Draft lesson.

Use the universe and choice convention of Universes and small categories. All indexing sets and Hom groups of the given source categories are small in that universe. The full additive-functor category can initially be formed in a larger ambient universe; Section 3 proves smallness of the Hom groups used in the Brown construction. An arbitrary coproduct means a coproduct indexed by a small set. Write \(X[n]\) for the specified integer powers of translation in a triangulated category.

The purpose of this lesson is to construct the abelian category in which a countable Brown approximation can be compared with its telescope. Its coproducts have a universal characterization inside that category. They need not agree with coproducts of the values of its functors.

1. A small family and a countable condition

Let \(\mathcal D\) be a locally small triangulated category with arbitrary coproducts. Let \(\mathcal F\) be a small family of objects. Consider these conditions:

1. If \(\operatorname{Hom}_{\mathcal D}(C,X)=0\) for every \(C\in\mathcal F\), then \(X\simeq0\).
2. For a countable family \(u_i:X_i\to Y_i\), if every \(\operatorname{Hom}_{\mathcal D}(C,u_i)\) is zero, then every \(\operatorname{Hom}_{\mathcal D}(C,\bigoplus_i u_i)\) is zero, for \(C\in\mathcal F\).

Call these the zero-detection condition and the countable zero-map condition, respectively. The second condition concerns arbitrary arrows, rather than only a particular sequence of objects.

For comparison, a family detects isomorphisms if a map \(f\) is invertible whenever all \(\operatorname{Hom}(C,f)\) are bijective. This is the componentwise conservative convention in Generators and small quotient families, Section 1. It is distinct from requiring the family to distinguish every pair of parallel arrows.

Lemma 1.1. Replace \(\mathcal F\) by the small family

Both stated conditions are preserved. For a family stable under shifts up to isomorphism, zero detection is equivalent to detection of isomorphisms.

Proof. The new family remains small. It includes the old family, so zero detection is preserved. Suppose \(u_i\) is zero on Hom from every shifted member. For fixed \(n\), the maps \(u_i[-n]\) are zero on Hom from every original member, by the translation equivalence. Apply condition 2 to this family. An equivalence preserves coproducts: transporting their legs gives the same Hom universal bijections, with the inverse equivalence giving the inverse comparison. Thus

This proves the new condition 2.

An isomorphism-detecting family detects zero objects by applying its test to \(0\to X\). Conversely suppose the family is shift-stable, detects zero objects, and \(\operatorname{Hom}(C,f)\) is bijective for every member. Complete \(f:X\to Y\) to a triangle with third object \(Z\). The Hom exact sequence contains

Its first and last displayed arrows are isomorphisms: the last one is tested by \(C[-1]\). Exactness gives \(\operatorname{Hom}(C,Z)=0\). Zero detection gives \(Z\simeq0\). Applying Hom from every object to a triangle with zero third object shows that \(\operatorname{Hom}(W,f)\) is bijective for all \(W\). In particular there is \(g:Y\to X\) with \(fg=1_Y\), and injectivity at \(W=X\) gives \(gf=1_X\). Thus \(f\) is invertible. This equivalence uses shift stability. \(\square\)

The Hom exactness just used, for both variances and all rotations, is proved in Triangles, finite universal maps and the limits of cohomology, Lemma 1.1. We shall henceforth use the shifted family. Zero detection will be needed for Brown generation; the constructions in Sections 2–6 use only condition 2.

2. Equivalent propagation and factorization conditions

Theorem 2.1. For a small family \(\mathcal F\), the following four conditions are equivalent:

• countable zero-map propagation;
• if every \(\operatorname{Hom}(C,u_i)\) is surjective, then every \(\operatorname{Hom}(C,\bigoplus_i u_i)\) is surjective;
• the analogous implication for injectivity;
• for countable \((X_i)\), every map \(f:C\to\bigoplus_i X_i\), with \(C\in\mathcal F\), factors as

where each \(S_i\) is a small coproduct of objects of \(\mathcal F\).

Proof of the first three equivalences. For each \(u_i\), choose a triangle

Let \(p_i:Z_i[-1]\to X_i\) be its actual preceding arrow obtained by inverse rotation. Consecutive maps compose to zero. Hom exactness says that \(\operatorname{Hom}(C,u_i)\) is surjective exactly when \(\operatorname{Hom}(C,v_i)=0\), and is injective exactly when \(\operatorname{Hom}(C,p_i)=0\).

The coproduct of the chosen triangles is distinguished, with its actual coproduct arrows and the canonical translation comparison. This universal-map theorem is proved in Translations, coherent powers and universal triangle comparisons, Section 7. Hence zero-map propagation applied to the \(v_i\) gives surjective propagation for the \(u_i\). Applied to the \(p_i\), it gives injective propagation.

