Draft lesson.

Use the universe and Choice convention of Universes and small categories. Hom groups and indexing sets are small in the stated universe; the collections of objects can be housed in a larger ambient universe. Choices of representing objects are made there. Translation has specified inverse comparisons, and \(X[n]\) denotes its coherent integer powers.

This lesson proves Brown representation from a small family satisfying a countable propagation condition. It also proves generation from those hypotheses, constructs products with their actual projections, and proves that both adjoints of a triangulated functor are triangulated. The family need not consist of compact objects.

1. The hypotheses and the abelian comparison category

Let \(\mathcal D\) be a locally small triangulated category with all small coproducts. Let \(\mathcal F\) be a small family such that:

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

We use the complete arguments in Product-preserving functors and countable Brown probes, with these precise interfaces:

• Lemma 1.1 permits replacement by the small shift family \(\{C[n]\}\), preserving both hypotheses. For that family the Hom tests also detect isomorphisms. Henceforth \(\mathcal F\) is shift-stable up to the specified isomorphisms.
• Let \(\mathcal S\) be the full subcategory of objects isomorphic to small coproducts of members of \(\mathcal F\). The category \(\mathcal B=\Pi(\mathcal S)\) consists of additive functors \(\mathcal S^{\mathrm{op}}\to\mathsf{Ab}\) taking source coproducts to products. Theorem 3.1 and its restriction argument prove that \(\mathcal B\) is locally small and abelian, with pointwise exactness. Restriction to \(\mathcal F\) is exact, faithful and conservative on \(\mathcal B\).
• For \(S\in\mathcal S\), write \(h_S=\operatorname{Hom}_{\mathcal S}(-,S)\). Theorem 4.1 gives a representable epimorphism onto every object of \(\mathcal B\), including every kernel in that category. Its additive Yoneda formula is \(\operatorname{Nat}(h_S,P)=P(S)\), with inverse \(x\mapsto(a\mapsto P(a)x)\).
• Theorem 5.1 constructs all small coproducts in \(\mathcal B\). They are internal coproducts, denoted \(\bigoplus^{\mathcal B}\). They need not be pointwise sums.
• The restricted Hom functor

is cohomological. Theorem 6.1 proves the canonical countable comparison

whose legs are precisely \(y(a_n)\) for the original coprojections \(a_n\). Precomposition by inverse translation defines an exact equivalence \(\mathsf T\) of \(\mathcal B\), with \(y(X[1])\simeq\mathsf T(yX)\), naturally in actual arrows.

• Theorem 8.1 is the following split calculation. If \(e_n:M_n\twoheadrightarrow A\), \(e_{n+1}u_n=e_n\), and \(u_n\) kills \(\ker e_n\), then, for the actual legs \(b_n:M_n\to M=\bigoplus_n^{\mathcal B}M_n\),

is split exact. The proof uses finite columns and a shear for the initial epimorphism, without assuming that initial epimorphism splits. In particular it uses no exactness assumption on arbitrary infinite coproducts or filtered colimits.

For arbitrary triangulated categories we use the actual Hom exact sequences in Triangles, finite universal maps and the limits of cohomology, Lemma 1.1. The completion, comparison and arbitrary coproduct triangle arguments are proved in Translations, coherent powers and universal triangle comparisons, Sections 4–7.

Theorem 1.1 (Brown representation and generation). Under the two hypotheses above:

1. Every additive contravariant cohomological functor \(H:\mathcal D^{\mathrm{op}}\to\mathsf{Ab}\) whose canonical maps

are isomorphisms is naturally isomorphic to \(\operatorname{Hom}_{\mathcal D}(-,Z)\) for some \(Z\in\mathcal D\). 2. If \(\mathcal K\) is a full triangulated subcategory containing the original family and closed under small ambient coproducts, then its inclusion into \(\mathcal D\) is an equivalence.

