Trace coreflections and balanced nonabelian categories

A chosen object can generate a useful full subcategory without generating the ambient category. Colimits survive inside that subcategory, while a kernel retains only the part reached by the chosen object. This lesson describes that change for an arbitrary cocomplete abelian category. It then computes a category in which every monomorphism is strict and every monomorphism that is also an epimorphism is invertible, yet a strict epimorphism has a pullback that is not even an epimorphism.

Use Coimages, images and composition of quotients for strict morphisms and the ordinary module interface, and Ideal-generated modules and nonstrict quotients for the polynomial ideal calculations. We retain the standard additive and abelian framework of the Stacks Project: additive categories, kernels, cokernels, images and coimages, and abelian categories. All Hom sets and coproduct indices below are small in a fixed universe.

1. The part of an object reached by a probe

Let \(\mathsf C\) be a locally small cocomplete abelian category, and fix any object \(G\). Say that \(Z\) is \(G\)-generated if some small coproduct of copies of \(G\) admits an epimorphism onto \(Z\). Write \(\mathsf C_G\) for the full subcategory of these objects and \(I:\mathsf C_G\to\mathsf C\) for inclusion.

For \(X\in\mathsf C\), take the evaluation morphism whose component at \(h:G\to X\) is \(h\), and factor it through its ambient image:

The first arrow is an epimorphism and \(\varepsilon_X\) is a monomorphism. The object \(T_GX\) is called the trace of \(G\) in \(X\). This definition uses no finite-presentation or projectivity hypothesis on \(G\).

Theorem 1.1. The trace defines an additive right adjoint \(T_G:\mathsf C\to\mathsf C_G\) to \(I\). Its counit is the monomorphism \(\varepsilon_X\). Its unit is an isomorphism, so \(T_G\) is idempotent up to the canonical isomorphism.

Proof. The epimorphism in (1.1) proves \(T_GX\in\mathsf C_G\). Let \(Z\in\mathsf C_G\), choose an epimorphism \(p:G^{(J)}\to Z\), and take \(a:Z\to X\). Every component of \(ap\) is one of the maps used in evaluation, so it factors through \(\varepsilon_X\).

Let \(c:X\to\operatorname{Coker}\varepsilon_X\) be the ambient cokernel. Then \(cap=0\), whence \(ca=0\) because \(p\) is an epimorphism. In the abelian category, the image monomorphism \(\varepsilon_X\) is a kernel of \(c\). Thus \(a\) factors uniquely as \(\varepsilon_X\bar a\). This gives the natural bijection

For \(f:X\to Y\), define \(T_Gf\) by \(\varepsilon_Y T_Gf=f\varepsilon_X\), using the factorization just proved. Uniqueness gives identities, composition and naturality. It also gives additivity: after composition with the monomorphism \(\varepsilon_Y\), both \(T_G(f+g)\) and \(T_Gf+T_Gg\) equal \((f+g)\varepsilon_X\).

When \(Z\) is generated, apply the factorization to \(\operatorname{id}_Z\). It produces \(\eta_Z:Z\to T_GZ\) with \(\varepsilon_Z\eta_Z=\operatorname{id}_Z\). Monicity of \(\varepsilon_Z\) then gives \(\eta_Z\varepsilon_Z=\operatorname{id}_{T_GZ}\). These maps are the unit and counit on generated objects. Since \(T_GX\) is itself generated, applying this observation to it proves idempotence. \(\square\)

In particular,

For \(G=0\), the trace is always zero and \(\mathsf C_G\) is the zero category; the construction and adjunction still apply.

2. Which kernels and colimits survive

Theorem 2.1. The category \(\mathsf C_G\) is additive and has all small colimits and all finite limits. Inclusion creates small colimits. For \(f:X\to Y\) between generated objects, let \(K\) and \(Q\) be its ambient kernel and cokernel. Then

The kernel arrow is \(T_GK\to K\to X\); the cokernel arrow is the ambient one.

Proof. Quotients of generated objects are generated, by composing their generating epimorphisms. Small coproducts of generated objects are also generated: take the coproduct of their generating maps. A coproduct of epimorphisms is an epimorphism, since equality of two maps after that coproduct implies equality on each target summand by cancellation.

Every small colimit in \(\mathsf C\) is a coequalizer of two maps between small coproducts of diagram objects. Its coequalizer arrow is an epimorphism, so its target is generated when all diagram objects are generated. Fullness then retains the entire universal property inside \(\mathsf C_G\). This includes the empty colimit.

Hom groups and bilinear composition restrict from \(\mathsf C\). Finite ambient direct sums remain generated and retain their product and coproduct properties, so \(\mathsf C_G\) is additive.

