A single object can determine an entire abelian category. Its endomorphisms supply the ring, and morphisms from that object supply the modules. We prove exactly which hypotheses make this work.
8.1. The hypotheses and the ring
Throughout this chapter, \(\mathcal A\) is an abelian category with coproducts indexed by every set. It is locally small: every collection \(\operatorname{Hom}_{\mathcal A}(X,Y)\) is a set. Its objects may form a proper class. All module underlying collections and coproduct indices are sets. We use sets and classes with global choice to fix coproducts and cokernels once and for all; alternatively these chosen constructions may be supplied as part of the data. No choice of a presentation will be made when defining the inverse functor.
Global choice here means a class assignment choosing a member of each nonempty set. Ordinary set choice alone does not silently choose constructions across a proper class of diagrams. None of the proofs of the Hom bijections below depends on the chosen representatives.
Alphabeta Math: small and locally small categories
Definition 8.1. Small projective generator
An object \(P\) is projective if, for every epimorphism \(e:X\to Y\) and morphism \(f:P\to Y\), some \(g:P\to X\) satisfies \(eg=f\). It is a generator if \(u,v:X\to Y\) and \(uf=vf\) for every \(f:P\to X\) imply \(u=v\). It is small if the canonical homomorphism
\[\bigoplus_{i\in I}\operatorname{Hom}(P,X_i)\longrightarrow\operatorname{Hom}\!\left(P,\coprod_{i\in I}X_i\right),\qquad (f_i)\longmapsto\sum_i\iota_i f_i,\]is bijective for every set-indexed family. The sum is finite because \((f_i)\) has finite support. “Small” is a property of this Hom functor; it does not mean that \(P\) has finitely many elements.
Put \(E=\operatorname{End}_{\mathcal A}(P)\), with addition from the additive category and multiplication \(ab=a\circ b\). Bilinearity of composition gives both distributive laws, associativity comes from composition, and \(1_P\) is the unit. Thus \(E\) is a ring. Set \(R=E^{\mathrm{op}}\), the same additive group with multiplication \(a^{\mathrm{op}}b^{\mathrm{op}}=(b\circ a)^{\mathrm{op}}\).
8.2. The Hom functor
Definition 8.2. The module attached to an object
Define \(H:\mathcal A\to R\text{-Mod}\) by
\[H(X)=\operatorname{Hom}_{\mathcal A}(P,X),\qquad a^{\mathrm{op}}\cdot f=f\circ a,\qquad H(u)(f)=u\circ f.\]The module addition is addition of morphisms. The two distributive laws follow from bilinear composition. Further,
\[a^{\mathrm{op}}\cdot(b^{\mathrm{op}}\cdot f)=f\circ b\circ a=(a^{\mathrm{op}}b^{\mathrm{op}})\cdot f.\]The identity acts as the identity, so this is a unital left module. Associativity also shows that \(H(u)\) is \(R\)-linear and that \(H\) preserves composition and identities. Bilinearity makes \(H\) additive.
Lemma 8.3. Exactness, detection, and coproducts
If \(P\) is a small projective generator, then \(H\) preserves kernels and cokernels, preserves set-indexed coproducts, is faithful, and reflects isomorphisms. Here faithful means that its maps on Hom sets are injective, and reflects isomorphisms means \(H(u)\) invertible implies \(u\) invertible.
Proof.
For a kernel \(k:K\to X\) of \(u:X\to Y\), the kernel universal property says exactly that composition with \(k\) identifies \(H(K)\) with \(\ker H(u)\). If \(e:X\to Y\) is epic, projectivity says \(H(e)\) is surjective.
For a cokernel \(c:Y\to Z\) of \(u:X\to Y\), Chapter 4 gives a factorization \(u=j p\), where \(p:X\to I\) is epic and \(j:I\to Y\) is the kernel of \(c\). Kernel preservation gives \(\ker H(c)=\operatorname{im}H(j)\); surjectivity of \(H(p)\) gives \(\operatorname{im}H(u)=\operatorname{im}H(j)\). Also \(H(c)\) is surjective. Thus \(H(Z)\), with \(H(c)\), is the cokernel of \(H(u)\) in modules. These are precisely the needed exactness assertions. Smallness is coproduct preservation by definition.