Closure in assertion 2 means that the coproducts can be chosen in \(\mathcal K\) and have their ambient universal property. We prove essential surjectivity, so the assertion also covers a subcategory not closed under all ambient isomorphisms. One need not assert literal equality of object collections.

2. A tower that kills every restricted kernel

Fix \(H\) as in the theorem, and initially fix such a \(\mathcal K\). Taking \(\mathcal K=\mathcal D\) always permits the construction. Restriction gives \(H_0=H|_{\mathcal S^{\mathrm{op}}}\in\mathcal B\).

Every object of \(\mathcal S\) used below has an isomorphic representative in \(\mathcal K\): choose the translates of the original family within \(\mathcal K\), and choose their labelled coproduct there. The translation comparison and the ambient coproduct universal property provide the isomorphism. Transport a cover through this isomorphism whenever necessary. Thus all the chosen representable donors can be taken in \(\mathcal K\).

Choose an epimorphism \(e_0:h_{X_0}\twoheadrightarrow H_0\), with \(X_0\in\mathcal S\cap\mathcal K\). Yoneda identifies it with an element \(h_0\in H(X_0)\). The same element defines a transformation on the entire category,

Its restriction is \(e_0\).

Suppose \(X_n\in\mathcal K\) and \(h_n\in H(X_n)\) have been chosen, with the corresponding epimorphism \(e_n:yX_n\twoheadrightarrow H_0\). Choose a cover

Yoneda for \(h_{Z_n}\) identifies the composite with an actual arrow \(c_n:Z_n\to X_n\). The equality of its image with \(\ker e_n\) is an equality of subobjects in \(\mathcal B\). It does not assert that \(c_n\) is a kernel in \(\mathcal D\).

Because \(e_ny(c_n)=0\), evaluation at \(Z_n\) on \(1_{Z_n}\) gives

Complete \(c_n\) inside \(\mathcal K\) to a distinguished triangle

Its ambient image is distinguished. Contravariant cohomology of \(H\) makes

exact. Choose \(h_{n+1}\) lifting \(h_n\), and let \(e_{n+1}:yX_{n+1}\to H_0\) be its restricted transformation. Naturality and (2.5) imply

Since \(e_n\) is epic, (2.6) makes \(e_{n+1}\) epic. Cohomology of \(y\) in (2.4), and the cover (2.2), give

Consequently \(y(u_n)\) kills \(\ker e_n\), and its image is canonically isomorphic to

Under that isomorphism the restriction of \(e_{n+1}\) to the image is the identity of \(H_0\), by (2.6). This specifies the image identification and the compatible maps, rather than only their abstract isomorphism types.

Induction therefore constructs a countable tower in \(\mathcal K\), together with compatible transformations \(\operatorname{Hom}(-,X_n)\to H\), for which (2.6)–(2.8) hold at every stage. Each initial index is a set of pairs \((C,x)\) with \(C\) in the small family and \(x\) in a small group. The same applies to each kernel cover. Countably many such choices remain in the stated universe.

3. The actual telescope and its split restricted sequence

Choose \(P=\bigoplus_{n\ge0}X_n\) in \(\mathcal K\), with its actual injections \(a_n\). Define

The coproduct universal property gives \(s\) uniquely. Equivalently it is the sum of the \(u_n\), followed by reindexing the tail and its canonical inclusion in \(P\). Choose a distinguished completion inside \(\mathcal K\),

Such a \(Z\) is a homotopy-colimit object for the chosen sequence. Two completions of \(d\) are isomorphic by completing the identity square on the first two vertices and applying the two-vertex comparison theorem. The resulting isomorphism need not be unique. Exercise 1 gives explicit distinct automorphisms fixing both other vertices of a cone triangle. A choice of cones in a bare triangulated category supplies no canonical functor on sequences from these axioms alone; constructions with further structure can supply functorial homotopy colimits.