Conversely suppose all \(\operatorname{Hom}(C,u_i)\) vanish. Exactness makes \(\operatorname{Hom}(C,p_i)\) surjective. Surjective propagation makes its coproduct map surjective. Since \((\bigoplus_i u_i)(\bigoplus_i p_i)=0\), every Hom map into \(\bigoplus_i X_i\) is killed by \(\bigoplus_i u_i\). This proves zero-map propagation from surjective propagation.

Alternatively \(\operatorname{Hom}(C,v_i)\) is injective when \(\operatorname{Hom}(C,u_i)=0\). Injective propagation makes \(\operatorname{Hom}(C,\bigoplus_i v_i)\) injective. Its composite with \(\operatorname{Hom}(C,\bigoplus_i u_i)\) is zero, so the latter is zero. This proves the implication from injective propagation. The signs in inverse rotation do not affect zero, injectivity or surjectivity; the maps used are the specified triangle maps.

Proof of the factorization equivalence. For any \(X\), put

with \(e_X\) equal to \(a\) on the copy labelled \((C,a)\). The index is small by local smallness and the universe convention. Every arrow from \(C\in\mathcal F\) to \(X\) lifts through its own labelled coprojection, so \(\operatorname{Hom}(C,e_X)\) is surjective. Surjective propagation gives the same assertion for \(\bigoplus_i e_{X_i}\). Lifting \(f\) through this map proves (2.1).

For the converse, assume (2.1) and that every \(\operatorname{Hom}(C,u_i)\) is zero. For each factor \(v_i:S_i\to X_i\), the map \(u_iv_i\) is zero on every generator coprojection of \(S_i\), hence zero. Therefore

This holds for every \(f\), proving zero-map propagation. \(\square\)

The stronger requirement that each donor \(S_i\) in (2.1) be one member of \(\mathcal F\) implies the fourth condition. It is not required here. Similarly compact objects satisfy zero-map propagation because maps from them into a coproduct are finite-support families of maps. The theorem does not assume compactness.

3. The abelian category of product-preserving functors

For this section let \(\mathcal S\) be any locally small additive category with arbitrary coproducts. Define \(\Pi(\mathcal S)\) to consist of additive functors \(P:\mathcal S^{op}\to\mathsf{Ab}\) such that the map given by the source coprojections is an isomorphism

Its arrows are natural transformations. The product in (3.1) is a product of abelian groups. The coproduct is in \(\mathcal S\).

Theorem 3.1. Kernels, cokernels, images and exactness in \(\Pi(\mathcal S)\) are computed pointwise. This is a full abelian subcategory of the additive-functor category, and it is closed under extensions.

Proof. Products of abelian-group exact sequences are exact. Kernels are coordinatewise; a tuple in the product of images lifts coordinatewise by choice. In particular a product of surjections is surjective. These facts identify both the kernel and cokernel of a product map with the product of its kernels and cokernels.

Let \(a:P\to Q\) be a natural transformation of the stated functors. Its pointwise kernel \(K(S)=\ker a_S\) is an additive functor: maps of kernels are the unique restrictions of the maps of \(P\), and the uniqueness proves composition and additivity. For a coproduct, (3.1) identifies its comparison with

Thus \(K\in\Pi(\mathcal S)\). The pointwise cokernel \(Q(S)/a_S(P(S))\), with its induced additive maps, also satisfies (3.1), by the corresponding product-cokernel comparison. Pointwise kernels and cokernels have the natural universal properties: component factors are natural by uniqueness. The abelian-group coimage-to-image isomorphisms are natural and pointwise invertible, so they give an isomorphism of these functors. Zero objects and finite biproducts satisfy (3.1). This proves the abelian assertions, including agreement of exactness with pointwise exactness. It also establishes the exact full embedding into the additive-functor category; see Pointwise abelian structure and natural splittings for the ambient construction.

To prove extension closure, take a pointwise short exact sequence \(0\to P\to R\to Q\to0\) of additive functors, where \(P,Q\) satisfy (3.1). Evaluation at \(\bigoplus_i S_i\) and the product of evaluations at \(S_i\) give two short exact rows and a natural comparison between them. Its outer maps are isomorphisms. Its middle map is injective: an element in its kernel maps to zero in \(Q\); it therefore comes from \(P\), where injectivity and the lower inclusion make it zero. It is surjective: map a desired lower middle element into the lower \(Q\), lift its inverse image through the upper surjection onto \(Q\), and correct the remaining difference using the isomorphism on \(P\). Thus \(R\) also satisfies (3.1). \(\square\)

In particular, in a pointwise exact five-term sequence \(P_1\to P_2\to P_3\to P_4\to P_5\), if all terms except \(P_3\) belong to \(\Pi(\mathcal S)\), then so does \(P_3\). Its short exact sequence has left term \(\operatorname{coker}(P_1\to P_2)\) and right term \(\ker(P_4\to P_5)\). These terms belong to the category by the kernel and cokernel assertions, and the extension argument applies.