The generator condition is faithfulness of \(H\). If \(H(Z)=0\), every morphism \(P\to Z\) is zero, so the generator condition applied to \(1_Z\) and \(0\) gives \(1_Z=0\). Every morphism into or out of \(Z\) is then zero, so \(Z\) is a zero object. If \(H(u)\) is invertible, kernel and cokernel preservation show \(H(\ker u)=H(\operatorname{coker}u)=0\). Therefore \(\ker u\) and \(\operatorname{coker}u\) are zero. By Chapter 4, \(u\) is both monic and epic, hence invertible.
□8.3. Presentations by copies of the generator
Lemma 8.4. Every object has a generator presentation
For every \(X\), there are sets \(I,J\) and a diagram
\[P^{(J)}\xrightarrow{d}P^{(I)}\xrightarrow{q}X\longrightarrow0\]in which \(q\) is the cokernel of \(d\). The notation \(P^{(I)}\) means the coproduct of \(I\) copies of \(P\).
Proof.
Take the set \(I=H(X)\). Define \(q:P^{(I)}\to X\) by \(q\iota_f=f\) for each morphism \(f:P\to X\). If \(vq=wq\), then \(vf=wf\) for every \(f\), hence \(v=w\). Thus \(q\) is epic, using its cancellation definition. Let \(k:K\to P^{(I)}\) be its kernel. Apply the same construction to \(K\): with \(J=H(K)\), there is an epimorphism \(q_K:P^{(J)}\to K\). Put \(d=kq_K\). A morphism annihilates \(d\) exactly when it annihilates \(k\), because \(q_K\) is epic. Since an epimorphism in an abelian category is the cokernel of its kernel, \(q\) is the cokernel of \(d\).
□Lemma 8.5. Morphisms are determined by module maps
For every \(X,Y\), composition induces a bijection
\[\operatorname{Hom}_{\mathcal A}(X,Y)\longrightarrow\operatorname{Hom}_R(H(X),H(Y)),\qquad u\longmapsto H(u).\]Thus \(H\) is fully faithful: its maps on Hom sets are bijections.
Proof.
First there is a left-module isomorphism \(R\to H(P)\), \(a^{\mathrm{op}}\mapsto a\): multiplication by \(b^{\mathrm{op}}\) corresponds to \(a\circ b\), exactly the prescribed action. Smallness therefore gives isomorphisms
\[\theta_I:R^{(I)}\xrightarrow{\sim}H(P^{(I)}),\qquad e_i\longmapsto\iota_i,\]where \(e_i\) is the basis vector with coordinate \(1_R\). A morphism \(v:P^{(I)}\to Y\) is specified by its restrictions \(v\iota_i\), and an \(R\)-linear map \(R^{(I)}\to H(Y)\) is specified by its values on \(e_i\). Consequently these two types of maps correspond bijectively through \(H(v)\theta_I\).
Now take a presentation \(P^{(J)}\xrightarrow dP^{(I)}\xrightarrow qX\to0\) from Lemma 8.4 and an \(R\)-linear map \(t:H(X)\to H(Y)\). The preceding bijection supplies a unique \(v:P^{(I)}\to Y\) with \(H(v)=tH(q)\). Then \(H(vd)=tH(qd)=0\). Faithfulness implies \(vd=0\). The cokernel property gives a unique \(u:X\to Y\) with \(uq=v\). Since \(H(q)\) is surjective, \(H(u)H(q)=tH(q)\) implies \(H(u)=t\). Faithfulness already gives uniqueness. This proves the claimed bijection without choosing representatives of module elements inside abstract objects.
□8.4. A functorial inverse
Theorem 8.6. Reconstruction from a small projective generator
With the hypotheses and chosen constructions of §8.1,
\[\boxed{\mathcal A\simeq R\text{-Mod},\qquad R=\operatorname{End}_{\mathcal A}(P)^{\mathrm{op}}.}\]An equivalence means functors \(H:\mathcal A\to R\text{-Mod}\), \(L:R\text{-Mod}\to\mathcal A\), and natural isomorphisms \(HL\cong1_{R\text{-Mod}}\), \(LH\cong1_{\mathcal A}\). The symbol \(\simeq\) does not assert equality or an isomorphism of categories.
Proof.
The canonical presentation. For a module \(M\), let \(I_M\) be its underlying set, let \(p_M:R^{(I_M)}\to M\) send \(e_m\) to \(m\), and let \(K_M=\ker p_M\), as a concrete submodule. Let \(J_M\) be the underlying set of \(K_M\), and let \(d_M:R^{(J_M)}\to R^{(I_M)}\) send \(e_k\) to \(k\). Then \(p_M\) is surjective, \(\operatorname{im}d_M=K_M\), and \(M\) is its cokernel. This includes zero modules and empty index sets.