Apply (1.2) to the tower. The induced isomorphism sends the \(n\)-th internal injection to \(y(a_n)\), so under that isomorphism \(y(s)\) is exactly the shift with columns \(y(a_{n+1})y(u_n)\). Define \(q:yP\to H_0\) by

Equations (2.6)–(2.7) verify every hypothesis of (1.3). Hence

is split exact, with these actual maps. In particular \(y(d)\) is monic and \(q\) is its cokernel. No compactness of \(C\) has been used. In general one cannot substitute \(\bigoplus_n\operatorname{Hom}(C,X_n)\) for \(yP(C)\); it is the internal comparison (1.2) that licenses the application of the telescope calculation.

The translation equivalence of \(\mathcal B\) preserves monomorphisms. Its natural comparison with \(y\) therefore makes \(y(d[1])\) monic as well. Apply \(y\) to (3.2):

Exactness and the last monomorphism imply \(y(w)=0\), so \(y(v)\) is epic and its kernel is the image of \(y(d)\). Thus (3.4) and (3.5) supply a unique isomorphism

This is a cokernel comparison with the specified arrows, not an application of faithfulness of \(y\). The foundation lesson exhibits a nonzero arrow killed by \(y\), so that stronger assertion would be false.

4. A representing element on the probes and the generation theorem

Product preservation of \(H\) gives a unique element \(h_P\in H(P)\) such that

Using the actual columns of \(d\), contravariance and additivity give

Since the product comparison is injective, \(H(d)h_P=0\). Cohomology of (3.2) yields \(h_Z\in H(Z)\) with \(H(v)h_Z=h_P\). Define on all of \(\mathcal D\)

The formula is additive and natural. Its restricted composite with \(y(v)y(a_n)\) is \(e_n\), by (4.1). Since (1.2) is a coproduct comparison with these legs, its restricted composite with \(y(v)\) is precisely \(q\). As \(y(v)\) is epic, (3.6) shows

In particular \(\theta_C\) is an isomorphism for every \(C\in\mathcal F\). This construction has kept \(Z\) in the chosen \(\mathcal K\).

We now prove generation before extending an arbitrary \(\theta\). Fix any \(W\in\mathcal D\), and apply the construction to

Hom exactness proves its contravariant cohomology, and the coproduct universal property proves (1.4). Yoneda identifies its element \(h_Z\) with an actual map \(f:Z\to W\); transformation (4.3) is postcomposition with \(f\). It is an isomorphism on the shift-stable family. The foundation's isomorphism-detection lemma makes \(f\) invertible. More explicitly, the cone of \(f\) has zero Hom from every member, because the five-term Hom sequence has invertible first and last maps; the last uses inverse shift of the test object. Zero detection makes the cone zero, and Hom from every object then constructs the two-sided inverse of \(f\).

Thus every \(W\) is isomorphic to an object \(Z\in\mathcal K\). The inclusion is fully faithful by fullness and essentially surjective by this construction, proving assertion 2 of Theorem 1.1. Literal membership of \(W\) in a nonreplete \(\mathcal K\) is unnecessary. No generation assumption was an input to the tower.

5. The all-shift comparison locus and representation on every object

We first give the abelian comparison needed for this step, with arbitrary abelian targets.

Lemma 5.1. In a commuting comparison of five-term exact sequences in an abelian category, if the four outside vertical maps are isomorphisms, then the middle one is an isomorphism.

Proof. Write a row as \(A_0\to A_1\to A_2\to A_3\to A_4\). Exactness at the three interior terms gives a short exact sequence

The first arrow is induced by the image factorization of \(A_1\to A_2\); the last is its image quotient under \(A_2\to A_3\). Kernel and cokernel universal properties make these identifications natural under a row comparison. Isomorphisms on \(A_0,A_1\) give an isomorphism on the first term of (5.1); isomorphisms on \(A_3,A_4\) give one on its last term. The short exact comparison lemma, proved for arbitrary abelian categories in Composition factors and uniform chain bounds, Lemma 1.1, now makes the middle map invertible. This argument does not assume that arbitrary objects have enough element-valued tests. \(\square\)

Let \(A,B\) be additive cohomological functors from a triangulated category to an abelian category, of the same variance, and \(\alpha:A\to B\) natural. Define

Lemma 5.2. This is a full replete triangulated subcategory. For closure under small coproducts of source objects, assume that the corresponding coproducts of values exist in the covariant case and that the canonical maps from these coproducts to the value at the source coproduct are invertible. In the contravariant case, assume instead that the corresponding products of values exist and that the canonical maps from the value at the source coproduct to these products are invertible.

