TRIANGULATED CATEGORIES WITHOUT MODELS
FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
Abstract. We exhibit examples of triangulated categories which are neither
the stable category of a Frobenius category nor a full triangulated subcategory
of the homotopy category of a stable model category. Even more drastically,
our examples do not admit any non-trivial exact functors to or from these
algebraic respectively topological triangulated categories.
Introduction. Triangulated categories are fundamental tools in both algebra and
topology. In algebra they often arise as the stable category of a Frobenius cat-
egory ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear
as a full triangulated subcategory of the homotopy category of a Quillen stable
model category [Hov99, 7.1]. The triangulated categories which belong, up to
exact equivalence, to one of these two families will be termed algebraic and topo-
logical, respectively. We borrow this terminology from [Kel06, 3.6] and [Sch06].
Algebraic triangulated categories are generally also topological, but there are many
well-known examples of topological triangulated categories which are not algebraic.
In the present paper we exhibit examples of triangulated categories which are
neither algebraic nor topological. As far as we know, these are the first examples
of this kind. Even worse (or better, depending on the perspective), our examples
do not even admit non-trivial exact functors to or from algebraic or topological
triangulated categories. In that sense, the new examples are completely orthogonal
to previously known triangulated categories.
Let (R,m) be a commutative local ring with m = (2) 6= 0 and m2 = 0. Examples
of this kind of rings are R = Z/4, or more generally R = W2(k) the 2-typical
Witt vectors of length 2 over a perfect field k of characteristic 2. There are also
examples which do not arise as Witt vectors, for instance the localization of the
polynomial ring Z/4[t] at the prime ideal (2). We denote by F(R) the category of
finitely generated free R-modules.
Theorem 1. The category F(R) has a unique structure of a triangulated category
with identity translation functor and such that the diagram
is an exact triangle.
Given an object X in an algebraic triangulated category T and an exact triangle
−→ A −→ C −→ ΣA,
1991 Mathematics Subject Classification. 18E30, 55P42.
Key words and phrases. Triangulated category, stable model category.
The first author was partially supported by the Spanish Ministry of Education and Science
under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship
EX2004-0616, and a Juan de la Cierva research contract.
http://arxiv.org/abs/0704.1378v2
2 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
the equation 2 · 1C = 0 holds, compare [Kel06, 3.6] and [Sch06]. Since the ring
R satisfies 2 · 1R 6= 0, the triangulation of the category F(R) is not algebraic. We
cannot rule out the possibility of a topological model for F(R) as easily: the classical
example of A = S the sphere spectrum in the stable homotopy category shows that
the morphism 2 · 1C can be nonzero in this more general context.
Nevertheless, F(R) is not topological either, which follows from Theorem 2. Here
we call an exact functor between triangulated categories trivial if it takes every
object to a zero object.
Theorem 2. Every exact functor from F(R) to a topological triangulated category
is trivial. Every exact functor from a topological triangulated category to F(R) is
trivial.
Acknowledgements. We are grateful to Bernhard Keller for helpful conversations
on the results of this paper, and to Amnon Neeman, who suggested the possibility
of constructing a triangulated structure on F(Z/4) by using Heller’s theory [Hel68].
In the original version of this note the first author alone constructed the tri-
angulation of the category F(Z/4) and proved that it does not admit any model.
The second author joined the project later by providing a simpler and more general
proof that the triangulation is not topological. The third author’s contribution
was an old preprint on the example considered in Remark 8, which provided some
guidance for the other results.
The triangulated categories. Let T be an additive category and let Σ: T
be a self-equivalence that we call translation functor. A candidate triangle (f, i, q)
in (T,Σ) is a diagram
(3) A
−→ ΣA,
where if , qi, and (Σf)q are zero morphisms. A morphism of candidate triangles
(α, β, γ) : (f, i, q) → (f ′, i′, q′) is a commutative diagram
// B′
// C′
// ΣA′
The category of candidate triangles is additive. The mapping cone of the morphism
(α, β, γ) is the candidate triangle
B ⊕A′
β f ′
// C ⊕B′
// ΣA⊕ C′
−Σf 0
Σα q′
// ΣB ⊕ ΣA′.
A homotopy (Θ,Φ,Ψ) from (α, β, γ) to (α′, β′, γ′) is given by morphisms
// B′
// C′
// ΣA′
such that
β′ − β = Φi+ f ′Θ, γ′ − γ = Ψq + i′Φ, Σ(α′ − α) = Σ(Θf) + q′Ψ.
TRIANGULATED CATEGORIES WITHOUT MODELS 3
We say in this case that the morphisms are homotopic. The mapping cones of
two homotopic morphisms are isomorphic. A contractible triangle is a candidate
triangle such that the identity is homotopic to the zero morphism. A homotopy
(Θ,Φ,Ψ) from 0 to 1 is called a contracting homotopy. Any morphism from or to
a contractible triangle is always homotopic to zero.
A triangulated category is a pair (T,Σ) as above together with a collection of
candidate triangles, called distinguished or exact triangles, satisfying the follow-
ing properties. The family of exact triangles is closed under isomorphisms. The
candidate triangle
(4) A
−→ A −→ 0 −→ ΣA,
is exact. Any morphism f : A → B in T can be extended to an exact triangle like
(3). A candidate triangle (3) is exact if and only if its translate
−→ ΣA
−→ ΣB,
is exact. Any commutative diagram
// ΣA
// B′
// C′
// ΣA′
whose rows are exact triangles can be extended to a morphism whose mapping
cone is also exact. This non-standard set of axioms for triangulated categories is
equivalent to the classical one, see [Nee01], and works better for the purposes of
this paper.
Now we are ready to prove Theorem 1.
Proof of Theorem 1. Given an object X in F(R) we consider the candidate triangle
X2 defined as
(5) X
−→ X.