Now assume there is a small full subcategory \(\mathcal S_0\) such that every object of \(\mathcal S\) is isomorphic to a small coproduct of its objects. Restriction to \(\mathcal S_0\) is exact, faithful and conservative on \(\Pi(\mathcal S)\). Indeed a natural transformation at any chosen coproduct is the product of its components on the summands, by naturality at the coprojections and (3.1). This determines that component, and isomorphisms transport the assertion to every object. Products of abelian-group sequences are exact, so exactness can likewise be tested on \(\mathcal S_0\). These are assertions about transformations of these functors, not about faithfulness of the later restricted Yoneda functor on all of \(\mathcal D\).

This also proves that each natural-transformation group is small. Restriction injects it into the small set

Thus \(\Pi(\mathcal S)\) is locally small even when the objects of \(\mathcal S\) do not form a small set.

4. Evaluation epimorphisms and representable presentations

Retain the small full \(\mathcal S_0\). For \(S\in\mathcal S\), write

The coproduct universal property in the first variable puts \(h_S\) in \(\Pi(\mathcal S)\). The additive Yoneda bijection is

Its inverse sends \(x\in P(S)\) to \(a:C\to S\longmapsto P(a)x\). This formula is additive in \(a\) and natural in \(C\); naturality at \(a\) proves that the two formulas are inverse. For \(P=h_T\), it says that \(h:\mathcal S\to\Pi(\mathcal S)\) is fully faithful and additive.

Theorem 4.1. Every \(P\in\Pi(\mathcal S)\) has a representable epimorphism \(h_Y\twoheadrightarrow P\), and an exact presentation

Proof. Form the small coproduct

Product preservation gives an element \(\xi\in P(Y)\) whose coordinate on the copy labelled \((C,x)\) is \(x\). Let \(\theta:h_Y\to P\) be its transformation (4.2). For the labelled injection \(j_{C,x}:C\to Y\),

Thus \(\theta_C\) is surjective for every \(C\in\mathcal S_0\). For a coproduct of these objects the component is a product of these surjections, hence surjective. Transporting across an isomorphism gives the same assertion at every object of \(\mathcal S\). Theorem 3.1 makes \(\theta\) an epimorphism in \(\Pi(\mathcal S)\).

Its kernel belongs to that category. Apply the same construction to the kernel to obtain \(h_X\twoheadrightarrow\ker\theta\). Composing with the kernel inclusion gives a transformation \(h_X\to h_Y\). Full faithfulness in (4.2) identifies it with one actual arrow \(u:X\to Y\). Its image is exactly \(\ker\theta\), proving (4.3). \(\square\)

Epimorphism here refers to the abelian functor category. It does not assert that \(u\), or any later arrow between objects of a triangulated category, is an epimorphism in that triangulated category.

5. Internal coproducts and all small colimits

Theorem 5.1. The functor \(h\) preserves arbitrary coproducts as coproducts in \(\Pi(\mathcal S)\). That abelian category has arbitrary coproducts and all small colimits.

Proof for representables. For a family \((S_i)\) and any target \(P\), the natural bijections

are restriction along the actual arrows \(h_{j_i}\), where \(j_i:S_i\to\bigoplus_i S_i\). Thus they prove the coproduct universal property with the specified legs. They do not assert a pointwise-sum formula.

Proof for arbitrary functors. Choose presentations

Take the pointwise cokernel in \(\Pi(\mathcal S)\)

The composite \(h_{Y_i}\to h_{\bigoplus_i Y_i}\to Q\) annihilates \(h_{u_i}\), so it gives a unique leg \(\ell_i:P_i\to Q\). A map \(Q\to P\) is a map \(h_{\bigoplus_i Y_i}\to P\) annihilating the displayed map from \(h_{\bigoplus_i X_i}\). By (5.1), this is precisely a family of maps \(h_{Y_i}\to P\), each annihilating \(h_{u_i}\). By the cokernel property in (5.2), this is precisely a family of maps \(P_i\to P\). All these identifications use the legs \(\ell_i\). Therefore \(Q\), with those legs, is \(\bigoplus_i^{\Pi}P_i\). The universal property also gives the canonical isomorphism between constructions using different presentations. No exactness of a pointwise direct-sum operation is used.