The object \(L(M)\). The coproduct bijection in Lemma 8.5 supplies a unique morphism \(\widetilde d_M:P^{(J_M)}\to P^{(I_M)}\) satisfying
\[H(\widetilde d_M)\theta_{J_M}=\theta_{I_M}d_M.\]Take the chosen cokernel \(c_M:P^{(I_M)}\to L(M)\). Exactness of \(H\) induces a unique isomorphism \(\nu_M:H(L(M))\to M\) with
\[\nu_M H(c_M)\theta_{I_M}=p_M.\]Indeed both sides exhibit the cokernel of \(d_M\); the unique comparison maps in both directions compose to identities by the cokernel universal property. Every module is therefore isomorphic to a module of the form \(H(X)\), which is essential surjectivity.
The morphism \(L(u)\). For \(u:M\to N\), define \(u_0:R^{(I_M)}\to R^{(I_N)}\) by \(e_m\mapsto e_{u(m)}\). The equality \(p_Nu_0=up_M\) sends \(K_M\) into \(K_N\). Define \(u_1:R^{(J_M)}\to R^{(J_N)}\) by \(e_k\mapsto e_{u_0(k)}\). Then \(u_0d_M=d_Nu_1\). Lift \(u_0,u_1\) uniquely to morphisms of the corresponding coproducts using Lemma 8.5. Faithfulness implies the lifted square commutes. The cokernel property defines a unique \(L(u)\) with
\[L(u)c_M=c_N\widetilde u_0.\]Identity module maps produce identity lifts. For composable \(u,v\), the formula on basis vectors gives \((vu)_0=v_0u_0\); uniqueness of lifts and of the cokernel factorization gives \(L(vu)=L(v)L(u)\). Thus \(L\) is an actual functor. The displayed formulas also give \(\nu_NH(L(u))=u\nu_M\): precompose with the surjection \(H(c_M)\theta_{I_M}\) to verify it. Hence \(\nu:HL\Rightarrow1\) is a natural isomorphism.
The other comparison. Full faithfulness gives a unique morphism \(\epsilon_X:L(H(X))\to X\) with \(H(\epsilon_X)=\nu_{H(X)}\). Lemma 8.3 implies \(\epsilon_X\) is invertible. For \(f:X\to Y\), naturality of \(\nu\) gives
\[H(f\epsilon_X)=H(\epsilon_YL(H(f))).\]Faithfulness gives \(f\epsilon_X=\epsilon_YL(H(f))\), so \(\epsilon:LH\Rightarrow1\) is natural. This proves the equivalence and closes the possible gap between constructing preimages separately and constructing a functor.
□Example 8.7. Abelian groups
In the category of abelian groups take \(P=\mathbb Z\). A homomorphism \(\mathbb Z\to A\) is determined by the image of \(1\), so evaluation gives \(H(A)\cong A\). Integer multiplication identifies \(\operatorname{End}(\mathbb Z)\) with \(\mathbb Z\), which is commutative and thus equals its opposite. Projectivity follows by lifting the image of \(1\) through a surjection; the generator property follows by testing each group element; smallness follows because each element of a direct sum has finite support. Reconstruction here is the familiar identification of abelian groups with \(\mathbb Z\)-modules.
Sources and further reading
- Alphabeta Math, Small projective generators and progenerators, module reconstruction with supplied copowers and cokernels, and equivalence and quasi-inverse. The proof above constructs its inverse from the canonical presentation of each module.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. Reconstructing with two copies of the integer generator
Applies to: Definition 8.1, Definition 8.2, Lemma 8.3.
Let \(\mathcal A=\mathrm{Ab},P=\mathbb Z^2\). A map \(P\to X\) is a row \((x_1,x_2)\) giving the images of its basis. Precomposition by \(A\in M_2(\mathbb Z)\) sends this row to \((x_1,x_2)A\); it is a left action of \(R=M_2(\mathbb Z)^{\mathrm{op}}\), since composing precompositions reverses the order. Maps out of \(P\) lift across a surjection by lifting its two basis images. They detect unequal maps by putting a test element in the first entry. The union of the supports of two entries in a direct sum is finite, proving smallness. Thus every generator hypothesis holds.