The object \(Q\) is a quotient of \(Y\), hence generated, and its cokernel property restricts by fullness. If \(a:Z\to X\), with \(Z\) generated, satisfies \(fa=0\), it factors uniquely through the ambient \(K\). Theorem 1.1 factors that map uniquely through \(T_GK\). This is exactly the kernel property in \(\mathsf C_G\).

More generally, suppose an ambient limit \(L\) of a diagram of generated objects exists. A cone from any generated \(Z\) gives a unique map \(Z\to L\), and (1.2) gives a unique map \(Z\to T_GL\). Composing the trace inclusion with the ambient projections therefore exhibits \(T_GL\) as the intrinsic limit. Ambient finite limits exist by the abelian framework, proving the remaining assertion. \(\square\)

The last statement is conditional for an arbitrary small diagram: cocompleteness of \(\mathsf C\) was assumed, whereas existence of all ambient small limits was not. Nor does Theorem 1.1 alone say that trace commutes with filtered colimits. The finitely presented module result in the prerequisite supplies additional hypotheses for that conclusion.

3. Strictness measures the missing part of a kernel

We first check that the additive definitions match the kernel-pair and self-pushout definitions in the strictness prerequisite.

In any additive category with kernels and cokernels, let \(i:K\to X\) be the kernel of \(f:X\to Y\). Its kernel pair is \(X\oplus K\), with projections

This is notation for biproduct morphisms. Given maps \(a,b:Z\to X\) with \(fa=fb\), their difference \(b-a\) factors uniquely through \(i\), giving the unique map to (3.1). A map out of \(X\) equalizes \(p_1,p_2\) precisely when it kills \(i\). Consequently their coequalizer is \(\operatorname{Coker}i\), the additive coimage.

Dually, if \(q:Y\to Q\) is the cokernel of \(f\), its self-pushout is \(Y\oplus Q\), with maps

Indeed, for \(a,b:Y\to Z\) satisfying \(af=bf\), the difference \(b-a\) factors uniquely through \(q\). The map from \(Y\oplus Q\) with components \(a\) and that factor is the required unique pushout map. The equalizer of \(j_1,j_2\) is \(\operatorname{Ker}q\), the additive image. Thus the canonical coimage-to-image map in either convention is the same map. Call \(f\) strict when this map is invertible.

Theorem 3.1. For every morphism \(f:X\to Y\) of \(\mathsf C_G\), let \(K=\operatorname{Ker}_{\mathsf C}f\) and \(J=\operatorname{Im}_{\mathsf C}f\). There are canonical identifications

Here \(X/T_GK\) means the ambient cokernel of \(T_GK\to K\to X\).

Proof. The ambient image \(J\) is an epimorphic image of the generated \(X\), so \(J\in\mathsf C_G\). By (2.1), the intrinsic cokernel of \(f\) is its ambient cokernel. Taking the intrinsic kernel of that cokernel gives \(T_GJ\). Since \(J\) is generated, that trace is \(J\). This proves the first formula.

The intrinsic kernel of \(f\) is \(T_GK\). Cokernels are ambient, giving the second formula. The abelian coimage-image isomorphism identifies \(J\) with \(X/K\). Hence the canonical map in (3.3) is the ambient quotient comparison

It is an epimorphism with ambient kernel \(K/T_GK\). By the abelian mono-epi criterion, (3.4) is invertible exactly when that kernel vanishes, equivalently when \(T_GK\to K\) is invertible. Formula (1.3) proves the stated criterion. Fullness makes ambient and intrinsic invertibility identical for these objects. \(\square\)

The criterion applies to arbitrary morphisms, including zero maps. It implies that \(\mathsf C_G\) is abelian exactly when every ambient kernel of a morphism between its objects is generated: the additive category already has kernels and cokernels, and the remaining abelian condition is invertibility of every coimage-image map.

4. A balanced category with a nonstrict quotient

Fix any field \(k\), let \(A=k[x,y]\), and put \(\mathfrak a=(x,y)\) and \(\overline k=A/\mathfrak a\). Take \(G=\mathfrak a\) in the ambient category of ordinary \(A\)-modules. Denote the resulting generated category by \(\mathsf C_0\).

The prerequisite proves that every map \(\mathfrak a\to A\) is multiplication by a unique polynomial, and that the same polynomials give \(\operatorname{End}_A(\mathfrak a)=A\). Therefore inclusion \(\mathfrak a\hookrightarrow A\) induces a bijection on \(\operatorname{Hom}_A(\mathfrak a,-)\), despite being noninvertible. The functor on all ambient modules is not conservative. It is not faithful either: the nonzero quotient \(A\to\overline k\) and the zero map become equal after every map \(\mathfrak a\to A\), since each such map has image in \(\mathfrak a\). In particular \(\mathfrak a\) is not a detecting generator for the ambient category.