For a small diagram \(D:I\to\Pi(\mathcal S)\), put

Let \(d:A\to B\) have column \(\iota_jD(a)-\iota_i\) on the summand labelled \(a:i\to j\). A map \(B\to P\) annihilates \(d\) exactly when its family of components satisfies \(t_jD(a)=t_i\) for every arrow. Thus \(\operatorname{coker}d\), with the induced object legs, has the cocone universal property of \(\operatorname{colim}_I D\). This includes the empty diagram. \(\square\)

Example 5.2 (a nonpointwise coproduct). Let \(\mathcal S\) be the full category of free abelian groups, of arbitrary small rank, and take \(\mathcal S_0=\{\mathbb Z\}\). Write \(L=\mathbb Z^{(\mathbb N)}\). Theorem 5.1 gives

At \(L\), the comparison from the pointwise sum is

A member on the left uses only finitely many target-coordinate injections, so its image is contained in the subgroup spanned by finitely many target basis vectors. The identity of \(L\) on the right has no such property. Hence (5.5) is not surjective. In matrix language the identity has one nonzero entry in each column, but infinitely many nonzero rows. A finite sum of the left-hand coordinate maps uses finitely many rows. This exhibits the failure without exchanging a product with a direct sum.

6. Restricted Yoneda and its countable comparison

Return to \(\mathcal D\) and the shifted family \(\mathcal F\), with the countable condition of Section 1. Let \(\mathcal S\) be its full subcategory of objects isomorphic to arbitrary coproducts of members of \(\mathcal F\). Coproduct reindexing follows from the universal property, so \(\mathcal S\) is additive and has arbitrary coproducts. The full subcategory on the members of \(\mathcal F\) is small and can serve as \(\mathcal S_0\).

Define

The Hom universal property in its source proves product preservation, regardless of whether \(C\) is compact. On \(\mathcal S\), this functor is \(h\); that restriction is fully faithful. Full faithfulness on all of \(\mathcal D\) is not assumed.

Theorem 6.1. The functor \(y\) is covariantly cohomological, preserves countable coproducts, and has the following propagation property: for countably many maps \(u_i:X_i\to X_i'\), if every \(y(u_i)\) is epic, then \(y(\bigoplus_i u_i)\) is epic.

Proof of cohomology and epimorphism propagation. Evaluating a triangle at each \(C\in\mathcal S\) gives its exact covariant Hom sequence. Pointwise exactness in Theorem 3.1 proves cohomology. Epicity of \(y(u_i)\) is equivalent to surjectivity of \(\operatorname{Hom}(C,u_i)\) for every \(C\in\mathcal F\), by the restriction test of Section 3. Theorem 2.1 gives the same surjectivity for the coproduct map, proving the epicity assertion.

Proof of the canonical coproduct comparison. Let \((X_i)\) be countable. Choose by Theorem 4.1 an epimorphism \(h_{Y_i}\twoheadrightarrow yX_i\), with \(Y_i\in\mathcal S\). Yoneda (4.2), now with target \(yX_i\), identifies this with an actual arrow \(b_i:Y_i\to X_i\) in \(\mathcal D\). Complete it to a triangle

Choose another epimorphism \(h_{Z_i}\twoheadrightarrow yW_i\), again corresponding to an actual arrow \(c_i:Z_i\to W_i\), with \(Z_i\in\mathcal S\). Exactness of \(y\) and epicity give presentations

Applying the construction of Theorem 5.1, their internal coproduct is the cokernel of

The two epicity assertions \(y(c_i)\) and \(y(b_i)\) propagate to their countable coproducts. Moreover the actual coproduct of (6.2) is a triangle. Its Hom exactness gives

The epic map \(h_{\bigoplus_i Z_i}\twoheadrightarrow y(\bigoplus_i W_i)\) shows that the image of its composite in (6.4) is the kernel of the second map in (6.5). Thus (6.4) has cokernel \(y(\bigoplus_i X_i)\). This identifies it with \(\bigoplus_i^{\Pi}yX_i\).

It remains to check which isomorphism was obtained. On the presentation summand \(h_{Y_i}\), the arrow in (6.5) is \(y(j_i)y(b_i)\), where \(j_i:X_i\to\bigoplus_i X_i\) is the actual coprojection. The map from the cokernel of (6.4) therefore has \(i\)-th leg exactly \(y(j_i)\), by epic cancellation against (6.3). Consequently it is the canonical comparison

This proves both existence of an isomorphism and its compatibility with every original coprojection. The empty countable family gives the zero-object comparison. \(\square\)

There is also the specified translation on \(\Pi(\mathcal S)\):

The shift closure of \(\mathcal S\) and its canonical coproduct comparisons put this functor in \(\Pi(\mathcal S)\). Precomposition with the opposite shift gives its inverse, using the specified unit and counit of translation. It is exact by pointwise exactness and preserves arbitrary internal coproducts because it is an equivalence. The Hom bijection gives a natural comparison \(y(X[1])\simeq\mathsf T(yX)\). Thus the entire shifted sequence in (6.5) is the actual Hom sequence transported by this comparison, rather than a separately chosen sequence of groups.

