Draft lesson.

Cauchy completion and Morita equivalence of many-object modules

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

The module category of a small preadditive category remembers its compact projective objects. These are the retracts of finite sums of representables. Adding those sums and retracts gives the Cauchy completion, and does not change the full module category. We prove this with explicit contravariant matrix and image formulas, then obtain the Morita criterion.

Prerequisites: Modules and tensor functors over small preadditive categories, Sections 1–2, Formal linear combinations and finite sums, Section 4, and Completing a category by retracts. The compact-object theorem for Ind categories has a different ambient category; projectivity is essential in the module classification below.

1. The small completion and its identities

Fix a universe and a small preadditive category \(\mathcal C\). Right modules are additive functors \(\mathcal C^{\mathrm{op}}\to\mathsf{Ab}\), with group-homomorphism components in every module morphism. An object is compact here when its Hom functor preserves small filtered colimits. Projectivity means that maps from it lift through epimorphisms. The module category has the pointwise abelian and colimit structures and representable probes \(h_c=\mathcal C(-,c)\) proved in the first prerequisite.

Let \(\mathcal B=\operatorname{Mat}(\mathcal C)\). Its objects are finite lists \(X=(c_1,\ldots,c_n)\); a map to \(Y=(d_1,\ldots,d_m)\) is a matrix \(a_{ji}:c_i\to d_j\). Composition is \((ba)_{\ell i}=\sum_j b_{\ell j}a_{ji}\). The empty list is zero, and concatenation gives the specified finite biproducts. The complete matrix identities and all additive transformation extensions are retained from the second prerequisite.

Define

An object is \((X,e)\), with \(e^2=e\in\mathcal B(X,X)\). A map \(a:(X,e)\to(Y,f)\) satisfies \(a=fae\), equivalently \(fa=a=ae\). Its identity is \(e\), and composition is matrix composition. Corner Hom groups are closed under addition. The biproduct of two objects is \((X\oplus Y,\operatorname{diag}(e,f))\): compressed block inclusions and projections have composites \(e,f\), zero cross terms, and sum equal to \(\operatorname{diag}(e,f)\), the completed identity.

This category is small: finite lists form a small set, their finite matrix Hom groups are small, and their idempotents form subsets of those groups. It is additive and idempotent complete. Indeed an idempotent \(b\) on \((X,e)\) satisfies \(eb=b=be\); the two corner maps represented by \(b\) split it through \((X,b)\). The canonical embedding \(i:\mathcal C\to\mathcal K_{\mathcal C}\) sends \(c\) to \(((c),1_c)\) and is fully faithful. We call (1.1) the Cauchy completion.

2. Which modules are compact projective?

Theorem 2.1. A right \(\mathcal C\)-module is compact projective if and only if it is a retract of a finite coproduct of representables.

Proof. A finite sum \(F=\bigoplus_{r=1}^n h_{c_r}\) is projective by lifting its finitely many component maps. It is compact because its Hom set is the finite product of the groups \(M(c_r)\); their filtered comparisons have common-stage representatives and equalities, including the zero-factor singleton. A retract \(P\xrightarrow s F\xrightarrow r P\), \(rs=1\), is projective: lift a map \(P\to N\) after precomposing with \(r\), then restrict the lift along \(s\).

It is compact as well. Given \(P\to\operatorname{colim}M_j\), precompose with \(r\) to obtain a map from \(F\). Compactness of \(F\) represents it at a stage; restricting along \(s\) represents the original map. If two maps from \(P\) at stages become equal, their precomposites with \(r\) from \(F\) become equal at a common later stage. Restricting along \(s\) proves equality of the original stage maps there. These are surjectivity and injectivity of the canonical compactness comparison for \(P\).

Conversely, use the actual epic presentation \(P_0=\bigoplus_{(c,m)}h_c\to P\) from the first prerequisite. Projectivity supplies a section \(s:P\to P_0\). The small directed set of finite subsets of the summand index presents \(P_0\) as the filtered colimit of its finite partial sums, with their inclusions: compatible maps on those partial sums are exactly a family of maps on all summands. Compactness factors \(s\) through one finite sum \(F\). Composing its inclusion with the original epimorphism gives \(r:F\to P\), with \(rs_F=1_P\). Thus \(P\) is the required retract. The empty finite sum includes the zero module. \(\square\)

Write \(\mathsf{CP}(\mathcal C)\) for the full category of these compact projectives. Define \(P_X=\bigoplus_i h_{c_i}\). Additive Yoneda identifies a matrix \(a:X\to Y\) with the module map \(P(a):P_X\to P_Y\) having the corresponding entries. These Hom identifications respect addition and composition, so \(X\mapsto P_X\) is fully faithful.

