THE ADDITIVITY OF TRACES IN TRIANGULATED
CATEGORIES
J. P. MAY
Abstract.We explain a fundamental additivity theorem for Euler charac-
teristics and generalized trace maps in triangulated categories. The pr*
*oof
depends on a refined axiomatization of symmetric monoidal categories wit*
*h a
compatible triangulation. The refinement consists of several new axioms *
*re-
lating products and distinguished triangles. The axioms hold in the exam*
*ples
and shed light on generalized homology and cohomology theories.
Contents
1. Generalized trace maps 2
2. Triangulated categories 6
3. Weak pushouts and weak pullbacks 9
4. The compatibility axioms 11
5. How to prove Verdier's axiom 17
6. How to prove the braid and additivity axioms 21
7. How to prove the braid duality axiom 22
8. The proof of the additivity theorem for traces 27
9. Homology and cohomology theories 29
References 31
Let C be a closed symmetric monoidal category with a compatible triangulation.
We shall give a precise definition that explains what we mean by this in x4. We*
* write
S for the unit object of C , ^ for the product, F for the internal hom functor,*
* and
DX = F (X, S) for the dual object of X. The reader so inclined should read fo*
*r ^
and Hom for F . For any object X, we have an evaluation map " : DX ^ X -! X.
As recalled in [11, x2], when X is dualizable we also have a coevaluation map
j : S -! X ^ DX. The Euler characteristic Ø(X) is then the composite
S __j_//_X ^ DX_fl_//DX ^ X_"__//S,
where fl is the commutativity isomorphism. We shall prove the following theorem.
Theorem 0.1. Assume given a distinguished triangle
(0.2) X __f_//_Yg__//Z_h_//_ X.
If X, Y , and therefore Z are dualizable, then Ø(Y ) = Ø(X) + Ø(Z).
____________
1991 Mathematics Subject Classification. Primary 18D10, 18E30, 18G55, 55U35.
It is a pleasure to thank Mark Hovey for the the proofs of Lemmas 3.6 and ot*
*her help and to
thank Halvard Fausk, John Greenlees, and Gaunce Lewis for careful reading and h*
*elpful comments.
1
2 J. P. MAY
Some of the significance of this basic result is discussed in [11]. In fact, *
*we shall
prove a more general additivity theorem of the same nature. We discuss generali*
*zed
trace maps and state the generalization in x1.
Philosophically, we view our additivity "theoremsä s basic results that must
hold in any closed symmetric monoidal category with a öc mpatible" triangulatio*
*n.
That is, our aim is less to prove the theorems than to explain the proper meani*
*ng
of the word öc mpatible".
In xx2,3, we define triangulated categories and briefly discuss homotopy push*
*outs
and pullbacks in such categories. We make heavy use of Verdier's axiom in our
work, and we take the opportunity to show that the axiom in the definition of a
triangulated category that is usually regarded as the most substantive one is i*
*n fact
redundant: it is implied by Verdier's axiom and the remaining, less substantial,
axioms. Strangely, since triangulated categories have been in common use for ov*
*er
thirty years, this observation seems to be new.
We explain our new axioms for the definition of a compatible triangulation on
a symmetric monoidal category and show how they imply Theorem 0.1 in x4. The
new axioms relate the product ^ and duality to distinguished triangles. The need
for the new axioms is not so strange, since the first published formulation of *
*com-
patibility conditions that I know of is only a few years old [6] and the new ax*
*ioms
are considerably less transparent than the others in this theory.
The axioms are folklore results in the stable homotopy category. They can also
be verified in the usual derived categories in algebraic geometry and homologic*
*al
algebra and in the Morel-Voevodsky A1-stable homotopy categories. We shall ex-
plain both intuitively and model theoretically what is involved in the verifica*
*tions in
xx5-7. The model theoretical material in those sections is the technical heart *
*of the
paper. A disclaimer may be in order. In view of what is involved in the verific*
*ation
of the axioms, they are unlikely to be satisfied except in triangulated categor*
*ies
that arise as the homotopy categories of suitable model categories. Nevertheles*
*s,
we shall see that, despite their complicated formulations, the axioms record in*
*for-
mation that is intuitively transparent. We show how to prove the generalization
Theorem 1.9 of Theorem 0.1 in x8.