Proof. Additive functors take zero to zero, so zero belongs. Naturality at an isomorphism proves repleteness. Coherent inverse powers of translation reindex the integers in (5.2), giving closure under both translation and its inverse.

For a triangle \(X\to Y\to Z\to X[1]\) with \(X,Y\in\mathcal L_\alpha\), the covariant exact rows have five positions

and the corresponding row for \(B\). All arrow squares commute by naturality, including the actual shifted last arrow. Four outside maps are invertible, so Lemma 5.1 makes the middle map invertible. For contravariant functors use the five positions

The same reasoning applies. Rotations and shift closure give closure when any other pair of the three vertices belongs.

Restrict translation and distinguished triangles to the full subcategory. Its identity triangles and rotations remain there. For a map between its objects, an ambient distinguished completion has its third object there by the preceding argument. Every ambient completion of a square remains a morphism in the subcategory by fullness. An ambient octahedron for composable maps uses cones of maps between its objects, so all its vertices belong. Consequently all triangle axioms restrict, proving that this is a triangulated subcategory.

For the coproduct assertion, an equivalence preserves the actual coproduct legs, so \((\bigoplus_iX_i)[n]\simeq\bigoplus_iX_i[n]\). Under the stated natural comparison maps, \(\alpha\) at this object is the coproduct, respectively product, of its components at \(X_i[n]\). A family of inverse component maps supplies its inverse by the same universal property. This proves the additional closure only under the stated preservation hypotheses. \(\square\)

Apply this lemma to \(A=\operatorname{Hom}(-,Z)\), \(B=H\), and \(\alpha=\theta\) of (4.3). Both are contravariantly cohomological and take actual coproducts to products. Equation (4.4), with shift stability, puts \(\mathcal F\) in \(\mathcal L_\theta\). It is a full triangulated coproduct-closed subcategory, so the generation theorem of Section 4 makes its inclusion an equivalence. Repleteness then implies that every ambient object belongs. Thus every \(\theta_W\) is an isomorphism, proving assertion 1 of Theorem 1.1.

A natural transformation with pointwise invertible components has a natural inverse: from \(\theta_XA(a)=B(a)\theta_Y\), compose with the two inverses to obtain its inverse naturality square. Hence this is representation as a functor, including all its maps.

6. Products and the right adjoint determined by universal elements

Corollary 6.1. The category \(\mathcal D\) of Theorem 1.1 has all small products. If \(F:\mathcal D\to\mathcal E\) is triangulated and its actual small coproduct comparisons are isomorphisms, it has a right adjoint.

Proof of products. For a small family \((Y_i)\), define

Products of exact sequences of abelian groups are exact: kernels are coordinatewise, and a tuple in an image has a preimage obtained by choosing one preimage in each coordinate. Applied to the Hom sequences, this proves cohomology of \(H\). For any small \((X_j)\), the two canonical rearrangements give

They agree with restriction along the actual injections, so prove the required canonical comparison. Brown gives an object \(P\) and a natural isomorphism \(\theta:\operatorname{Hom}(-,P)\to H\). Put \(p_i:P\to Y_i\) equal to the \(i\)-th component of \(\theta_P(1_P)\). Naturality gives

Its bijectivity is precisely the product universal property with projections \(p_i\). For the empty family \(H\) is the zero functor, and the resulting product is a terminal object, hence a zero object in this additive category.