7. A noncompact family that satisfies the hypotheses

Let \(k\) be a small field and let \(\mathcal D=K(\mathsf{Vect}_k)\) be the homotopy category of arbitrary cochain complexes of small vector spaces. Its translation, cone triangles and full triangulated structure are constructed in Chain homotopies, mapping cones and triangles.

Termwise sums are actual coproducts in this homotopy category. A family of morphism classes has a family of chain-map representatives, and their maps assemble on the termwise injections. If a chain map on the sum is null-homotopic on each summand, choose the corresponding homotopies and assemble them degree by degree. The homotopy equation holds on every injection, so holds on the sum. This proves both existence and uniqueness for the coproduct of morphism classes.

Put \(C=k^{(\mathbb N)}[0]\) and \(\mathcal F=\{C[n]:n\in\mathbb Z\}\). A chain map \(k[n]\to X\) is a cycle in degree \(-n\); two such maps are homotopic exactly when their cycles differ by a boundary. Hence

The second equality is the actual coproduct universal property in the source, not a commutation assertion in the target.

If all groups (7.1) vanish, every \(H^r(X)\) vanishes. Such a vector-space complex is contractible, including when it is unbounded. To see this explicitly, write \(B^r=\operatorname{im}d^{r-1}\) and choose a complement \(L^r\) to \(B^r=\ker d^r\) in \(X^r\). The restriction \(d^r:L^r\to B^{r+1}\) is an isomorphism. Define

Then \(d^{r-1}s^r(b+\ell)=b\) and \(s^{r+1}d^r(b+\ell)=\ell\). Thus \(ds+sd=1_X\), proving that \(X\) is zero in \(K\). The choices are made for all degrees; no induction from a first or last degree is required. This proves zero detection, and Lemma 1.1 gives isomorphism detection for this shifted family.

Cohomology commutes with arbitrary termwise direct sums of vector-space complexes. A finite-support tuple is a cycle exactly when each coordinate is a cycle. A finite-support tuple of boundaries has a finite-support family of preimages. Consequently both cycles and boundaries are the respective direct sums, and their quotient is

with the actual coprojection maps. If every \(\operatorname{Hom}_{K}(C[n],u_i)\) is zero, (7.1) shows that every \(H^{-n}(u_i)\) is zero: put any desired element in one coordinate and zero in the others. Formula (7.3) makes \(H^{-n}(\bigoplus_i u_i)\) zero. Applying (7.1) again proves zero-map propagation, even for arbitrary small families. In particular the countable condition holds.

Nevertheless \(C\) is not compact. In

every left-hand element has image in finitely many target coordinates. The identity of \(C\) is not of this form. All these complexes are concentrated in degree zero, so no nonzero homotopy can change their maps. This gives a concrete noncompact family satisfying both hypotheses, without passing to a derived localization or identifying an acyclic complex with zero by declaration.

Example 7.2 (restricted Yoneda need not be faithful). Take \(\mathcal D=D(\mathsf{Ab})\), with the shift and full triangle structure in Triangles after quasi-isomorphism localization. Use \(\mathcal F=\{\mathbb Z[n]:n\in\mathbb Z\}\). The bounded-projective test in Derived morphisms through quasi-isomorphism roofs, Theorems 5.1–5.2 gives

Here \(\mathbb Z\) is projective because lifting the image of 1 through a surjection lifts the entire homomorphism. Thus the stated test applies to its degree-zero complex and every shift.

We verify the arbitrary coproducts used by this example. Termwise sums are actual coproducts in \(K(\mathsf{Ab})\), by the representative and homotopy assembly already given. They preserve quasi-isomorphisms because the finite-support cycle and boundary argument of (7.3) also works for abelian groups. In the derived localization these sums still have the universal property. For existence, choose an incoming roof for each prescribed map \(QX_i\to QY\), where \(Q:K\to D\) is localization. Sum their denominators and assemble their numerators by the coproduct property in \(K\). The sum denominator is a quasi-isomorphism, so the resulting roof defines a map from \(Q(\bigoplus_iX_i)\) with the prescribed restrictions.

For uniqueness, suppose a derived map from this sum restricts to zero on every \(X_i\). Represent it by \(\bigoplus_iX_i\xleftarrow{s}V\xrightarrow{f}Y\). For each injection \(j_i\), the incoming square lemma for roofs gives a denominator \(a_i:U_i\to X_i\) and a map \(b_i:U_i\to V\) with \(sb_i=j_ia_i\) in \(K\). The zero restriction and the incoming equality test give a further denominator \(c_i:U_i'\to U_i\) with \(fb_ic_i=0\). Assemble \(b_ic_i\) to \(b:\bigoplus_iU_i'\to V\). Then \(sb\) is the sum of the denominators \(a_ic_i\), under the actual injections, hence a quasi-isomorphism. Since \(s\) is one too, its cohomology equations imply that \(b\) is a quasi-isomorphism. Also \(fb=0\) by the injections. Refining the original roof by \(b\) proves that it is zero. Applied to a difference of two maps, this proves uniqueness. Both square and equality tests are proved in Derived morphisms through quasi-isomorphism roofs, Sections 1–2.

