L_{\mathcal D}(\theta_{H_{\mathcal A}(X)})\). Naturality and preservation of identities and composition follow from functoriality. Lemma 10.3 ensures all these functors satisfy the three required hypotheses.
Write \(\epsilon_{\mathcal A}:L_{\mathcal A}H_{\mathcal A}\Rightarrow1_{\mathcal A}\) and \(\nu_{\mathcal A}:H_{\mathcal A}L_{\mathcal A}\Rightarrow1_{R\text{-Mod}}\), with analogous notation for \(\mathcal D\). The comparisons
\[UW(F)=L_{\mathcal D}H_{\mathcal D}F L_{\mathcal A}H_{\mathcal A}\xrightarrow{\epsilon_{\mathcal D}}F L_{\mathcal A}H_{\mathcal A}\xrightarrow{F\epsilon_{\mathcal A}}F,\] \[WU(T)=H_{\mathcal D}L_{\mathcal D}T H_{\mathcal A}L_{\mathcal A}\xrightarrow{\nu_{\mathcal D}}T H_{\mathcal A}L_{\mathcal A}\xrightarrow{T\nu_{\mathcal A}}T\]are invertible at every object, natural in their object arguments, and natural in \(F\) or \(T\). The last assertion is precisely naturality of a transformation at the comparison morphisms. Thus \(U,W\) are inverse up to natural isomorphism. Compose this equivalence with Corollary 9.6. Evaluation of \(W(F)\) at \(R\) gives exactly \(B_F\), proving the displayed formulas and the transformation correspondence.
For composable \(F:\mathcal A\to\mathcal D\) and \(G:\mathcal D\to\mathcal E\), the comparison \(L_{\mathcal D}H_{\mathcal D}\cong1_{\mathcal D}\) gives \(W(G)W(F)\cong W(GF)\). Chapter 9 and Theorem 10.1 therefore identify \(B_{GF}\) with \(B_G\otimes_SB_F\) as an outer bimodule, using the transformation classification to obtain that bimodule isomorphism. For an identity functor, \(H_{\mathcal A}L_{\mathcal A}\cong1_{R\text{-Mod}}\) identifies its bimodule with the regular \(R\)-\(R\) bimodule. Composition and identities have now been checked at both levels.
□The distinction between the levels matters. The theorem does not identify abelian categories themselves with individual bimodules. It identifies suitable abelian categories with module categories; for each pair of them, it identifies their permitted functors with bimodules and their transformations with bimodule maps. Different choices of generators can produce different rings with equivalent module categories.
10.3. When is a tensor functor an equivalence?
Theorem 10.5. Morita equivalence is a pair of inverse bimodules
Rings \(R,S\) are Morita equivalent when \(R\text{-Mod}\) and \(S\text{-Mod}\) are equivalent. This holds exactly when there are bimodules \({}_SB_R\), \({}_RC_S\) and bimodule isomorphisms
\[\boxed{C\otimes_SB\cong{}_RR_R,\qquad B\otimes_RC\cong{}_SS_S.}\]In that case \(T_B\) and \(T_C\) are quasi-inverse equivalences.
Proof.
If \(F:R\text{-Mod}\to S\text{-Mod}\) is an equivalence with quasi-inverse \(G\), Lemma 10.3 says both are additive, right exact, and coproduct-preserving. Theorem 9.4 gives \(F\cong T_B\), \(G\cong T_C\), where \(B=F(R)\) and \(C=G(S)\). Their inverse-functor comparisons and Theorem 10.1 give natural isomorphisms
\[T_{C\otimes_SB}\cong T_C T_B\cong GF\cong1\cong T_R,\] \[T_{B\otimes_RC}\cong T_B T_C\cong FG\cong1\cong T_S.\]Theorem 9.5 identifies these with bimodule morphisms. Applying the same correspondence to each inverse natural transformation proves those bimodule morphisms are isomorphisms.
Conversely, if the two displayed bimodule isomorphisms are given, tensoring them with each module, then applying associativity and the unit maps, gives natural isomorphisms \(T_C T_B\cong T_R\cong1\) and \(T_B T_C\cong T_S\cong1\). Thus these tensor functors are quasi-inverses by the definition of equivalence in Chapter 8. No compatibility beyond these two natural isomorphisms is required by that definition.