We are going to prove that the category F(R) has a triangulated category struc-
ture with identity translation functor where the exact triangles are the candidate
triangles isomorphic to the direct sum of a contractible triangle and a candidate
triangle of the form (5).
The family of exact triangles is closed under isomorphisms by definition. The
candidate triangle (4) is contractible, and hence exact. The ring R is a quotient of a
discrete valuation ring with maximal ideal generated by 2, see [Coh46, Corollary 3];
therefore any morphism f : A→ B in F(R) can be decomposed up to isomorphism
1 0 0
0 2 0
0 0 0
 : A =W ⊕X ⊕ Y −→W ⊕X ⊕ Z = B.
Then f is extended by the direct sum of (5) and the contractible triangle
W ⊕ Y
// W ⊕ Z
// Y ⊕ Z
// W ⊕ Y.
4 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
The translate of a contractible triangle is also contractible, and the triangle (5) is
invariant under translation. This proves that the translate of an exact triangle is
exact. Translating a candidate triangle six times yields the original one, therefore
if a candidate triangle has an exact translate then the original candidate triangle
is also exact.
We say that a candidate triangle A
→ A is a quasi-exact triangle if
−→ B,
is an exact sequence of R-modules. The exact triangles are all quasi-exact.
Now we are going to show that any diagram of candidate triangles
(6) A
// B′
// C′
// A′
with exact rows can be completed to a morphism with exact mapping cone.
Suppose that the upper row in (6) is contractible and the lower row is quasi-
exact. Since f ′α = βf then f ′αq = 0; since C is projective, there exists γ′ : C → C′
such that q′γ′ = αq. Let (Θ,Φ,Ψ) be a contracting homotopy for the upper row.
Then γ = γ′ + (i′β − γ′i)Φ completes (6) to a morphism of candidate triangles.
If the upper row in (6) is quasi-exact and the lower row is contractible then
(6) can also be completed to a morphism. This can be shown directly, but it also
follows from the previous case since we have a duality functor
HomR(−, R) : F(R)
−→ F(R)op,
which preserves contractible triangles and quasi-exact triangles. Here we use that
R is injective as an R-module, see [Lam99, Example 3.12].
If the upper and the lower rows in (6) areX2 and Y2, respectively, then γ = β+2δ
extends (6) to a morphism of candidate triangles for any δ : X → Y .
This proves that any diagram like (6) with exact rows can be completed to a
morphism ϕ = (α, β, γ). Now we have to check that the completion can be done
in such a way that the mapping cone is exact. Suppose that the upper and the
lower rows are X2 ⊕ T and Y2 ⊕ T
′, respectively, with T and T ′ contractible. The
morphism ϕ is given by a matrix of candidate triangle morphisms
ϕ11 ϕ12
ϕ21 ϕ22
: X2 ⊕ T −→ Y2 ⊕ T
where ϕij = (αij , βij , γij). Here ϕ12, ϕ21 and ϕ22 are homotopic to 0 since either the
source or the target is contractible, therefore the mapping cone of ϕ is isomorphic
to the mapping cone of
ϕ11 0
: X2 ⊕ T −→ Y2 ⊕ T
which is the direct sum of the mapping cone of ϕ11 and two contractible triangles,
T ′ and the translate of T .
TRIANGULATED CATEGORIES WITHOUT MODELS 5
We can suppose that
α11 =
1 0 0
0 2 0
0 0 0
 : X = L⊕M ⊕N −→ L⊕M ⊕ P = Y.
Moreover, as we have seen above we can take γ11 = β11 + 2δ for
0 0 0
0 1 0
0 0 0
 : X = L⊕M ⊕N −→ L⊕M ⊕ P = Y.
We have 2β11 = 2α11, therefore β11 = α11 + 2Φ for some Φ: X → Y . Now we
observe that (δ,Φ, 0) is a homotopy from ϕ11 to ζ = (α11 + 2δ, α11 + 2δ, α11 + 2δ),
so the mapping cone of ϕ11 is isomorphic to the mapping cone of ζ.
The mapping cone of ζ is clearly the direct sum of five candidate triangles, namely
M2, N2, M2 (once again), P2, and the mapping cone of the identity 1 : L2 → L2,
which is contractible. Therefore the mapping cone of ζ is exact, and also the
mapping cone of ϕ11, ψ and ϕ.
It remains to show the uniqueness claim in Theorem 1. In any triangulation, all
contractible candidate triangles are exact [Nee01, 1.3.8]. The triangle X2 is a finite
direct sum of copies of R2. Hence every triangulation of (F(R), Id) which contains
R2 contains all the exact triangles which we considered above. Two triangulations
with the same translation functor necessarily agree if one class of triangles is con-
tained in the other, so there is only one triangulation in which R2 is exact. This
completes the proof. �
Remark 7. The exact triangles in F(R) can be characterized more intrinsically as
follows. Let T be a quasi-exact triangle, which we can regard as a Z/3-graded chain
complex of free R-modules with H∗(T ) = 0. As T is free we have a short exact
sequence
2T →֒ T
։ 2T,
and the resulting long exact sequence in homology reduces to an isomorphism
σ : H∗(2T ) → H∗−1(2T ). As the grading is 3-periodic we can regard σ
3 as an
automorphism of H∗(2T ). We claim that T is exact if and only if σ
3 = 1. One
direction is straightforward: if T is contractible then H∗(2T ) = 0, and if T = X2
then Hi(2T ) = 2X for all i and σ is the identity. The converse is more fiddly and
we will not go through the details. It would be nice to give a proof of Theorem 1
based directly on this definition of exactness, but we do not know how to do so.
Remark 8. Let k be a field of characteristic 2. The same arguments as in the proof
of Theorem 1 show that the category F(k[ε]/ε2) of finitely generated free modules
over the algebra k[ε]/ε2 of dual numbers admits a triangulation with the identity
translation functor and such that the diagram
k[ε]/ε2
−→ k[ε]/ε2
−→ k[ε]/ε2
−→ k[ε]/ε2
is an exact triangle. However, this triangulated category is both algebraic and
topological, and hence, from our current perspective, less interesting.