Consequently these are actual arbitrary coproducts in \(D\), with cohomology (7.3). Formula (7.5) detects zero objects: if all cohomology groups vanish, the arrow \(X\to0\) is a quasi-isomorphism, hence invertible in \(D\). The same formula and the sum cohomology calculation give zero-map propagation. Thus this family satisfies both hypotheses of Section 1.

Now let \(M=(\mathbb Z/2)[0]\), and put \(P=(\mathbb Z\xrightarrow{2}\mathbb Z)\) in degrees minus one and zero. Reduction gives a quasi-isomorphism \(p:P\to M\). The chain map \(a:P\to\mathbb Z[1]\) with identity component in degree minus one represents

It is nonzero. Indeed the bounded-projective test identifies \(\operatorname{Hom}_D(P,\mathbb Z[1])\) with its homotopy group. Chain maps are integers in degree minus one, and homotopies change that integer by an even integer. Thus this group is \(\mathbb Z/2\), and \(a\) has class 1. Inverting \(Qp\) cannot turn that class into zero.

Every cohomology map of \(\delta\) is zero, because its source and target cohomology occur in distinct degrees. Hence \(\operatorname{Hom}_D(C,\delta)=0\) for every \(C\in\mathcal F\), by (7.5). It is also zero for every \(S\in\mathcal S\): Hom out of a coproduct is the product of the Hom groups from its summands, and isomorphisms transport this statement. Therefore \(y(\delta)=0\) although \(\delta\ne0\). The restricted functor \(y\) is not faithful in this example. This proves the distinction while retaining zero detection and the countable condition.

8. A stabilized-image telescope in an abelian category

Here is the abelian calculation that will apply to a Brown tower. It is valid even when coproducts in the abelian category are not pointwise.

Theorem 8.1. Let \(\mathcal B\) be an abelian category with countable coproducts. Suppose there are arrows \(u_n:M_n\to M_{n+1}\) and epimorphisms \(e_n:M_n\twoheadrightarrow H\), for \(n\ge0\), such that

For the actual coprojections \(a_n:M_n\to P=\bigoplus_{n\ge0}M_n\), define \(s:P\to P\) and \(q:P\to H\) by

Then

is split exact. The compatible legs \(e_n\) also exhibit \(H\) as the colimit of this sequence of \(M_n\).

Proof: the forced sections. Because \(e_n\) is the cokernel of its kernel, the second equation of (8.1) gives a unique \(i_n:H\to M_{n+1}\) with \(u_n=i_ne_n\). The first equation and epic cancellation give \(e_{n+1}i_n=1_H\). For \(n\ge1\), put \(K_n=\ker e_n\), with inclusion \(k_n\). The map \(1-i_{n-1}e_n\) factors uniquely as \(k_nr_n\). Monic cancellation gives \(r_nk_n=1\) and \(r_ni_{n-1}=0\). Therefore

In these coordinates \(u_n(h,k)=(h,0)\) for \(n\ge1\), whereas \(u_0\) is the column \((e_0,0)\). A splitting of \(e_0\) has not been assumed.

Proof: the constant identity telescope. Put \(U=\bigoplus_{n\ge1}H\), with legs \(j_n\). Let \(s_Hj_n=j_{n+1}\), \(d_H=1-s_H\), and \(cj_n=1_H\). Define \(r:U\to U\) by its finite columns

The coproduct universal property gives this map. No infinite sum in a Hom group is proposed. On each \(j_n\), subtracting consecutive finite sums proves

For example, \(d_Hrj_n=-\sum_{t<n}(j_t-j_{t+1})=j_n-j_1\); this also holds for \(n=1\).

Proof: one shear for the initial epimorphism. Reindexing the coproduct using (8.4) identifies

This reindexing is justified by the Hom-out universal properties: on either side a map specifies one map from \(M_0\), one from each copy of \(H\), and one from each \(K_n\). Put \(d=1-s\). Its columns under (8.7) are \((1,-j_1e_0,0)\) on \(M_0\), \((0,j_n-j_{n+1},0)\) on the \(n\)-th copy of \(H\), and the identity injection into \(V\) on \(K_n\).