Here \(H(X)=X^2\) as an abelian group, and \(H(f)=(f,f)\). Kernels and cokernels are computed entrywise, so \(H\) is exact and commutes with arbitrary direct sums. If \(H(f)\) is bijective, injectivity follows by testing \((x,0)\), and surjectivity follows by taking a preimage of \((y,0)\); hence \(f\) is an isomorphism. This checks the action, preservation, and detection assertions with a different generator from \(\mathbb Z\).
Worked example 2. A presentation by the chosen generator
Applies to: Lemma 8.4.
Continue with \(P=\mathbb Z^2\). The sequence \(P\xrightarrow{d}P\xrightarrow q\mathbb Z/3\to0\), \(d(a,b)=(3a,b)\), \(q(a,b)=\bar a\), is exact: the image of \(d\) is \(3\mathbb Z\times\mathbb Z=\ker q\), and \(q\) is onto. Any map killing \(d\) vanishes on \((3,0),(0,1)\), so it factors uniquely through \(\bar a\). Thus it is the required cokernel presentation using copies of this \(P\), rather than merely some free presentation.
Worked example 3. Recovering maps and objects from the matrix action
Applies to: Lemma 8.5, Theorem 8.6.
A module map \(\Phi:X^2\to Y^2\) commuting with all right integer matrices must commute with \(E_{11},E_{22}\); hence \(\Phi(x,0)=(u(x),0)\), \(\Phi(0,x)=(0,v(x))\) for additive maps \(u,v\). Commutation with \(E_{12}\) forces \(u=v\), so \(\Phi=H(u)\), uniquely. This directly proves full faithfulness in this model.
To check that every left \(M_2(\mathbb Z)^{\mathrm{op}}\)-module occurs, regard it as a right matrix module \(N\), and put \(X=NE_{11}\). The map \(X^2\to N\), \((x,y)\mapsto x+yE_{12}\), has inverse \(n\mapsto(nE_{11},nE_{21})\), because \(E_{21}E_{12}=E_{22}\) and \(E_{12}E_{21}=E_{11}\). Matrix multiplication verifies that the first map is right-linear for the row action. Its two composites are identities, and the formulas commute with module maps. Hence this explicitly supplies the object reconstruction and natural comparison isomorphisms in Theorem 8.6.
Further proof details
Here is the module-linearity calculation in the object reconstruction. For \(x,y\in NE_{11}\) and \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), the row action gives \((x,y)A=(xa+yc,xb+yd)\). On the other side,
\[(x+yE_{12})A=xa+xbE_{12}+yc+ydE_{12},\]because \(x=xE_{11},y=yE_{11}\), \(E_{11}A=aE_{11}+bE_{12}\), and \(E_{12}A=cE_{11}+dE_{12}\). This is exactly the image of that row. The proposed inverse lands in \(NE_{11}\) because \(E_{21}E_{11}=E_{21}\). Its composite with the forward map is \(nE_{11}+nE_{21}E_{12}=n(E_{11}+E_{22})=n\). The reverse composite sends \(x+yE_{12}\) to \((x,y)\), using \(E_{12}E_{11}=E_{11}E_{21}=0\), \(E_{12}E_{21}=E_{11}\). A module map commutes with each multiplication by a matrix unit, which proves naturality of both formulas.
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. A projective generator need not be small
Related statements: Definition 8.1, Lemma 8.3, Theorem 8.6.
Claim refuted. The word “small” can be dropped from reconstruction by a projective generator.
Proof.
In \(\mathrm{Ab}\) let \(P=\bigoplus_{n\ge1}\mathbb Z e_n\). A map \(P\to Y\) lifts across a surjection \(X\to Y\) by choosing a preimage of the image of each basis element and extending by finite sums. This is projectivity (with the usual choice convention for modules). It is a generator: if \(a,b:X\to Y\) differ on \(x\), the map sending \(e_1\) to \(x\) and all other basis vectors to zero detects their difference.
But view \(P\) itself as the coproduct \(\bigoplus_n\mathbb Z\). The identity \(P\to\bigoplus_n\mathbb Z\) has a nonzero component in every summand. Every element of \(\bigoplus_n\operatorname{Hom}(P,\mathbb Z)\) has only finitely many nonzero component maps, so its image has zero components outside one fixed finite set of coordinates. It cannot be the identity. Therefore the smallness comparison is not onto, and \(\operatorname{Hom}(P,-)\) does not preserve this coproduct. Any equivalence would preserve it by transporting the universal property. Thus the omitted hypothesis really prevents this particular reconstruction functor from being an equivalence.
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