Indeed F(k[ε]/ε2) is an algebraic and topological triangulated category for any
field k. The translation functor Σ = τ∗ is the restriction of scalars along the
k-algebra automorphism τ : k[ε]/ε2 → k[ε]/ε2 with τ(ε) = −ε, and
(9) k[ε]/ε2
−→ k[ε]/ε2
−→ k[ε]/ε2
−→ τ∗k[ε]/ε2
6 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
is an exact triangle. An algebraic model for this triangulated category was obtained
by Keller in [Kel05]. Keller’s model is a differerential graded (dg) k-category. Here
we exhibit an alternative model, which is a dg k-algebra A such that F(k[ε]/ε2) is
exact equivalent to the category of compact objects in the derived categoryD(A) of
dg (right) A-modules. This shows that F(k[ε]/ε2) is both algebraic and topological.
Let A = k〈a, u, v, v−1〉/I be the free graded k-algebra generated by a, u, v and
v−1 in degrees |a| = |u| = 0 and |v| = −1 modulo the two-sided homogeneous ideal
I generated by
a2, au+ ua+ 1, av + va and uv + vu.
The differential d : A→ A is determined by
d(a) = u2v, d(u) = 0, d(v) = 0,
and the Leibniz rule. The ungraded algebra H0(A) is isomorphic to the dual num-
bers k[ε]/ε2, where ε = [u] is the homology class of the cycle u. The graded algebra
H∗(A) is determined by this isomorphism since [v] is a unit in degree −1 such that
ε · [v] + [v] · ε = 0.
We claim that the 0-dimensional homology functor
H0 : D
c(A) −→ F(k[ε]/ε2)
is an equivalence of categories, where the left hand side is the full subcategory of
those dg A-modules whose H0 is finitely generated over k[ε]/ε
Let M be any dg A-module and let [x] ∈ H0(M) be a homology class with
[x] · ε = 0. We choose a representing cycle x and an element y with d(y) = xu; then
the element z = yuv−xa is a cycle with x = zu−d(ya), so [x] = [z] ·ε in homology.
So every homology class which is annihilated by ε is also divisible by ε, which proves
that H0(M) is a free k[ε]/ε
2-module. Moreover, the translation functor in D(A) is
the usual shift of complexes M 7→M [1] and the natural isomorphism
τ∗H0(M) ∼= H0(M [1]) = H−1(M)
is given by [x] 7→ [xv].
The universal case of this is M = A{x, y}, the free graded right module over
the underlying graded algebra of A with |x| = 0 and |y| = 1. We can endow M
with a dg A-module structure with d(x) = 0 and d(y) = xu, so that M is just the
mapping cone of the chain map A
→ A. The cycle z = yuv − xa ∈ M gives a
quasiisomorphism A→M . Using this, we obtain an exact triangle
−→ A[1]
in Dc(A) which maps to the exact triangle (9). The rest of the proof that H0 is
an exact equivalence from Dc(A) to F(k[ε]/ε2) is relatively straightforward, and we
omit it.
We still owe the proof that the triangulated category F(R) does not admit non-
trivial exact functors to or from a topological triangulated category. For this pur-
pose we introduce two intrinsic properties that an object A of a triangulated cate-
gory may have.
A Hopf map for an object A is a morphism η : ΣA → A which satisfies 2η = 0
and such that for some (hence any) exact triangle
(10) A
−→ ΣA
TRIANGULATED CATEGORIES WITHOUT MODELS 7
we have iηq = 2 · 1C . An object which admits a Hopf map will be termed hopfian.
We note that the class of hopfian objects is closed under isomorphism, suspension
and desuspension. If F is an exact functor with natural isomorphism τ : ΣF ∼= FΣ
and η : ΣA → A a Hopf map for A, then the composite F (η)τ : ΣF (A) −→ F (A)
is a Hopf map for F (A).
We call an object E exotic if there exists an exact triangle
(11) E
−→ ΣE
for some morphism h : E → ΣE. We note that the class of exotic objects is
closed under isomorphism, suspension and desuspension. Every exact functor takes
exotic objects to exotic objects. Every object of the triangulated category F(R) of
Theorem 1 is exotic.
We remark without proof that the morphism h which makes (11) exact is unique
and natural for morphisms between exotic objects. We show below that h is of the
form h = 2ψ for an isomorphism ψ : E → ΣE.
Remark 12. The integer 2 plays a special role in the definition of exotic objects,
which ultimately comes from the sign which arises in the rotation of a triangle. In
more detail, suppose that there is an exact triangle
(13) E
−→ ΣE
for some integer n. We claim that if E is nonzero, then n ≡ 2 mod 4 and 4 ·1E = 0,
so that the triangle (13) equals the ‘exotic’ triangle (11) with n = 2. Indeed, we
can find a morphism ψ : E → ΣE which makes the diagram
// ΣE
// ΣE
commute, and ψ is an isomorphism. We have nψ = h = −nψ which gives 2nψ = 0.
Since ψ is an isomorphism, this forces 2n · 1E = 0. Exactness of (13) lets us choose
a morphism f : E → E with 2 · 1E = n · f . But then 4 · 1E = n
2f2 = 0. So if n
is divisible by 4, then E = 0. If n is odd, then E is anhihilated by 4 and the odd
number n2, so also E = 0.
Hopf maps are incompatible with the property of being exotic in the sense that
these two classes of objects are orthogonal.
Proposition 14. Let T be a triangulated category, A a hopfian object and E an
exotic object. Then the morphism groups T(A,E) and T(E,A) are trivial. In par-
ticular, every exotic and hopfian object is a zero object.