There is nevertheless a weaker detection property on all ambient modules:

To check the exact scope, take \(0\ne m\in M\). If \(\mathfrak a m\ne0\), multiplication \(a\mapsto am\) gives a nonzero map \(\mathfrak a\to M\). If \(\mathfrak a m=0\), the map \(\overline k\to M\), \(1\mapsto m\), is injective because a nonzero field scalar cannot annihilate \(m\). Compose it with the surjection \(\mathfrak a\to\overline k\) sending \(x\) to \(1\) and \(y\) to zero, verified from the ideal presentation in the prerequisite. This again gives a nonzero map. Thus (4.1) holds. As a result,

The equality follows by summing the images \(t\mathfrak a\), including \(t=1\). The trace implication follows because every probe map factors through the trace.

The prerequisite also proves that every module annihilated by \(\mathfrak a\) belongs to \(\mathsf C_0\): for each of its elements \(m\), use the map \(x\mapsto m,\ y\mapsto0\), and take the coproduct of these maps. In particular \(\overline k\) is a nonzero object of \(\mathsf C_0\).

Proposition 4.1. A morphism of \(\mathsf C_0\) is a monomorphism or epimorphism exactly when it is one in the ambient module category. Every monomorphism of \(\mathsf C_0\) is strict. The category is balanced: a morphism that is both mono and epi is invertible.

Proof. In an additive category with kernels, a morphism is mono exactly when its kernel is zero. Indeed, a zero kernel forces every difference killed by the morphism to vanish. Conversely, monicity makes the kernel arrow zero; its universal property then forces the identity of the kernel object to be zero, making that object zero. The dual argument applies to cokernels and epis.

The intrinsic kernel of \(f\) is \(T_{\mathfrak a}(\ker f)\). By (4.2), it vanishes exactly when the ambient kernel vanishes. The intrinsic cokernel is already the ambient one, so its vanishing also agrees in both categories. Ordinary module monos and epis are injective and surjective respectively. A mono has ambient kernel zero, which is generated, and Theorem 3.1 makes it strict. Finally, an ambient injective surjective module map is invertible, and its inverse is a morphism in the full subcategory. \(\square\)

Balancedness does not make every morphism strict. Consider the quotient \(u:\mathfrak a\to\mathfrak a/(Ax)\). Its target is generated because it is a quotient of \(\mathfrak a\). Its ambient kernel is \(Ax\simeq A\), under multiplication by \(x\), so its intrinsic kernel is \(x\mathfrak a\). The complete intrinsic factorization is

The canonical map between the last two objects has ambient kernel \(Ax/(x\mathfrak a)\simeq\overline k\ne0\). It is not invertible. Thus this balanced additive category with kernels and cokernels is not abelian.

5. A strict epimorphism whose pullback loses epimorphy

Consider

The second map is \(A\)-linear because \(\mathfrak a\) annihilates the class of \(x\). All these objects are generated. The ambient kernel of \(v\) is the square ideal \(\mathfrak a^2\), which is generated by images of multiplication maps \(\mathfrak a\to\mathfrak a^2\). Theorem 3.1 therefore makes \(v\) a strict epimorphism.

Its ambient pullback along \(w\) is

Projection onto \(b\) identifies this module with \(J=Ax+\mathfrak a^2=(x,y^2)\). Indeed, the displayed condition says exactly that the degree-one part of \(b\) is a scalar multiple of \(x\), and that scalar uniquely determines \(c\). Under this identification the map \(P\to\overline k\) is the coefficient of \(x\) in that degree-one part.

Every map \(\mathfrak a\to J\), composed with \(J\hookrightarrow A\), is multiplication by a polynomial \(t\). It lands in \(J\) exactly when \(t\mathfrak a\subset J\). Write \((J:\mathfrak a)\) for those polynomials. The condition \(tx\in J\) is automatic. Reducing \(ty\in J\) modulo \(x\) says that \(y\bar t\) is divisible by \(y^2\) in \(k[y]\). Cancellation gives divisibility of \(\bar t\) by \(y\), so \(t\in(x,y)=\mathfrak a\). Conversely \(t\in\mathfrak a\) gives \(t\mathfrak a\subset\mathfrak a^2\subset J\). Hence

The intrinsic pullback is the trace of (5.2), by Theorem 2.1. Its two structural maps are thus inclusion \(\mathfrak a^2\hookrightarrow\mathfrak a\) and the zero map \(\mathfrak a^2\to\overline k\). They form the actual pullback square

Its top arrow is not an epimorphism: the identity and zero endomorphisms of the nonzero \(\overline k\) agree after it. This is a failure of pullback stability for strict epimorphisms. It coexists with the exact filtered-colimit and filtered base-change properties proved for \(\mathsf C_0\) in the prerequisite; those concern a different comparison.