For an idempotent \(e\) on \(X\), put \(J(X,e)=\operatorname{im}P(e)\), with inclusion \(\iota_e\) and corestricted projection \(\pi_e\). Pointwise idempotency gives \(\pi_e\iota_e=1\) and \(\iota_e\pi_e=P(e)\). For a corner map, set

The corner equations make identities and composition hold: the inserted middle factor \(\iota_f\pi_f=P(f)\) is absorbed by the adjacent corner maps. This gives a functor to \(\mathsf{CP}(\mathcal C)\).

Theorem 2.2. The functor \(J:\mathcal K_{\mathcal C}\to\mathsf{CP}(\mathcal C)\) is an additive equivalence.

Proof. For a module map \(v:J(X,e)\to J(Y,f)\), lift it to \(\iota_fv\pi_e:P_X\to P_Y\). Matrix Yoneda supplies its unique matrix \(a\). The two splitting identities make \(P(fae)=P(a)\), so faithfulness gives \(fae=a\). Restricting this matrix by (2.1) returns \(v\). Conversely lifting (2.1) for a corner matrix gives \(P(fae)=P(a)\), recovering the matrix. These are inverse Hom maps, additive and compatible with composition. Theorem 2.1 represents every compact projective as a retract of some \(P_X\); the associated idempotent \(sr\) has image isomorphic to that retract by the given \(s,r\). This proves essential surjectivity. \(\square\)

Thus \(\mathsf{CP}(\mathcal C)\) is essentially small even when the full module category is not small. This classification concerns compact projective modules. Exercise 2 exhibits a compact module outside it.

3. Extending right modules through finite sums

Let \(s:\mathcal C\to\mathcal B\) be the singleton embedding. For \(M\in\mathsf{Mod}\text{-}\mathcal C\), define

The direction is \(\widehat M(Y)\to\widehat M(X)\). The matrix multiplication and contravariant law give \(\widehat M(ba)=\widehat M(a)\widehat M(b)\): in coordinate \(i\), both sides are \(\sum_{j,\ell}M(a_{ji})M(b_{\ell j})z_\ell\), since additivity identifies this with \(\sum_\ell M(\sum_jb_{\ell j}a_{ji})z_\ell\). Identity matrices give identities, including the identity of the empty zero group. Entrywise sums give additivity. A module morphism \(\theta:M\to N\) extends by the direct sum of its component maps; its matrix naturality is the original naturality at each entry.

Theorem 3.1. Restriction \(s^*:\mathsf{Mod}\text{-}\mathcal B\to\mathsf{Mod}\text{-}\mathcal C\) and (3.1) are inverse equivalences, including all module morphisms.

Proof. Restricting (3.1) to singleton objects gives the canonical identification with \(M\), naturally on its maps. For a right \(\mathcal B\)-module \(N\), the comparison

has components \(N(i_i)\), where \(i_i:(c_i)\to X\) are the block inclusions. Its inverse is \(\sum_iN(p_i)\), using the block projections \(p_i:X\to(c_i)\). Contravariance and the biproduct equations prove both inverse identities. In the empty case both groups are zero because an additive functor sends the zero identity to zero.

Every matrix is \(a=\sum_{j,i}i_j a_{ji}p_i\). Applying the additive contravariant \(N\) shows that (3.2) intertwines its action with (3.1), so it is natural on every matrix. Naturality of a module transformation at each \(i_i,p_i\) makes the comparisons natural in \(N\). Their singleton comparisons and the comparison of an already extended module are the identity after the canonical one-term-sum identifications. Thus the two functors with these comparisons are inverse equivalences on objects and all maps. \(\square\)

4. Extending through every projector

Let \(j:\mathcal B\to\operatorname{Kar}(\mathcal B)\) send \(X\) to \((X,1_X)\). For a right \(\mathcal B\)-module \(N\), define

Since \(a=fae\), contravariance gives \(N(a)=N(e)N(a)N(f)\), so its image is in \(\operatorname{im}N(e)\). The identity \(e\) acts as identity on its image, and restricted composition is composition. Addition is inherited from the ambient groups. Thus (4.1) is an additive right module. A module transformation \(\theta:N\to N'\) maps each image into the corresponding image by its naturality at \(e\), and its restrictions define the extended transformation. The zero projector gives the zero group.

Theorem 4.1. Restriction \(j^*\) and (4.1) are inverse module-category equivalences with their actual comparison maps.

Proof. Restriction of (4.1) gives \(N\), since the image of its identity is its whole group. Conversely let \(Q\) be a module on the completion. The corner maps

are both represented by \(e\). They satisfy \(p_ei_e=1_{(X,e)}\) and \(i_ep_e=e\) as an endomorphism of \((X,1_X)\). Consequently \(Q(p_e)\) is an injection with image exactly \(\operatorname{im}Q(e)\); its inverse there is \(Q(i_e)\). These are the comparison between \(Q(X,e)\) and \(\widetilde{j^*Q}(X,e)\), with both inverse identities furnished by the two displayed retract equations.