The biproduct-matrix automorphism

has inverse with a minus sign. Testing its columns gives

Set \(R=\operatorname{diag}(1_{M_0},r,1_V)\), \(Q=(0,c,0)\), and \(J=(0,j_1,0)\). Equations (8.6) give

Moreover \(q=Qb\): on \(M_0\) this is \(e_0\), on each \(H\)-copy it is the identity, and on every \(K_n\) it is zero. These are precisely the original \(e_n\) in (8.4). Define

Conjugating the preceding identities gives

In particular \(d\) is monic and \(q\) is split epic. If \(qv=0\), then \(v=dtv\), and its factor through \(d\) is unique because \(d\) is monic. If \(wd=0\), then \(w=wzq\), again with unique factor because \(q\) is epic. This proves (8.3), including its actual kernel and cokernel maps. The section \(z\) uses the forced \(H\)-summand of \(M_1\); the initial \(e_0\) can be nonsplit.

A map \(v:P\to A\) gives a cocone on the \(M_n\) exactly when \(va_{n+1}u_n=va_n\) for all \(n\). This is equivalent to \(vd=0\). The cokernel property just proved gives its unique factor through \(q\), with the prescribed components through \(e_n\). Thus \(H\), with those legs, is their sequential colimit. If \(H=0\), (8.1) forces \(u_n=0\), and the proof gives \(d=1_P\), as required. \(\square\)

Every additive functor preserves the four equations (8.10), hence this split exact sequence. This observation does not assert that the functor preserves arbitrary nonsplit exact sequences or arbitrary filtered colimits. In \(\Pi(\mathcal S)\), (6.6) supplies the countable coproduct interface, and (6.7) supplies its exact shifted interface. Those are the specific comparisons needed when applying (8.3) to a triangulated telescope.

9. Four graded exercises with complete solutions

Exercise 1 (basic: one representable presentation). Let \(\mathcal S\) be the category of free abelian groups and let \(A=\mathbb Z/2\). For \(P_A(S)=\operatorname{Hom}_{\mathbb Z}(S,A)\), prove that \(P_A\in\Pi(\mathcal S)\) and compute a presentation by two copies of \(h_{\mathbb Z}\). Verify its exactness also at a free group of infinite rank.

Solution. The Hom-out coproduct property gives \(P_A(\bigoplus_iS_i)=\prod_iP_A(S_i)\); additivity follows from bilinear composition. Reduction \(\rho:\mathbb Z\to A\) gives the natural transformation \(h_{\mathbb Z}\to P_A\). At \(S=\mathbb Z^{(J)}\), its component is

It is surjective: choose for each coordinate the representative 0 or 1. Its kernel is \(\prod_j2\mathbb Z\), the image of coordinatewise multiplication by two. That multiplication is injective. Thus the following sequence is exact at every free group, including infinite rank:

Theorem 3.1 therefore makes it an exact sequence in \(\Pi(\mathcal S)\). The arrow \(h_2\) is induced by the actual multiplication-by-two map \(\mathbb Z\to\mathbb Z\), and the quotient leg is induced by actual reduction. Evaluation at \(\mathbb Z\) already tests exactness because it is the chosen \(\mathcal S_0\); the explicit infinite-rank calculation checks the product comparison that permits this test.

Exercise 2 (intermediate: quotient coproducts). In the setting of Exercise 1, compute \(Q=\bigoplus_{n\in\mathbb N}^{\Pi}P_A\). At \(L=\mathbb Z^{(\mathbb N)}\), show that the map from the pointwise direct sum into \(Q(L)\) is not surjective. Does this contradict pointwise computation of cokernels in Theorem 3.1?

Solution. Taking the internal coproduct of the presentations in (9.1), Theorem 5.1 gives

Let \(B=(\mathbb Z/2)^{(\mathbb N)}\), and let \(\rho_L:L\to B\) reduce every coordinate. For any free \(S\), a homomorphism \(S\to B\) lifts to \(L\): choose a finite-support integral lift for the image of each basis vector of \(S\), then extend linearly. If \(S\to L\) is killed by \(\rho_L\), every basis image belongs to \(2L\), so division by two gives a unique homomorphism \(S\to L\). Hence reduction induces a natural isomorphism

Under (9.3), the internal legs \(P_A\to Q\) are the maps induced by the coordinate injections \(A\to B\). A member of \(\bigoplus_nP_A(L)\) has only finitely many of these coordinates, so its image in \(\operatorname{Hom}(L,B)\) is contained in finitely many target-coordinate summands. The map sending the \(n\)-th basis vector of \(L\) to the \(n\)-th basis vector modulo two has infinitely many target coordinates. It is in \(Q(L)\) and is not in the pointwise comparison image.