In particular \(\mathsf C_0\) is not quasi-abelian: pullback stability of strict epimorphisms is one of the defining axioms in Schneiders, Definition 1.1.3, printed page 8.

The signs in (5.2) matter. As a kernel in ambient modules, the pullback is \(\ker(v-w)\). A kernel of \(v+w\) describes an isomorphic object only after negating the \(\overline k\) coordinate, with the corresponding change to its projection.

6. Graded exercises with complete solutions

Exercise 1 — Foundation: combine two probes. For objects \(G,H\) in a locally small cocomplete abelian category, prove that \(T_{G\oplus H}X\) is the sum of the subobjects \(T_GX\) and \(T_HX\) of \(X\). Show that \(\mathsf C_{G\oplus H}\) is the smallest full subcategory closed under isomorphisms, small coproducts and ambient quotients and containing both \(G\) and \(H\).

Solution. Interpret the sum as the ambient image of \(T_GX\oplus T_HX\to X\). A map \(G\oplus H\to X\) has two components. Each component lands in the corresponding trace, so every such map lands in their sum. Hence the trace of \(G\oplus H\) is contained in that sum. Conversely, a map \(G\to X\) extends to \(G\oplus H\) by zero on \(H\); a map from \(H\) extends by zero on \(G\). Thus both traces, and therefore their sum, lie in \(T_{G\oplus H}X\). The two subobject inclusions prove equality.

The generated category is closed under coproducts and quotients by Theorem 2.1 and under isomorphisms by its definition. It contains \(G,H\), since the projections from \(G\oplus H\) onto them are split epimorphisms. Any subcategory with the stated closure and containing \(G,H\) contains \(G\oplus H\), all its small coproducts, and all their epimorphic images. Those are exactly the generated objects, proving minimality. Zero objects or zero probes cause no exception; the empty coproduct is included.

Exercise 2 — Intermediate: an idempotent probe and failure of balancedness. Let \(R\) be any unital ring, \(e^2=e\), and \(P=Re\) as a left module. Set \(L=ReR\), the two-sided ideal of finite sums of terms \(res\). Prove

Now let \(R\) be the ring of lower triangular three-by-three matrices over a field, take \(e=e_{11}\), and consider \(f:Re_{11}\to Re_{11}/ke_{31}\). Prove that this is a mono and epi in \(\mathsf C_P\) and is not invertible.

Solution. A homomorphism \(h:Re\to M\) is determined by \(m=h(e)\), and \(em=h(e^2)=m\). Conversely \(m=em\) defines \(h_m(re)=rm\). This is well-defined: if \(re=r'e\), then \((r-r')m=(r-r')em=0\). These assignments are inverse and natural.

The images of these maps sum to \(R(eM)\). This is \(LM\): each term \(resm\) lies in \(R(eM)\), and each \(rem\) is obtained with \(s=1\). The generated-object criterion follows from (1.3). Apply Theorem 3.1 to the ambient kernel for the strictness criterion. No commutation of ring coefficients has been used.

For the matrix example, \(P\) is the space of first columns. Left multiplication sends \(e_{31}\) to a scalar multiple of \(e_{31}\), so \(ke_{31}\) is a submodule. The source and its quotient belong to \(\mathsf C_P\); the quotient map is an intrinsic epi. But \(e_{11}(ke_{31})=0\), so the Hom description gives \(T_P(ke_{31})=0\). Its intrinsic kernel is zero, making it a mono by the additive kernel criterion. Its ambient kernel is the nonzero \(ke_{31}\), so it cannot have a module inverse and is not invertible in the full subcategory.

Thus generated subcategories need not be balanced. The weak zero-detection property (4.1), which gave balancedness for the polynomial ideal probe, fails here on \(ke_{31}\).

Exercise 3 — Advanced: strict pullback failures in every degree. Keep \(A=k[x,y]\) and \(\mathfrak a=(x,y)\). Let \(d\ge2\), and take a nonzero homogeneous polynomial \(p\) of degree \(d-1\). Let

Prove that \(v_d\) is a strict epi, calculate the ambient and intrinsic pullbacks along \(w_p\), and show that the intrinsic projection onto \(\overline k\) is zero.