For a corner map \(a:(X,e)\to(Y,f)\), the identity \(ap_e=p_fj(a)\) holds as a map \((X,1_X)\to(Y,f)\), since its two underlying matrices are \(ae=a=fa\). Applying \(Q\) gives precisely naturality of the comparison. Naturality of a module morphism at \(i_e,p_e\) proves naturality in \(Q\). On \(jX\) the comparison is the identity; for a module already constructed by (4.1), it is identity on the indicated image subgroup. Thus restriction and image extension, with these comparisons, are inverse on all objects and all transformations. No choice of a splitting in the original category is required. \(\square\)

Combining Theorems 3.1 and 4.1 gives a canonical equivalence

Its extension evaluates at \((X,e)\) as \(\operatorname{im}\widehat M(e)\), with the contravariant action (3.1). The module need not be compact or projective to extend this way.

5. The Morita criterion

Theorem 5.1. For small preadditive categories \(\mathcal C,\mathcal D\), their right-module categories are equivalent if and only if \(\mathcal K_{\mathcal C}\) and \(\mathcal K_{\mathcal D}\) are additively equivalent.

Proof, forward. An ordinary module-category equivalence preserves all existing limits and colimits by their universal properties. It is additive: zero maps and the canonical biproduct projections, injections, diagonals and codiagonals recover the sum \(f+g=\nabla(f\oplus g)\Delta\), and an equivalence preserves these structures. Epimorphisms are preserved and reflected by cancellation, so their lifting property preserves and reflects projective objects. The natural Hom bijections and preservation of filtered cocones likewise preserve and reflect compactness. Hence the equivalence restricts to an additive equivalence of compact-projective subcategories. Theorem 2.2 identifies these with the two Cauchy completions, proving the assertion.

Proof, reverse. Choose an additive equivalence \(U:\mathcal K_{\mathcal C}\to\mathcal K_{\mathcal D}\) and an additive inverse \(V\) with their natural inverse comparisons. Precomposition by \(V^{\mathrm{op}}\) sends right modules on \(\mathcal K_{\mathcal C}\) to right modules on \(\mathcal K_{\mathcal D}\); precomposition by \(U^{\mathrm{op}}\) gives its inverse. Whiskering the chosen inverse comparisons gives the natural inverse module comparisons at every object and every group-homomorphism component. Compose these equivalences with (4.3) on both sides. Explicitly, extend a \(\mathcal C\)-module by the matrix and image formulas, precompose with \(V^{\mathrm{op}}\), and restrict along \(\mathcal D\to\mathcal K_{\mathcal D}\). The opposite construction is its inverse by Theorems 3.1–4.1 and the comparisons for \(U,V\). This proves the module equivalence on all maps. \(\square\)

An ordinary equivalence between the additive Cauchy categories is automatically additive by the same finite-biproduct argument, so that phrasing gives the same criterion. If the categories are linear over a specified field, ordinary equivalence does not by itself preserve that field's scalar action; Exercise 4 separates these requirements. Empty \(\mathcal C\) has a zero module category and a Cauchy completion with its single zero object, and is covered by all the arguments above.

6. Four graded exercises with complete solutions

Exercise 1 — basic. Let \(\mathcal C\) be the one-object category of a field \(k\). For the two-term list \(X\), take

Find \(J(X,e)\), the value of an extended right \(k\)-module \(V\) at \((X,e)\), and the corner endomorphism ring. Keep the covariant realization and contravariant module action distinct.

Solution. The realization \(P_X=k^2\) uses column vectors, so \(P(e)(a,b)=(a+b,0)\). Its image is the first axis and is isomorphic to the one-dimensional right module \(k\). The contravariant formula (3.1), on the other hand, sends \((v_1,v_2)\in V^2\) to \((v_1,v_1)\). Its image is the diagonal copy of \(V\). A corner endomorphism is a matrix \(a=eae\). Multiplying shows that every such matrix is \(\lambda e\), \(\lambda\in k\), and these are all the possibilities. Addition and multiplication correspond to those of \(k\), with completed identity \(e\), not the two-by-two identity matrix. On the diagonal module value, \(\lambda e\) acts by scalar \(\lambda\). The two different ambient images describe respectively the projective module and the maps out of that module; they are compatible with the contravariant variance.

Exercise 2 — intermediate. For the one-object category of \(\mathbb Z\), show that \(P=\mathbb Z/d\mathbb Z\), \(d\ge2\), is compact in its module category but is not in the image of \(J\).