Proof. Let η : ΣA → A be a Hopf map. Given any morphism f : E → A there
exists g : E → C such that (f, f, g) is a morphism from (11) to (10), and hence
if = 2g = iηqg = iη(Σf)h = iη(Σf)2ψ = 0. Here we use the notation of Remark
12 for n = 2 and the fact that 2η = 0. Moreover, (10) is exact, so f = 2f ′ for some
f ′ : E → A. This equation follows for any morphism f : E → A, hence f is divisible
by any power of 2, but 4 · 1E = 0, so f = 0.
The proof of T(A,E) = 0 is similar. Alternatively, we can reduce this statement
to the previous one by observing that the properties of being exotic and hopfian
8 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
are self-dual. In other words, an object E is exotic in a triangulated category T if
and only if E is exotic as an object of the opposite category Top with the opposite
triangulation, and similarly for Hopf maps. �
Proposition 15. Every object of a topological triangulated category is hopfian.
Proof. We can assume that the topological triangulated category is HoM for a
stable model category M. We use that for every object A of HoM there exists an
exact functor F : HoSp → HoM from the stable homotopy category which takes
the sphere spectrum to an object isomorphic to A. Here Sp is the category of
‘sequential spectra’ of simplicial sets with the stable model structure of Bousfield
and Friedlander [BF78, Sec. 2]. To construct F we let X be a cofibrant-fibrant
object of the model category M which is isomorphic to A in the homotopy category
HoM. The universal property of the model category of spectra [SS02, Thm. 5.1
(1)] provides a Quillen adjoint functor pair
Hom(X,−)
whose left adjoint X∧ takes the sphere spectrum S to X , up to isomorphism. The
left derived functor of the left Quillen functor X ∧ − : Sp → M is exact and can
serve as the required functor F .
Since exact functors preserve Hopf maps it thus suffices to treat the ‘universal
example’, i.e., to exhibit a Hopf map for the sphere spectrum as an object of the
stable homotopy category. The stable homotopy class η : ΣS → S of the Hopf map
from the 3-sphere to the 2-sphere precisely has this property, hence the name. In
more detail, we have an exact triangle
−→ S/2
−→ ΣS
in the stable homotopy category, where S/2 is the mod-2 Moore spectrum; then
the morphism 2 · 1S/2 factors as iηq, and moreover 2η = 0. �
In topological triangulated categories, something a little stronger than Proposi-
tion 15 is true in that Hopf maps can be chosen naturally for all objects. However,
we don’t need this and so we omit the details. Now we can give the
Proof of Theorem 2. Every object of the triangulated category F(R) is exotic and
every object of a topological triangulated category is hopfian. So an exact functor
from one type of triangulated category to the other hits objects which are both
exotic and hopfian. But such objects are trivial by Proposition 14. �
Remark 16. The only special thing we use in the proof of Theorem 2 about topolog-
ical triangulated categories is that therein every object has a Hopf map. Hopf maps
can also be obtained from other kinds of structure that were proposed by different
authors in order to ‘enrich’ or ‘enhance’ the notion of a triangulated category. So
our argument also proves that the triangulated category F(R) of Theorem 1 does
not admit such kinds of enrichments, and every exact functors to or from such
enriched triangulated categories is trivial. For example, if T is an algebraic trian-
gulated category, then for some (hence any) exact triangle (10) we have 2 · 1C = 0;
so the zero map is a Hopf map.
Another example of such extra structure is the notion of a triangulated derivator,
due to Grothendieck [Gro90], and the closely related notions of a stable homotopy
TRIANGULATED CATEGORIES WITHOUT MODELS 9
theory in the sense of Heller [Hel88, Hel97] or a system of triangulated diagram
categories in the sense of Franke [Fra96]. In each of these settings, the stable
homotopy category is the underlying category of the free example on one gener-
ator (the sphere spectrum). We do not know a precise reference of this fact for
triangulated derivators, but we refer to [Cis02, Cor. 4.19] for the ‘unstable’ (i.e.,
non-triangulated) analog. In Franke’s setting the universal property is formulated
as Theorem 4 of [Fra96]. These respective universal properties in the enhanced
context provide, for every object A, an exact functor (Ho Sp)cp → T which takes
the sphere spectrum S to A, up to isomorphism. This functors sends the classical
Hopf map for the sphere spectrum to a Hopf map for A.
Another kind of structure which underlies many triangulated categories is that of
a stable infinity category as investigated by Lurie in [Lur06]. The appropriate uni-
versal property of the infinity category of spectra is established in [Lur06, Cor. 17.6],
so again every object of the homotopy category of any stable, presentable infinity
category has a Hopf map.
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[BF78] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisim-
plicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill.,
1977), II. Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130.
[Cis02] D.-C. Cisinski, Propriétés universelles et extensions de Kan dérivées, Preprint (2002).
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Math. Soc. 59 (1946), 54–106.
[Fra96] J. Franke, Uniqueness theorems for certain triangulated categories possessing an Adams
spectral sequence, K-theory Preprint Archives #139 (1996).
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[GM03] S. I. Gelfand and Y. I. Manin, Methods of homological algebra, second ed., Springer
Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[Gro90] A. Grothendieck, Dérivateurs, manuscript, around 1990, partially available from
http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html
[Hel68] A. Heller, Stable homotopy categories, Bull. Amer. Math. Soc. 74 (1968), 28–63.
[Hel88] A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (1988), no. 383, vi+78 pp.
[Hel97] A. Heller, Stable homotopy theories and stabilization, J. Pure Appl. Algebra 115 (1997),
113-130.
[Hov99] M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American
Mathematical Society, Providence, RI, 1999.
[Kel05] B. Keller, On triangulated orbit categories, Doc. Math. 74 (2005), 551–581.
[Kel06] B. Keller, On differential graded categories, Proceedings of the International Congress
of Mathematicians, Madrid, Spain, 2006, vol. II, European Mathematical Society, 2006,
pp. 151–190.
[Lam99] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, 189.