Solution. Every positive ideal power is generated by \(\mathfrak a\), as proved by the degree-monomial multiplication maps in the prerequisite. Hence the kernel \(\mathfrak a^d\) of \(v_d\) is generated, making \(v_d\) strict. Multiplication by \(\mathfrak a\) takes \(p\) into \(\mathfrak a^d\), so \(w_p\) is \(A\)-linear. Since \(p\) is nonzero of degree \(d-1\), its class is nonzero.

The ambient pullback identifies by first projection with \(J_p=Ap+\mathfrak a^d\). Modulo \(\mathfrak a^d\), a polynomial multiple of \(p\) is just its constant coefficient times \(p\). That coefficient is uniquely determined, because the class of \(p\) is nonzero. It is the second pullback projection to \(\overline k\).

We claim

Suppose a nonzero \(t\) lies in the left side, and let \(t_j\) be its lowest nonzero homogeneous component. If \(j<d-2\), the nonzero component \(xt_j\) has degree less than \(d-1\), whereas \(J_p\) has no components in those degrees. This is impossible. If \(j=d-2\), membership of both \(xt\) and \(yt\) gives \(xt_j=cp\) and \(yt_j=c'p\) for scalars \(c,c'\). Multiply the first equation by \(y\) and the second by \(x\), obtaining \((cy-c'x)p=0\). The polynomial ring is a domain and \(p\ne0\); independence of \(x,y\) then gives \(c=c'=0\), contradicting \(t_j\ne0\). Therefore all components of \(t\) have degree at least \(d-1\). Conversely, any \(t\in\mathfrak a^{d-1}\) satisfies \(t\mathfrak a\subset\mathfrak a^d\subset J_p\). This proves (6.3), including \(t=0\).

The multiplication description of maps \(\mathfrak a\to A\) now gives \(T_{\mathfrak a}J_p=\mathfrak a\mathfrak a^{d-1}=\mathfrak a^d\). Thus the intrinsic pullback is \(\mathfrak a^d\), with inclusion into \(\mathfrak a\) and zero projection onto \(\overline k\). That projection is not epi. The argument works over every field and for every nonzero homogeneous \(p\) specified in the question.

Exercise 4 — Expert: an extension condition and its limits. In the setting of Theorem 1.1, suppose \(\mathsf C_G\) is closed under ambient extensions: if an ambient short exact sequence has both end objects in \(\mathsf C_G\), then its middle object is also in \(\mathsf C_G\). Prove that every pullback of a strict epimorphism in \(\mathsf C_G\) is a strict epimorphism. Does this stability imply ambient extension closure?

Solution. Let \(q:X\to Y\) be a strict epi in \(\mathsf C_G\), and let \(a:Z\to Y\) be another morphism there. Intrinsic epis are ambient epis in every \(\mathsf C_G\): their intrinsic cokernels are ambient cokernels, and zero cokernel characterizes epimorphy in the additive and abelian settings. By Theorem 3.1, the ambient kernel \(K\) of \(q\) is generated.

Form the ambient pullback \(P=X\times_Y Z\). We retain Stacks, Lemma 12.5.14 for stability of ambient epimorphisms under pullback, and Lemma 12.5.12 for the induced kernel isomorphism. These exact interfaces give the ambient short exact sequence

Extension closure makes \(P\) generated. Thus its trace is itself, so this ambient pullback is the intrinsic pullback with the same projections. Its projection onto \(Z\) is an epi with generated ambient kernel \(K\). Theorem 3.1 makes it strict, as required.

The converse fails. Let \(R=k[\epsilon]/(\epsilon^2)\) and \(G=R/(\epsilon)\). A \(G\)-generated module is annihilated by \(\epsilon\). Conversely, an annihilated module is a \(k\)-vector space, and copies of \(G=k\) indexed by its elements surject onto it. Thus \(\mathsf C_G\) is precisely the full category of modules annihilated by \(\epsilon\), equivalently ordinary \(k\)-vector spaces. Its intrinsic kernels, cokernels and pullbacks are the vector-space ones, all morphisms are strict, and epis remain epis under pullback.

But the ambient short exact sequence

has generated ends and a middle object that is not generated, since \(\epsilon\cdot1\ne0\) in \(R\). Its trace is the proper ideal \((\epsilon)\): maps from \(k\) land in the annihilator of \(\epsilon\), which is that ideal, and every element of the ideal is reached. Hence ambient extension closure is sufficient for the asserted strict pullback stability and is not necessary.

Self-checked; no independent review has occurred. Original exposition and exercise text are dedicated to the public domain under CC0 1.0.