The axioms give information that has been used in stable homotopy theory for
decades. Adams' 1971 Chicago lectures [1, IIIx9] gave a systematic account of p*
*rod-
ucts in homology and cohomology theories that implicitly used one version of th*
*ese
axioms, and I first formulated some of the axioms in forms similar to those giv*
*en
here in unpublished notes written soon after. In x9, I will briefly indicate th*
*e role
the axioms play in generalized homology and cohomology theories. The discussion
applies to any symmetric monoidal category with a compatible triangulation.
One moral of this paper is that the types of structured categories we conside*
*r are
still not well understood, despite their ubiquitous appearance in algebraic top*
*ology,
homological algebra, and algebraic geometry. We will leave several problems abo*
*ut
them unresolved.
1. Generalized trace maps
We recall the following definition from [8, III.7.1]. We do not need the tria*
*ngu-
lation of C here, just the closed symmetric monoidal structure.
Definition 1.1. Let X be a dualizable object of C with a self-map f : X -! X.
Let C be any object of C and suppose given a map = X : X -! X ^C. Define
THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES 3
the trace of f with respect to , denoted ø(f), to be the composite
S __j_//_X ^ DX_fl_//DX ^ Xid^f//_DX ^ Xid^//_DX ^ X ^ C"^id//_S ^ C ~=C.
Since (f ^ id) O j = (id^Df) O j and Ö (id^f) = Ö (Df ^ id), easy diagram
chases show that the same map ø(f) is obtained if we insert any of the following
four composites between fl O j and " ^ id:
id^f // id^
DX ^ X __________//_DX ^ X________//_DX ^ X ^ C
Df^id
id^f^id //
DX ^ X ___id^____//DX ^ X ^ C_________//_DX ^ X ^ C.
Df^id^ id
If C = S and is the unit isomorphism X ~=X ^ S, then ø(f) is denoted Ø(f)
and is called the trace or Lefschetz constant of f. The trace of the identity m*
*ap is
the Euler characteristic of X. If C = X, then is thought of as a diagonal map
and ø(id) : S -! X is called the transfer map of X with respect to .
The definition includes a variety of familiar maps in algebra, algebraic geom*
*etry,
and algebraic topology. If C is the category of vector spaces over a field and*
* X
is a finite dimensional vector space, then Ø(f) is just the classical trace of *
*the
linear transformation f. If X is graded, then Ø(X) is just the classical Euler
characteristic. The classical (reduced) Euler characteristics and Lefschetz num*
*bers
in algebraic topology are also special cases. The essential point in the verifi*
*cation
of assertions such as these is the additivity theorem that we prove in this pap*
*er.
In the most interesting situations, C is a comonoid (or coalgebra) with copro*
*duct
: C -! C ^ C and counit , : C -! S and : X -! X ^ C is a coaction of C
on X, meaning that the following diagrams commute:
X ________//_X ^ C and X GGG
GGGGG
|| id^|| || GGGGGGGG
fflffl| fflffl| fflffl|GG
X ^ C __^id//_X ^ C ^ C X ^ C id^,//_X.
The second diagram implies the commutativity of the diagram
Ø(f)
_______________________________________________*
*______________________________________________________________@
____________%%_____________________________________*
*_____________________________
S _ø(f)//_C_,_//_S,
which is familiar and important in a variety of contexts. We recall the followi*
*ng
further formal properties of generalized trace maps from [8, IIIx7]. The proofs*
* are
easy diagram chases, some of which use the alternative descriptions of ø(f) giv*
*en
in Definition 1.1. Assume that X and Y are dualizable.