□Example 10.6. A field and its two-by-two matrix ring
Let \(k\) be a field and \(S=M_2(k)\). Matrix addition is entrywise and \((AB)_{ij}=\sum_{\ell=1}^2A_{i\ell}B_{\ell j}\). Distributivity follows entrywise; associativity follows because both triple-product entries are \(\sum_{\ell,p}A_{i\ell}B_{\ell p}C_{pj}\); the identity matrix is a unit. Hence \(S\) is a ring.
Take \(B=k^2\), columns with left matrix action and right scalar action, and \(C=k^{1\times2}\), the space of two-entry rows with left scalar action and right matrix action. The actions commute by matrix associativity. Column-row multiplication gives an \(S\)-\(S\) bimodule map
\[B\otimes_kC\longrightarrow S,\qquad b\otimes c\longmapsto bc.\]For standard columns \(e_i\), standard rows \(e_j^*\), and matrix units \(E_{ij}\), this sends \(e_i\otimes e_j^*\) to \(E_{ij}\). Its inverse sends \(A\) to \(\sum_{i,j}A_{ij}e_i\otimes e_j^*\). Expanding any column and row in these two coordinates verifies both composites. Multiplication respects both outer matrix actions, so the isomorphism and its inverse are bimodule maps.
Row-column multiplication is balanced over \(S\), since \((cA)b=c(Ab)\). It gives a \(k\)-\(k\) bimodule map \(C\otimes_SB\to k\), \(c\otimes b\mapsto cb\). Every tensor expands into sums of scalar multiples of \(e_i^*\otimes e_j\), and balancing gives
\[e_i^*\otimes e_j=e_1^*E_{1i}\otimes e_j=e_1^*\otimes E_{1i}e_j=\delta_{ij}e_1^*\otimes e_1.\]Here \(\delta_{ij}\) is \(1\) for \(i=j\) and \(0\) otherwise. Thus the inverse is \(a\mapsto a e_1^*\otimes e_1\); the preceding equation and \(e_1^*e_1=1\) verify both composites. We have proved both inverse-bimodule identities, so Theorem 10.5 gives \(k\text{-Mod}\simeq M_2(k)\text{-Mod}\).
Sources and further reading
- Alphabeta Math, Morita equivalence and invertible bimodules, especially the Eilenberg–Watts dictionary with composition, Morita equivalence and inverse bimodules, and the explicit matrix-ring pair.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. Composition multiplies the bimodule maps
Applies to: Theorem 10.1, Proposition 10.2.
Take all three rings to be \(\mathbb Z\), \(B=\mathbb Z/6,C=\mathbb Z/4\). Their tensor is \(\mathbb Z/2\): its generator is killed by 6 and 4, hence by 2, and the balanced product modulo two shows it has order 2. Thus \(T_C(T_B(M))\cong M/(6M+4M)=M/2M\), since \(6M+4M=2M\), exactly the functor of \(C\otimes B\). For \(\beta(b)=3b\), \(\gamma(c)=2c\), the tensor map multiplies the generator by 6 and is zero. Acting successively on \(c\otimes(b\otimes m)\) also gives \(2c\otimes(3b\otimes m)=6c\otimes(b\otimes m)=0\). This checks the composition formula for transformations. The identity bimodule \(\mathbb Z\) acts by \(n\otimes m\mapsto nm\), with inverse \(m\mapsto1\otimes m\), verifying the identity clause.
Worked example 2. Transporting the quotient functor through an equivalence
Applies to: Lemma 10.3, Theorem 10.4.
Use \(\mathcal A=\mathrm{Ab},P=\mathbb Z\) and \(\mathcal D=\mathbb F_2\text{-}\mathrm{Mod},Q=\mathbb F_2\). Evaluation at 1 identifies both reconstruction functors with the usual underlying modules; their inverses equip a \(\mathbb Z\)-module with its abelian-group structure, or keep the \(\mathbb F_2\)-module structure. These are additive equivalences. They preserve kernels, quotient cokernels, and arbitrary sums because the objects and structure maps are the same. In particular the image of the regular generator still lifts elements, detects maps, and has finite support on sums.