Proof of the adjoint. For \(Y\in\mathcal E\), put

The distinguished image triangle and Hom exactness prove cohomology. The specified coproduct comparison for \(F\), followed by the Hom universal map, proves its product preservation. Choose a representing object \(GY\) and an additive natural isomorphism

Set \(\epsilon_Y=\beta_{GY,Y}(1_{GY})\). Naturality in \(X\) gives \(\beta_{X,Y}(a)=\epsilon_YF(a)\). For \(b:Y\to Y'\), define \(G(b)\) to be the unique arrow satisfying

Bijectivity in (6.5) gives existence and uniqueness. Equation (6.6) gives identities and composition by unique cancellation in that bijection, so \(G\) is a functor and \(\epsilon\) is natural. Additivity of (6.5) shows that \(G(b+b')=G(b)+G(b')\). Equation (6.6) also makes (6.5) natural in \(Y\).

Define \(\eta_X:X\to GFX\) by \(\beta_{X,FX}(\eta_X)=1_{FX}\). It is natural, and

The first identity is the definition. For the second, apply \(\beta_{GY,Y}\), use (6.6) for \(b=\epsilon_Y\), and use the first identity at \(GY\); the image is \(\epsilon_Y\), which is the image of \(1_{GY}\). Injectivity proves the second identity. These units, counits and Hom bijections establish \(F\dashv G\). The following section proves its full triangulated structure. \(\square\)

7. Every right adjoint of a triangulated functor is triangulated

The statement here requires neither Brown hypotheses nor coproduct preservation. Let \(F:\mathcal D\to\mathcal E\) be triangulated and \(F\dashv G\) any ordinary adjunction. Write its translation comparison as

First \(G\) is additive. A right adjoint preserves terminal objects and binary products by its Hom adjunction, with the actual product projections. Thus \(G0\) is zero and the comparison \(G(Y\oplus Y')\simeq GY\oplus GY'\) preserves projections. Zero maps factor through zero, so they are preserved. The two product equations for each biproduct injection now show that injections are also preserved. Diagonals are determined by their projections and codiagonals by their injection composites. The identity

therefore proves preservation of addition. The Hom adjunction is an isomorphism of groups: its two directions are \(a\mapsto\epsilon_YF(a)\) and \(b\mapsto G(b)\eta_X\), both additive.

The translation comparison for \(G\) is the natural isomorphism

defined by the following natural Hom bijections, using the specified inverse of translation and the comparison for \(F\):

Yoneda realizes these bijections as postcomposition with \(\sigma_Y\). Their inverses make it invertible. They are natural in both variables, so \(\sigma\) is natural. Equivalently, its actual counit equation is

Indeed the adjoint transpose of \(\epsilon_Y[1]\tau_{GY}:F((GY)[1])\to Y[1]\) is the inverse of (7.3), by reversing (7.4). This gives (7.5) and checks all its source and target types.

Take a distinguished triangle in \(\mathcal E\),

Choose in \(\mathcal D\) a distinguished completion

Apply \(F\) with last map \(\tau_{GY_0}Fh\). Naturality of the counit gives the first square to (7.6). The triangle completion axiom supplies \(c:FC\to Y_2\) such that

Let \(\bar c:C\to GY_2\) be its adjoint transpose. The first equality and naturality of adjunction give \(\bar c j=Gg\). Taking transposes in the second equality, using (7.5), gives

To check that last inference directly, the transpose of \(\sigma_{Y_0}^{-1}h\) is \(\epsilon_{Y_0[1]}F(\sigma_{Y_0}^{-1})Fh =\epsilon_{Y_0}[1]\tau_{GY_0}Fh\), exactly the second right side in (7.8). Thus uniqueness in the adjunction proves (7.9).

For any \(W\in\mathcal D\), the sequence

is exact. Adjunction identifies its first three terms with Hom from \(FW\) to \(Y_0,Y_1,Y_2\), and (7.4) identifies the last two with Hom from \(FW\) to \(Y_0[1],Y_1[1]\). Equation (7.5) identifies its connecting map with \(\delta\); naturality of \(\sigma\) identifies its next map with \(f[1]\). It is therefore the actual exact Hom sequence of (7.6).