Solution. A map \(P\to M\) is determined by an element \(m\in M\) with \(dm=0\). In a filtered group colimit, represent \(m\) at a stage. Its relation \(dm=0\) becomes true at a later stage by the zero-representative criterion, giving the required factor from \(P\). Equality of two such maps becomes equality of their single generator images at a common stage. Thus the canonical filtered Hom comparison is bijective and \(P\) is compact. It is not projective: the identity \(P\to P\) cannot lift through \(\mathbb Z\to P\), since every homomorphism \(P\to\mathbb Z\) is zero by torsion-freeness of \(\mathbb Z\). Its identity is nonzero. Theorem 2.2 therefore excludes it from \(J\)'s image. Compactness without projectivity does not characterize the module Cauchy completion.

Exercise 3 — advanced. Let \(R=k\times k\), and let \(\mathcal C\) be its one-object category. Let \(\mathcal D\) have two objects \(p,q\), each with endomorphism ring \(k\), and zero cross Hom groups. Give a module equivalence, describe their Cauchy completions, and explain why their original categories and finite matrix categories need not be equivalent.

Solution. For \(e=(1,0)\), a right \(R\)-module \(M\) decomposes as \(Me\oplus M(1-e)\). The map \(m\mapsto(me,m(1-e))\) has inverse \((x,y)\mapsto x+y\), by the two orthogonal idempotent identities. These are natural on all module maps. They give an equivalence with pairs of \(k\)-vector spaces, the right \(\mathcal D\)-modules, whose inverse gives componentwise \(R\)-action. The input categories cannot be equivalent: \(\mathcal C\) has one object, while \(p,q\) are not isomorphic, since all cross maps are zero and their identities are nonzero.

Both Cauchy completions identify with pairs of finite-dimensional vector spaces and their componentwise linear maps. For \(\mathcal C\), an idempotent on a finite free \(R^n\) is a pair of idempotent matrices, whose images are finite-dimensional pairs. Conversely a pair \((k^a,k^b)\) is the image of the diagonal projector on \(R^{a+b}\) using \(e\) in \(a\) entries and \(1-e\) in \(b\) entries. The corner Hom identification in Theorem 2.2 gives all component maps. For \(\mathcal D\), finite sums of \(h_p=(k,0)\) and \(h_q=(0,k)\) already give all finite-dimensional pairs; their projectors add no further isomorphism classes. This gives their Cauchy equivalence.

Finite sums alone do not give the same result. Every object of \(\operatorname{Mat}(\mathcal C)\) corresponds to a pair \((k^n,k^n)\). Its projector \(e\) on the one-term list has image \((k,0)\), which is not any such object; it cannot split in that matrix category, since a splitting object would realize its image. In \(\operatorname{Mat}(\mathcal D)\) all idempotents split by taking their componentwise image vector spaces, which again have finite dimensions and are represented by finite lists. Idempotent completeness is preserved by equivalence. Thus these two matrix categories are not equivalent, despite equivalence after the required Cauchy completion.

Exercise 4 — synthesis. Take \(k=\mathbb C\) and its conjugation automorphism \(\sigma\). On right \(k\)-modules, keep the underlying abelian groups and maps but change the action to \(m\star a=m\sigma(a)\). Prove that this is an ordinary additive module-category equivalence, and describe its corresponding operation on matrix and corner data. Is it \(\mathbb C\)-linear?

Solution. The new action satisfies the module axioms because conjugation preserves sums, products and one. Old module maps respect the new action, so this defines a functor. Applying it twice gives the original action and maps because \(\sigma^2=1\). Thus it is an additive equivalence. On the Cauchy description, conjugate every matrix entry, sending \((X,e)\) to \((X,\sigma(e))\) and \(a\) to \(\sigma(a)\). Conjugation respects matrix sums, composition, idempotency and the equation \(a=fae\), so this too is an additive involutive equivalence. Restricting right modules along the one-object conjugation functor gives precisely the stated action twist; the matrix and image extensions retain it on the completion.

It is not \(\mathbb C\)-linear. In the original Hom group, multiplying a module map \(f\) by \(a\) means \(m\mapsto f(m)a\). Its image under the twist is that same underlying map. Scalar multiplication by \(a\) in the new target Hom group instead gives \(m\mapsto f(m)\sigma(a)\). Taking \(f=1_k\) and \(a=i\) makes these different maps. Thus ordinary additive Morita equivalence and equivalence preserving a chosen ground-field action are different requirements. The criterion above states the former; a field-linear version must require field-linear equivalences throughout.

References

• Manuel Saorín, Carlos Parra and Simone Virili, Torsion pairs in categories of modules over a preadditive category, Corollaries 1.5–1.6, for the projective-completion and module Morita framework. The proofs here supply the compactness factors, all contravariant extension formulas and the actual module comparison maps.
• The linked course prerequisites contain the many-object module presentations, finite matrix construction and full Karoubi functor universal property. The Ind compact theorem concerns its stated formal completion, rather than every finitely presented module.

Original exposition and worked solutions in this lesson: CC0-1.0.