For \(F(M)=M/2M\), the transported bimodule is \(H_{\mathcal D}(F(L_{\mathcal A}(\mathbb Z)))=\mathbb F_2\), with right integer action given by reduction. Reversing the dictionary produces \(\mathbb F_2\otimes_{\mathbb Z}M\cong M/2M\) by the explicit inverse formulas of chapter 6. Thus the complete dictionary computes both actions and recovers this functor after transport, with its hypotheses checked rather than assumed.
Further proof details
The regular generators used here have the required properties by their one-element descriptions. In \(\mathbb Z\)-modules and in \(\mathbb F_2\)-modules, a map out of the regular module is uniquely \(r\mapsto rm\). If the target map is onto, a chosen preimage of \(m\) defines a lift. If two maps differ at \(m\), the regular-module map \(1\mapsto m\) detects that difference. For a map into a direct sum, its value at 1 has finite support, and its coordinate maps are therefore finitely supported and uniquely determined. These prove projectivity, the generator condition, and smallness directly. Under the identity descriptions of the equivalences, they apply unchanged, and kernel and quotient factorization maps are literally retained.
Worked example 3. The two-by-two Morita pair, checked by matrix units
Applies to: Theorem 10.5.
Let \(R=k\) be a field, \(S=M_2(k)\), \({}_SB_R=k^2\) (columns), and \({}_RC_S=(k^2)^*\) (rows). Matrix associativity verifies the commuting actions. The map \(C\otimes_S B\to k\), \(r\otimes v\mapsto rv\), is balanced. Writing the standard column and row bases as \(e_i,e_i^*\), balancing gives \(e_i^*\otimes e_j=0\) for \(i\ne j\), by inserting \(E_{ii}\), and \(e_2^*\otimes e_2=e_1^*\otimes e_1\), by inserting \(E_{12}\). Hence the map is an isomorphism, inverse \(a\mapsto a e_1^*\otimes e_1\). The other map \(B\otimes_k C\to S\), \(v\otimes r\mapsto vr\), takes the four basis tensors to the four matrix units and respects both \(S\)-actions, so is a bimodule isomorphism. Associativity and these two maps give \(C\otimes_S(B\otimes_k V)\cong V\) and \(B\otimes_k(C\otimes_S W)\cong W\), naturally. This verifies every inverse-bimodule hypothesis and the resulting equivalence.
Counterexamples and limits of the hypotheses
Each construction below refutes the explicitly stated stronger claim. It does not refute the related theorem with its original hypotheses.
Counterexample 1. One inverse tensor identity is not enough
Related statements: Theorem 10.5.
Claim refuted. It suffices to require only C⊗B to be the regular bimodule in the Morita criterion.
Proof.
Let \(R=k,S=k\times k\), and \(B=C=k\), with types \({}_SB_R\), \({}_RC_S\). Let both \(S\)-actions use the first projection: \((a,b)x=ax\) and \(x(a,b)=xa\); the \(k\)-actions are ordinary scalar multiplication. They commute and are unital. The map \(C\otimes_SB\to k\), \(x\otimes y\mapsto xy\), is balanced and has inverse \(z\mapsto1\otimes z\), since \(x\otimes y=1\otimes xy\). Thus \(C\otimes_SB\cong R\).
But \(B\otimes_kC\cong k\) has the second idempotent \(e_2=(0,1)\) acting as zero, whereas it acts nontrivially on \({}_SS_S\), since \(e_2e_2=e_2\ne0\). A bimodule isomorphism would preserve that action, so no such isomorphism exists. The functors show the same failure: an \(S\)-module decomposes as \(e_1W\oplus e_2W\) because \(e_1+e_2=1,e_1e_2=0\). Tensoring by \(C\) keeps only \(e_1W\); tensoring back by \(B\) cannot recover the nonzero module \((0,k)\). Both inverse equations in the theorem are necessary.
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