Springer-Verlag, New-York, 1999.
[Lur06] J. Lurie, Derived algebraic geometry I: Stable infinity categories.
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[Nee01] A. Neeman, Triangulated Categories, Annals of Mathematics Studies, vol. 148, Princeton
University Press, Princeton, NJ, 2001.
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239 (2002), 803-828.
10 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
Universitat de Barcelona, Departament d’Àlgebra i Geometria, Gran Via de les
Corts Catalanes, 585, 08007 Barcelona, Spain
E-mail address: fmuro@ub.edu
Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany
E-mail address: schwede@math.uni-bonn.de
Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield
Road, Sheffield S3 7RH, UK
E-mail address: n.p.strickland@sheffield.ac.uk
	Introduction
	Acknowledgements
	The triangulated categories
	References

  We exhibit examples of triangulated categories which are neither the stable
category of a Frobenius category nor a full triangulated subcategory of the
homotopy category of a stable model category. Even more drastically, our
examples do not admit any non-trivial exact functors to or from these algebraic
respectively topological triangulated categories.

Introduction. Triangulated categories are fundamental tools in both algebra and
topology. In algebra they often arise as the stable category of a Frobenius cat-
egory ([Hel68, 4.4], [GM03, IV.3 Exercise 8]). In topology they usually appear
as a full triangulated subcategory of the homotopy category of a Quillen stable
model category [Hov99, 7.1]. The triangulated categories which belong, up to
exact equivalence, to one of these two families will be termed algebraic and topo-
logical, respectively. We borrow this terminology from [Kel06, 3.6] and [Sch06].
Algebraic triangulated categories are generally also topological, but there are many
well-known examples of topological triangulated categories which are not algebraic.
In the present paper we exhibit examples of triangulated categories which are
neither algebraic nor topological. As far as we know, these are the first examples
of this kind. Even worse (or better, depending on the perspective), our examples
do not even admit non-trivial exact functors to or from algebraic or topological
triangulated categories. In that sense, the new examples are completely orthogonal
to previously known triangulated categories.
Let (R,m) be a commutative local ring with m = (2) 6= 0 and m2 = 0. Examples
of this kind of rings are R = Z/4, or more generally R = W2(k) the 2-typical
Witt vectors of length 2 over a perfect field k of characteristic 2. There are also
examples which do not arise as Witt vectors, for instance the localization of the
polynomial ring Z/4[t] at the prime ideal (2). We denote by F(R) the category of
finitely generated free R-modules.
Theorem 1. The category F(R) has a unique structure of a triangulated category
with identity translation functor and such that the diagram
is an exact triangle.
Given an object X in an algebraic triangulated category T and an exact triangle
−→ A −→ C −→ ΣA,
1991 Mathematics Subject Classification. 18E30, 55P42.
Key words and phrases. Triangulated category, stable model category.
The first author was partially supported by the Spanish Ministry of Education and Science
under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship
EX2004-0616, and a Juan de la Cierva research contract.
http://arxiv.org/abs/0704.1378v2
2 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
the equation 2 · 1C = 0 holds, compare [Kel06, 3.6] and [Sch06]. Since the ring
R satisfies 2 · 1R 6= 0, the triangulation of the category F(R) is not algebraic. We
cannot rule out the possibility of a topological model for F(R) as easily: the classical
example of A = S the sphere spectrum in the stable homotopy category shows that
the morphism 2 · 1C can be nonzero in this more general context.
Nevertheless, F(R) is not topological either, which follows from Theorem 2. Here
we call an exact functor between triangulated categories trivial if it takes every
object to a zero object.
Theorem 2. Every exact functor from F(R) to a topological triangulated category
is trivial. Every exact functor from a topological triangulated category to F(R) is
trivial.
Acknowledgements. We are grateful to Bernhard Keller for helpful conversations
on the results of this paper, and to Amnon Neeman, who suggested the possibility
of constructing a triangulated structure on F(Z/4) by using Heller’s theory [Hel68].
In the original version of this note the first author alone constructed the tri-
angulation of the category F(Z/4) and proved that it does not admit any model.
The second author joined the project later by providing a simpler and more general
proof that the triangulation is not topological. The third author’s contribution
was an old preprint on the example considered in Remark 8, which provided some
guidance for the other results.
The triangulated categories. Let T be an additive category and let Σ: T
be a self-equivalence that we call translation functor. A candidate triangle (f, i, q)
in (T,Σ) is a diagram
(3) A
−→ ΣA,
where if , qi, and (Σf)q are zero morphisms. A morphism of candidate triangles
(α, β, γ) : (f, i, q) → (f ′, i′, q′) is a commutative diagram
// B′
// C′
// ΣA′
The category of candidate triangles is additive. The mapping cone of the morphism
(α, β, γ) is the candidate triangle
B ⊕A′
β f ′
// C ⊕B′
// ΣA⊕ C′
−Σf 0
Σα q′
// ΣB ⊕ ΣA′.
A homotopy (Θ,Φ,Ψ) from (α, β, γ) to (α′, β′, γ′) is given by morphisms
// B′
// C′
// ΣA′
such that
β′ − β = Φi+ f ′Θ, γ′ − γ = Ψq + i′Φ, Σ(α′ − α) = Σ(Θf) + q′Ψ.
TRIANGULATED CATEGORIES WITHOUT MODELS 3
We say in this case that the morphisms are homotopic. The mapping cones of
two homotopic morphisms are isomorphic. A contractible triangle is a candidate
triangle such that the identity is homotopic to the zero morphism. A homotopy
(Θ,Φ,Ψ) from 0 to 1 is called a contracting homotopy. Any morphism from or to
a contractible triangle is always homotopic to zero.