Compare (7.10) with the Hom sequence of (7.7). The middle comparison is \(\operatorname{Hom}(W,\bar c)\); the four outside comparisons are identities. Equations (7.8)–(7.9) give the squares, so Lemma 5.1 makes that middle comparison invertible for every \(W\). Surjectivity at \(W=GY_2\) supplies \(b:GY_2\to C\) with \(\bar c b=1\). Injectivity at \(W=C\) then gives \(b\bar c=1\). Thus \(\bar c\) is an isomorphism. Replacing the third vertex in (7.7) by it proves that

is distinguished. This proves exactness with the specified comparison (7.3), rather than only exactness of a collection of cohomology groups. Exercise 3 proves the left-adjoint assertion, with its rotation sign.

8. A concrete setting with noncompact probes

Let \(k\) be a small field and take \(\mathcal D=K(\mathsf{Vect}_k)\). For \(C=k^{(\mathbb N)}[0]\), the family \(\{C[n]\}\) satisfies both hypotheses although \(C\) is not compact. The complete calculation in the foundation lesson, Section 7 proves termwise coproducts in \(K\), computes

constructs a contraction for every unbounded acyclic vector-space complex, and proves the countable condition from finite-support cycles and boundaries. The identity of \(k^{(\mathbb N)}\) has infinitely many target coordinates and is absent from the finite-target-coordinate image of the compactness comparison.

Theorem 1.1 therefore represents every cohomological product-preserving functor on this category using these noncompact probes. It also shows that any full triangulated coproduct-closed subcategory containing \(C[n]\) is equivalent to \(K(\mathsf{Vect}_k)\).

Here the product obtained by Corollary 6.1 can be described explicitly: for complexes \(Y_i\), it is their termwise vector-space product. A chain map into that product is a family of chain maps into the \(Y_i\). For morphism classes, choose representatives of a family to prove surjectivity of the comparison. If a chain map to the product has every component null-homotopic, choose each component homotopy and assemble the degreewise maps into the product. The homotopy equation holds in every coordinate, giving null-homotopy of the original map. Hence

with the actual coordinate projections. This argument uses products in the target of a chain map and does not turn the noncompact source probes into compact objects.

9. Four graded exercises with complete solutions

Exercise 1 (basic: two distinct cone comparisons). In \(K(\mathsf{Vect}_k)\), take \(A=k[0]\), \(B=k[1]\), and the zero map \(A\to B\). Exhibit two distinct automorphisms of its cone triangle fixing \(A\) and \(B\). Explain the limitation this puts on choosing homotopy-colimit comparison maps from triangle completion alone.

Solution. With the cochain cone convention of Chain homotopies, mapping cones and triangles, this cone is \(B\oplus A[1]\), with zero differential. Its triangle is

Identify \(A[1]=B=k[1]\). Besides the identity, take \(r(b,a)=(b+a,a)\). Its inverse is \((b,a)\mapsto(b-a,a)\). It fixes \(i_B\), and \(p_{A[1]}r=p_{A[1]}\); thus it gives an automorphism of the entire distinguished triangle with identity on the first two vertices. It differs from the identity in \(K\): all these complexes are concentrated in degree \(-1\), so any chain homotopy between their endomorphisms has zero components, and the off-diagonal identity cannot be null-homotopic. Consequently an isomorphism between completions fixing their first two vertices is not necessarily unique. Completion and its comparison axiom give choices for a telescope, but no canonical assignment of compatible comparison maps. This does not preclude a functorial telescope supplied by an enhancement or other additional construction.

Exercise 2 (intermediate: the two adjoints of the diagonal). Under Theorem 1.1, let \(I\) be a small set. In the pointwise triangulated category \(\mathcal D^I\), show that the diagonal \(\Delta:\mathcal D\to\mathcal D^I\) has left adjoint \(\Sigma\) and right adjoint \(\Pi\). Identify all structural maps, including the empty-set case. Prove that these adjoints are triangulated.