There is no contradiction. The cokernel of the specific transformation between the two objects \(h_L\) in (9.2) is pointwise. The earlier formation of these representable coproducts was internal, as proved in (5.1). A pointwise direct sum of the \(P_A\) need not belong to \(\Pi(\mathcal S)\), so it cannot be substituted for this construction. Even evaluation at the generating object \(\mathbb Z\), where the pointwise comparison is an isomorphism, would not make that substitution legitimate: the restriction test of Section 3 concerns arrows between objects already in \(\Pi(\mathcal S)\).

Exercise 3 (advanced: arbitrarily large noncompact probes). Let \(J\) be any small infinite set, put \(C_J=k^{(J)}[0]\), and let \(\mathcal F_J=\{C_J[n]:n\in\mathbb Z\}\) in \(K(\mathsf{Vect}_k)\). Prove zero detection and zero-map propagation for arbitrary small families. Prove that \(C_J\) is not compact. Explain why zero-map propagation does not assert that Hom from \(C_J\) preserves coproducts.

Solution. The actual source-coproduct calculation gives

As \(J\) is nonempty, this product is zero exactly when its factor is zero: insert a factor element in any fixed coordinate and zero in all others. Vanishing for every \(n\) thus means that \(X\) is acyclic. Formula (7.2) contracts it, so it is zero in \(K\). This proves zero detection, and the shift-stable test in Lemma 1.1 gives isomorphism detection.

If \(\operatorname{Hom}(C_J[n],u_i)=0\), (9.4) implies \(H^{-n}(u_i)=0\). The finite-support proof of (7.3) is valid for any small index set. It follows that \(H^{-n}(\bigoplus_i u_i)=0\), and then (9.4) gives \(\operatorname{Hom}(C_J[n],\bigoplus_i u_i)=0\). This proves the stronger arbitrary-family zero-map implication without exchanging a product with a direct sum.

Finally consider the family \((k[0])_{j\in J}\). A member of \(\bigoplus_j\operatorname{Hom}_{K}(C_J,k[0])\) uses finitely many target summands. The identity \(C_J\to k^{(J)}[0]\) has image in all target basis directions, so it is outside that comparison image. There are no degree-minus-one homotopies for these degree-zero complexes. Thus \(C_J\) is not compact. Zero-map propagation tests families of zero-induced arrows. It asserts neither surjectivity of this Hom-coproduct comparison nor an exchange between the product in (9.4) and the sum in (7.3).

Exercise 4 (advanced: a nonsplit first leg). In \(\mathsf{Ab}\), take \(M_0=\mathbb Z\), \(M_n=\mathbb Z/2\) for \(n\ge1\), and \(H=\mathbb Z/2\). Let \(e_0\) be reduction, \(e_n=1\) for \(n\ge1\), \(u_0=e_0\), and \(u_n=1\) for \(n\ge1\). Compute the maps in (8.3), a left inverse of \(1-s\), and a section of \(q\). Prove the colimit property with the original legs although \(e_0\) has no section.

Solution. Compatibility \(e_{n+1}u_n=e_n\) holds. The kernel of \(e_0\) is \(2\mathbb Z\), which \(u_0\) kills; the other kernels are zero. Thus (8.1) holds. Put \(U=(\mathbb Z/2)^{(\{1,2,\ldots\})}\) with coprojections \(j_n\), and use \(d_H,c,r\) from (8.5)–(8.6). Then \(P=\mathbb Z\oplus U\), and the actual maps are

In particular \(qa_0=e_0\) and \(qa_n=e_n\) for all \(n\ge1\). Define

Because \(rj_1=0\) and \(rd_H=1_U\), (9.5) gives \(td(m,u)=(m,u)\). Also

Their sum is \((m,u)\). The equations \(qz=1_H\) and \(qd=0\) follow respectively from \(cj_1=1\), and from \(cd_H=0\) with cancellation of \(\overline m\). These are all four split identities, proving the sequence with its actual maps.

If a family \(v_n:M_n\to A\) is compatible, its coproduct map \(v:P\to A\) has \(vd=0\). The equation \(dt+zq=1\) gives \(v=vzq\). Thus \(vz:H\to A\) is its factor with all original legs. The factor is unique since \(qz=1\). This proves the colimit property. A homomorphism \(\mathbb Z/2\to\mathbb Z\) is zero because its image is killed by two, so \(e_0\) has no section. The section \(z\) instead uses \(M_1\), exactly as in the general proof.

10. Freely accessible references

Henning Krause, Derived categories, resolutions, and Brown representability, Section 4, discusses the general perfect-generator condition and the functor-category approach to Brown representability. The constructions and universal-map verifications needed here are supplied in Sections 1–8.

The Stacks Project, Lemma 13.38.1, gives Brown representability for compactly generated triangulated categories. That compact hypothesis is sufficient for the countable condition of Section 2; the examples in Sections 7 and 9 show why the present condition also covers noncompact probes.