A triangulated category is a pair (T,Σ) as above together with a collection of
candidate triangles, called distinguished or exact triangles, satisfying the follow-
ing properties. The family of exact triangles is closed under isomorphisms. The
candidate triangle
(4) A
−→ A −→ 0 −→ ΣA,
is exact. Any morphism f : A → B in T can be extended to an exact triangle like
(3). A candidate triangle (3) is exact if and only if its translate
−→ ΣA
−→ ΣB,
is exact. Any commutative diagram
// ΣA
// B′
// C′
// ΣA′
whose rows are exact triangles can be extended to a morphism whose mapping
cone is also exact. This non-standard set of axioms for triangulated categories is
equivalent to the classical one, see [Nee01], and works better for the purposes of
this paper.
Now we are ready to prove Theorem 1.
Proof of Theorem 1. Given an object X in F(R) we consider the candidate triangle
X2 defined as
(5) X
−→ X.
We are going to prove that the category F(R) has a triangulated category struc-
ture with identity translation functor where the exact triangles are the candidate
triangles isomorphic to the direct sum of a contractible triangle and a candidate
triangle of the form (5).
The family of exact triangles is closed under isomorphisms by definition. The
candidate triangle (4) is contractible, and hence exact. The ring R is a quotient of a
discrete valuation ring with maximal ideal generated by 2, see [Coh46, Corollary 3];
therefore any morphism f : A→ B in F(R) can be decomposed up to isomorphism
1 0 0
0 2 0
0 0 0
 : A =W ⊕X ⊕ Y −→W ⊕X ⊕ Z = B.
Then f is extended by the direct sum of (5) and the contractible triangle
W ⊕ Y
// W ⊕ Z
// Y ⊕ Z
// W ⊕ Y.
4 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
The translate of a contractible triangle is also contractible, and the triangle (5) is
invariant under translation. This proves that the translate of an exact triangle is
exact. Translating a candidate triangle six times yields the original one, therefore
if a candidate triangle has an exact translate then the original candidate triangle
is also exact.
We say that a candidate triangle A
→ A is a quasi-exact triangle if
−→ B,
is an exact sequence of R-modules. The exact triangles are all quasi-exact.
Now we are going to show that any diagram of candidate triangles
(6) A
// B′
// C′
// A′
with exact rows can be completed to a morphism with exact mapping cone.
Suppose that the upper row in (6) is contractible and the lower row is quasi-
exact. Since f ′α = βf then f ′αq = 0; since C is projective, there exists γ′ : C → C′
such that q′γ′ = αq. Let (Θ,Φ,Ψ) be a contracting homotopy for the upper row.
Then γ = γ′ + (i′β − γ′i)Φ completes (6) to a morphism of candidate triangles.
If the upper row in (6) is quasi-exact and the lower row is contractible then
(6) can also be completed to a morphism. This can be shown directly, but it also
follows from the previous case since we have a duality functor
HomR(−, R) : F(R)
−→ F(R)op,
which preserves contractible triangles and quasi-exact triangles. Here we use that
R is injective as an R-module, see [Lam99, Example 3.12].
If the upper and the lower rows in (6) areX2 and Y2, respectively, then γ = β+2δ
extends (6) to a morphism of candidate triangles for any δ : X → Y .
This proves that any diagram like (6) with exact rows can be completed to a
morphism ϕ = (α, β, γ). Now we have to check that the completion can be done
in such a way that the mapping cone is exact. Suppose that the upper and the
lower rows are X2 ⊕ T and Y2 ⊕ T
′, respectively, with T and T ′ contractible. The
morphism ϕ is given by a matrix of candidate triangle morphisms
ϕ11 ϕ12
ϕ21 ϕ22
: X2 ⊕ T −→ Y2 ⊕ T
where ϕij = (αij , βij , γij). Here ϕ12, ϕ21 and ϕ22 are homotopic to 0 since either the
source or the target is contractible, therefore the mapping cone of ϕ is isomorphic
to the mapping cone of
ϕ11 0
: X2 ⊕ T −→ Y2 ⊕ T
which is the direct sum of the mapping cone of ϕ11 and two contractible triangles,
T ′ and the translate of T .
TRIANGULATED CATEGORIES WITHOUT MODELS 5
We can suppose that
α11 =
1 0 0
0 2 0
0 0 0
 : X = L⊕M ⊕N −→ L⊕M ⊕ P = Y.
Moreover, as we have seen above we can take γ11 = β11 + 2δ for
0 0 0
0 1 0
0 0 0
 : X = L⊕M ⊕N −→ L⊕M ⊕ P = Y.
We have 2β11 = 2α11, therefore β11 = α11 + 2Φ for some Φ: X → Y . Now we
observe that (δ,Φ, 0) is a homotopy from ϕ11 to ζ = (α11 + 2δ, α11 + 2δ, α11 + 2δ),
so the mapping cone of ϕ11 is isomorphic to the mapping cone of ζ.
The mapping cone of ζ is clearly the direct sum of five candidate triangles, namely
M2, N2, M2 (once again), P2, and the mapping cone of the identity 1 : L2 → L2,
which is contractible. Therefore the mapping cone of ζ is exact, and also the
mapping cone of ϕ11, ψ and ϕ.
It remains to show the uniqueness claim in Theorem 1. In any triangulation, all
contractible candidate triangles are exact [Nee01, 1.3.8]. The triangle X2 is a finite
direct sum of copies of R2. Hence every triangulation of (F(R), Id) which contains
R2 contains all the exact triangles which we considered above. Two triangulations
with the same translation functor necessarily agree if one class of triangles is con-
tained in the other, so there is only one triangulation in which R2 is exact. This
completes the proof. �
Remark 7. The exact triangles in F(R) can be characterized more intrinsically as
follows. Let T be a quasi-exact triangle, which we can regard as a Z/3-graded chain
complex of free R-modules with H∗(T ) = 0. As T is free we have a short exact
sequence
2T →֒ T
։ 2T,
and the resulting long exact sequence in homology reduces to an isomorphism
σ : H∗(2T ) → H∗−1(2T ). As the grading is 3-periodic we can regard σ
3 as an
automorphism of H∗(2T ). We claim that T is exact if and only if σ
3 = 1. One
direction is straightforward: if T is contractible then H∗(2T ) = 0, and if T = X2
then Hi(2T ) = 2X for all i and σ is the identity. The converse is more fiddly and
we will not go through the details. It would be nice to give a proof of Theorem 1
based directly on this definition of exactness, but we do not know how to do so.