Solution. Set \(\Sigma(X_i)=\bigoplus_iX_i\) and \(\Pi(X_i)=\prod_iX_i\), using Corollary 6.1 for the latter. On a family \((f_i)\), their maps are uniquely specified by

Uniqueness proves identities and composition. The two actual universal bijections are

The first restrictions are along \(j_i\), and the second composites are with \(p_i\). They are natural by (9.2). The first adjunction has unit with components \(j_i\) and counit the fold map \(\bigoplus_iY\to Y\). The second has unit the diagonal map \(Y\to\prod_iY\) and counit with components \(p_i\). Their triangular identities can be checked on the same legs, or from the mutually inverse bijections (9.3).

Pointwise translation and distinguished triangles make \(\mathcal D^I\) triangulated: every axiom is solved coordinatewise, with the specified set of choices. The diagonal is triangulated with coordinatewise identity comparison. Section 7 makes its right adjoint \(\Pi\) triangulated. Exercise 3 makes its left adjoint \(\Sigma\) triangulated; alternatively the actual arbitrary triangle-coproduct theorem proves this directly. For \(I=\varnothing\), the category of families has one object and one arrow, both functors to \(\mathcal D\) select zero, and (9.3) reduces to the singleton zero Hom groups. Both resulting image triangles are zero triangles.

Exercise 3 (advanced: a left adjoint and its last-arrow sign). Let \(L\dashv F\), where \(F:\mathcal D\to\mathcal E\) is triangulated with comparison \(\tau\) as in (7.1). Construct a comparison \(\lambda_Y:L(Y[1])\to(LY)[1]\) and prove that \(L\) is triangulated. Together with Section 7, this must apply to either adjoint without Brown assumptions.

Solution. A left adjoint preserves the initial zero object and binary coproducts with their injections. It preserves zero maps; the biproduct projection equations then make it preserve projections. Formula (7.2) again proves additivity. Its ordinary Hom adjunction is a group isomorphism, by the unit and counit formulas.

Write the unit as \(\upsilon_Y:Y\to FLY\). Translation and the adjunction give natural bijections

Using their inverse direction, the covariant Yoneda formula realizes an isomorphism \(\lambda_Y:L(Y[1])\to(LY)[1]\). Equivalently it is the adjoint transpose of \(\tau_{LY}^{-1}\upsilon_Y[1]:Y[1]\to F((LY)[1])\). Thus its actual mate equation is

The bijections (9.4) prove invertibility and naturality, with the specified inverse-shift comparisons.

Take a distinguished triangle \(A\xrightarrow fB\xrightarrow gC\xrightarrow\delta A[1]\) in \(\mathcal E\), and choose a triangle in \(\mathcal D\)

Apply \(F\). Compare the distinguished rotation \(B\xrightarrow gC\xrightarrow\delta A[1]\xrightarrow{-f[1]}B[1]\) to that image triangle, using \(\upsilon_B,\upsilon_C\) on its first two vertices. Completion gives \(c:A[1]\to FQ\) such that

Its left-adjoint transpose \(\bar c:L(A[1])\to Q\) satisfies \(\bar c L\delta=k\). The second square, equation (9.5), and naturality of the unit give

For a direct verification, transpose the right side by applying \(F\) and composing with \(\upsilon_{A[1]}\). After postcomposition with \(\tau_{LB}\), naturality of \(\tau\) and (9.5) change it to \(-F(Lf)[1]\upsilon_A[1]=-\upsilon_B[1]f[1]\), which is exactly (9.7). The bijection of the adjunction proves (9.8).