Remark 8. Let k be a field of characteristic 2. The same arguments as in the proof
of Theorem 1 show that the category F(k[ε]/ε2) of finitely generated free modules
over the algebra k[ε]/ε2 of dual numbers admits a triangulation with the identity
translation functor and such that the diagram
k[ε]/ε2
−→ k[ε]/ε2
−→ k[ε]/ε2
−→ k[ε]/ε2
is an exact triangle. However, this triangulated category is both algebraic and
topological, and hence, from our current perspective, less interesting.
Indeed F(k[ε]/ε2) is an algebraic and topological triangulated category for any
field k. The translation functor Σ = τ∗ is the restriction of scalars along the
k-algebra automorphism τ : k[ε]/ε2 → k[ε]/ε2 with τ(ε) = −ε, and
(9) k[ε]/ε2
−→ k[ε]/ε2
−→ k[ε]/ε2
−→ τ∗k[ε]/ε2
6 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
is an exact triangle. An algebraic model for this triangulated category was obtained
by Keller in [Kel05]. Keller’s model is a differerential graded (dg) k-category. Here
we exhibit an alternative model, which is a dg k-algebra A such that F(k[ε]/ε2) is
exact equivalent to the category of compact objects in the derived categoryD(A) of
dg (right) A-modules. This shows that F(k[ε]/ε2) is both algebraic and topological.
Let A = k〈a, u, v, v−1〉/I be the free graded k-algebra generated by a, u, v and
v−1 in degrees |a| = |u| = 0 and |v| = −1 modulo the two-sided homogeneous ideal
I generated by
a2, au+ ua+ 1, av + va and uv + vu.
The differential d : A→ A is determined by
d(a) = u2v, d(u) = 0, d(v) = 0,
and the Leibniz rule. The ungraded algebra H0(A) is isomorphic to the dual num-
bers k[ε]/ε2, where ε = [u] is the homology class of the cycle u. The graded algebra
H∗(A) is determined by this isomorphism since [v] is a unit in degree −1 such that
ε · [v] + [v] · ε = 0.
We claim that the 0-dimensional homology functor
H0 : D
c(A) −→ F(k[ε]/ε2)
is an equivalence of categories, where the left hand side is the full subcategory of
those dg A-modules whose H0 is finitely generated over k[ε]/ε
Let M be any dg A-module and let [x] ∈ H0(M) be a homology class with
[x] · ε = 0. We choose a representing cycle x and an element y with d(y) = xu; then
the element z = yuv−xa is a cycle with x = zu−d(ya), so [x] = [z] ·ε in homology.
So every homology class which is annihilated by ε is also divisible by ε, which proves
that H0(M) is a free k[ε]/ε
2-module. Moreover, the translation functor in D(A) is
the usual shift of complexes M 7→M [1] and the natural isomorphism
τ∗H0(M) ∼= H0(M [1]) = H−1(M)
is given by [x] 7→ [xv].
The universal case of this is M = A{x, y}, the free graded right module over
the underlying graded algebra of A with |x| = 0 and |y| = 1. We can endow M
with a dg A-module structure with d(x) = 0 and d(y) = xu, so that M is just the
mapping cone of the chain map A
→ A. The cycle z = yuv − xa ∈ M gives a
quasiisomorphism A→M . Using this, we obtain an exact triangle
−→ A[1]
in Dc(A) which maps to the exact triangle (9). The rest of the proof that H0 is
an exact equivalence from Dc(A) to F(k[ε]/ε2) is relatively straightforward, and we
omit it.
We still owe the proof that the triangulated category F(R) does not admit non-
trivial exact functors to or from a topological triangulated category. For this pur-
pose we introduce two intrinsic properties that an object A of a triangulated cate-
gory may have.
A Hopf map for an object A is a morphism η : ΣA → A which satisfies 2η = 0
and such that for some (hence any) exact triangle
(10) A
−→ ΣA
TRIANGULATED CATEGORIES WITHOUT MODELS 7
we have iηq = 2 · 1C . An object which admits a Hopf map will be termed hopfian.
We note that the class of hopfian objects is closed under isomorphism, suspension
and desuspension. If F is an exact functor with natural isomorphism τ : ΣF ∼= FΣ
and η : ΣA → A a Hopf map for A, then the composite F (η)τ : ΣF (A) −→ F (A)
is a Hopf map for F (A).
We call an object E exotic if there exists an exact triangle
(11) E
−→ ΣE
for some morphism h : E → ΣE. We note that the class of exotic objects is
closed under isomorphism, suspension and desuspension. Every exact functor takes
exotic objects to exotic objects. Every object of the triangulated category F(R) of
Theorem 1 is exotic.
We remark without proof that the morphism h which makes (11) exact is unique
and natural for morphisms between exotic objects. We show below that h is of the
form h = 2ψ for an isomorphism ψ : E → ΣE.
Remark 12. The integer 2 plays a special role in the definition of exotic objects,
which ultimately comes from the sign which arises in the rotation of a triangle. In
more detail, suppose that there is an exact triangle
(13) E
−→ ΣE
for some integer n. We claim that if E is nonzero, then n ≡ 2 mod 4 and 4 ·1E = 0,
so that the triangle (13) equals the ‘exotic’ triangle (11) with n = 2. Indeed, we
can find a morphism ψ : E → ΣE which makes the diagram
// ΣE
// ΣE
commute, and ψ is an isomorphism. We have nψ = h = −nψ which gives 2nψ = 0.
Since ψ is an isomorphism, this forces 2n · 1E = 0. Exactness of (13) lets us choose
a morphism f : E → E with 2 · 1E = n · f . But then 4 · 1E = n
2f2 = 0. So if n
is divisible by 4, then E = 0. If n is odd, then E is anhihilated by 4 and the odd
number n2, so also E = 0.
Hopf maps are incompatible with the property of being exotic in the sense that
these two classes of objects are orthogonal.
Proposition 14. Let T be a triangulated category, A a hopfian object and E an
exotic object. Then the morphism groups T(A,E) and T(E,A) are trivial. In par-
ticular, every exotic and hopfian object is a zero object.
Proof. Let η : ΣA → A be a Hopf map. Given any morphism f : E → A there
exists g : E → C such that (f, f, g) is a morphism from (11) to (10), and hence
if = 2g = iηqg = iη(Σf)h = iη(Σf)2ψ = 0. Here we use the notation of Remark
12 for n = 2 and the fact that 2η = 0. Moreover, (10) is exact, so f = 2f ′ for some
f ′ : E → A. This equation follows for any morphism f : E → A, hence f is divisible
by any power of 2, but 4 · 1E = 0, so f = 0.
The proof of T(A,E) = 0 is similar. Alternatively, we can reduce this statement
to the previous one by observing that the properties of being exotic and hopfian
8 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
are self-dual. In other words, an object E is exotic in a triangulated category T if
and only if E is exotic as an object of the opposite category Top with the opposite
triangulation, and similarly for Hopf maps. �
Proposition 15. Every object of a topological triangulated category is hopfian.
Proof. We can assume that the topological triangulated category is HoM for a
stable model category M. We use that for every object A of HoM there exists an
exact functor F : HoSp → HoM from the stable homotopy category which takes
the sphere spectrum to an object isomorphic to A. Here Sp is the category of
‘sequential spectra’ of simplicial sets with the stable model structure of Bousfield
and Friedlander [BF78, Sec. 2]. To construct F we let X be a cofibrant-fibrant
object of the model category M which is isomorphic to A in the homotopy category
HoM. The universal property of the model category of spectra [SS02, Thm. 5.1
(1)] provides a Quillen adjoint functor pair
Hom(X,−)
whose left adjoint X∧ takes the sphere spectrum S to X , up to isomorphism. The
left derived functor of the left Quillen functor X ∧ − : Sp → M is exact and can
serve as the required functor F .
Since exact functors preserve Hopf maps it thus suffices to treat the ‘universal
example’, i.e., to exhibit a Hopf map for the sphere spectrum as an object of the
stable homotopy category. The stable homotopy class η : ΣS → S of the Hopf map
from the 3-sphere to the 2-sphere precisely has this property, hence the name. In
more detail, we have an exact triangle
−→ S/2
−→ ΣS
in the stable homotopy category, where S/2 is the mod-2 Moore spectrum; then
the morphism 2 · 1S/2 factors as iηq, and moreover 2η = 0. �
In topological triangulated categories, something a little stronger than Proposi-
tion 15 is true in that Hopf maps can be chosen naturally for all objects. However,
we don’t need this and so we omit the details. Now we can give the
Proof of Theorem 2. Every object of the triangulated category F(R) is exotic and
every object of a topological triangulated category is hopfian. So an exact functor
from one type of triangulated category to the other hits objects which are both
exotic and hopfian. But such objects are trivial by Proposition 14. �
Remark 16. The only special thing we use in the proof of Theorem 2 about topolog-
ical triangulated categories is that therein every object has a Hopf map. Hopf maps
can also be obtained from other kinds of structure that were proposed by different
authors in order to ‘enrich’ or ‘enhance’ the notion of a triangulated category. So
our argument also proves that the triangulated category F(R) of Theorem 1 does
not admit such kinds of enrichments, and every exact functors to or from such
enriched triangulated categories is trivial. For example, if T is an algebraic trian-
gulated category, then for some (hence any) exact triangle (10) we have 2 · 1C = 0;
so the zero map is a Hopf map.
Another example of such extra structure is the notion of a triangulated derivator,
due to Grothendieck [Gro90], and the closely related notions of a stable homotopy
TRIANGULATED CATEGORIES WITHOUT MODELS 9
theory in the sense of Heller [Hel88, Hel97] or a system of triangulated diagram
categories in the sense of Franke [Fra96]. In each of these settings, the stable
homotopy category is the underlying category of the free example on one gener-
ator (the sphere spectrum). We do not know a precise reference of this fact for
triangulated derivators, but we refer to [Cis02, Cor. 4.19] for the ‘unstable’ (i.e.,
non-triangulated) analog. In Franke’s setting the universal property is formulated
as Theorem 4 of [Fra96]. These respective universal properties in the enhanced
context provide, for every object A, an exact functor (Ho Sp)cp → T which takes
the sphere spectrum S to A, up to isomorphism. This functors sends the classical
Hopf map for the sphere spectrum to a Hopf map for A.
Another kind of structure which underlies many triangulated categories is that of
a stable infinity category as investigated by Lurie in [Lur06]. The appropriate uni-
versal property of the infinity category of spectra is established in [Lur06, Cor. 17.6],
so again every object of the homotopy category of any stable, presentable infinity
category has a Hopf map.
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10 FERNANDO MURO, STEFAN SCHWEDE, AND NEIL STRICKLAND
Universitat de Barcelona, Departament d’Àlgebra i Geometria, Gran Via de les
Corts Catalanes, 585, 08007 Barcelona, Spain
E-mail address: fmuro@ub.edu
Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany
E-mail address: schwede@math.uni-bonn.de
Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield
Road, Sheffield S3 7RH, UK
E-mail address: n.p.strickland@sheffield.ac.uk
	Introduction
	Acknowledgements
	The triangulated categories
	References