For any \(W\in\mathcal D\), the Hom-into-\(W\) sequence of

has the five exact positions \((LC)[1],(LB)[1],L(A[1]),LC,LB\), in that order. The Hom adjunction, inverse translation and (9.5) identify it with the exact contravariant Hom sequence into \(FW\) of the rotated original triangle. Its preceding map is the actual shifted \(Lg\), and the mate identity supplies its last-arrow comparison. Compare this sequence with Hom into \(W\) for (9.6). Precomposition with \(\bar c\) is the middle map; the other four comparisons are identities. Equations (9.7)–(9.8) give all squares. Lemma 5.1 makes \(\operatorname{Hom}(Q,W)\to\operatorname{Hom}(L(A[1]),W)\) an isomorphism for every \(W\). The covariant Yoneda inverse test makes \(\bar c\) invertible: at \(W=L(A[1])\) obtain a left inverse, and injectivity at \(W=Q\) gives the right inverse.

Replacing \(Q\) by \(L(A[1])\) in (9.6), and then by \((LA)[1]\) through \(\lambda_A\), proves that

is distinguished. Inverse rotation, with the specified translation inverse, now gives the distinguished triangle \(LA\xrightarrow{Lf}LB\xrightarrow{Lg}LC\xrightarrow{\lambda_A L\delta}(LA)[1]\). The minus sign in (9.7)–(9.10) is the rotation sign and is essential. This proves the left-adjoint assertion and completes the assertion for either adjoint.

Exercise 4 (advanced: why every integer shift is in the test). Let \(A,B:\mathcal D\to\mathcal A\) be covariantly cohomological additive functors, where \(\mathcal A\) is any abelian category, and let \(\alpha:A\to B\) be natural. Prove that the objects on which every \(\alpha_{X[n]}\) is an isomorphism form a triangulated subcategory, and show it is closed under existing retracts. Give an example where testing only \(n=0\) is not shift-closed. State the extra hypothesis needed to assert coproduct closure.

Solution. The zero, repleteness, both-shift and cone arguments of Lemma 5.2 apply with this variance and this arbitrary abelian target. Its proof uses the kernel/cokernel construction (5.1) and the arbitrary-abelian short exact comparison, so it also covers targets without a faithful element test. The explicit restriction of completion, rotation, square completion and octahedra in that lemma proves all the triangulated axioms.

For a retract \(i:X\to Y\), \(p:Y\to X\), with \(pi=1_X\) and \(Y\) in the locus, an inverse to \(\alpha_X\) is

Naturality gives \(\alpha_X A(p)=B(p)\alpha_Y\) and \(B(i)\alpha_X=\alpha_YA(i)\). The two composites of (9.11) with \(\alpha_X\) are therefore \(B(pi)=1\) and \(A(pi)=1\). Apply the same computation to every shifted retract, proving the required all-shift assertion.

For failure of the one-shift test, take \(\mathcal D=K(\mathsf{Vect}_k)\), \(A=H^0\), \(B=0\), and \(\alpha=0\). Cohomology is additive and cohomological. The object \(k[1]\) has zero \(H^0\), so passes the \(n=0\) isomorphism test to the zero group. Its inverse shift \(k[0]\) has \(H^0=k\), so fails. Thus that smaller test does not define a translation subcategory.

For coproduct closure in the covariant case, one additionally requires that the relevant coproducts in the abelian target exist and that both canonical comparisons from the coproducts of the values to the value at the source coproduct be invertible. Then, after shifting the actual coprojections, \(\alpha\) on their sum is the sum of the component isomorphisms, as in Lemma 5.2. For contravariant functors the corresponding additional assumption is comparison to products. Cohomology and naturality alone assert neither preservation property.

10. Freely accessible references

Henning Krause, Derived categories, resolutions, and Brown representability, version 3, Sections 4.4–4.5, develops Brown representation under a countable propagation condition. The construction here uses the explicitly proved product-preserving-functor category and its split tower comparison from the linked foundation lesson.

The Stacks Project, Brown representability, Lemma 13.38.1, proves the compactly generated case using successive kernel-killing triangles. Its compactness hypothesis gives a different Hom–telescope interface. Sections 2–5 above provide the general argument using internal coproducts and prove generation as well as representation.