Lemma 1.2 (Unit property). For any map f : S -! S, Ø(f) = f.
4 J. P. MAY
Lemma 1.3 (Fixed point property). If h : C -! C is a map such that the following
diagram commutes, then h O ø(f) = ø(f):
X _____//X ^ C
f || |f^h|
fflffl| fflffl|
X _____//X ^ C.
For example, when C = X and is a diagonal of the usual sort, we have
(f ^ f) O = O f and can take h = f. This property is closely related to the
Lefschetz fixed point theorem.
Lemma 1.4 (Invariance under retraction). Let i : X -! Y and r : Y -! X be a
retraction, r O i = id. Let X : X -! X ^ C, Y : Y -! Y ^ D, and h : C -! D
be maps such that the following diagram commutes:
X
X _____//X ^ C
i|| |i^h|
fflffl| fflffl|
Y __Y__//Y ^ D.
Then h O ø(f) = ø(i O f O r) for any map f : X -! X.
For example, we can take C = D and Y = (i ^ id) O X O r. When i is an
isomorphism with inverse r, this gives invariance under isomorphism.
Duality and traces are natural with respect to symmetric monoidal functors, by
[8, III.1.9, III.7.7].
Proposition 1.5. Let F : C -! D be a symmetric monoidal functor such that the
unit map ~ : T -! F S is an isomorphism, where T is the unit object of D. Let X
be a dualizable object of C such that the product map
OE : F X ^ F D(X) -! F (X ^ DX)
is an isomorphism. Then F X is dualizable in D, the natural map F DX -! DF X
is an isomorphism, and
OE : F X ^ F Z -! F (X ^ Z)
is an isomorphism for every object Z of C . Given X : X -! X ^ C, define
FX = OE-1 O F X : F X -! F X ^ F C. Then, regarding ~ as an identification,
ø(F f) = F ø(f) : T -! F C for any map f : X -! X.
Returning to the algebraic properties of trace maps, we first record their be*
*havior
with respect to ^-products, coproducts, and suspension, and then formulate our
additivity theorem.
Lemma 1.6 (Commutation with ^-products). Given maps X : X -! X ^ C
and Y : Y -! Y ^ D, define
X^Y = (id^fl^id)O( X ^ Y ) : X^Y -! (X^C)^(Y ^D) -! (X^Y )^(C^D).
Then ø(f ^ g) = ø(f) ^ ø(g) : S -! C ^ D for any f : X -! X and g : Y -! Y .
Now assume that C is additive with coproduct _; it follows that ^ is bilinear.
THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES 5
Lemma 1.7 (Commutation with sums). Given maps X : X - ! X ^ C and
Y : Y -! Y ^ C, define
X_Y = X _ Y : X _ Y -! (X ^ C) _ (Y ^ C) ~=(X _ Y ) ^ C
Then ø(h) = ø(f) + ø(g) : S -! C for any map h : X _ Y - ! X _ Y , where
f : X -! X and g : Y -! Y are obtained from h by restriction and retraction.
That is, as one would expect of a trace, the cross terms X -! Y and Y -! X
of h make no contribution. Now assume our original hypothesis that C has a
triangulation compatible with its symmetric monoidal structure. A diagram chase
from (TC1) of Definition 4.1 gives the following generalization of [11, 4.7].
Lemma 1.8 (Anticommutation with suspension). Given X : X -! X^C, define
X : X -! ( X) ^ C by suspending X and using the canonical isomorphism
(X ^ C) ~=( X) ^ C. Then ø( f) = -ø(f) for any map f : X -! X.
The following result is our generalization of Theorem 0.1. For reasons that w*
*ill
become clear in x8, we now assume that C is the homotopy category of a closed
symmetric monoidal model category B that satisfies the usual properties that le*
*ad
to a triangulation on C that is compatible with its smash product. These proper*
*ties
are made precise at the start of xx5, 6.
Theorem 1.9 (Additivity on distinguished triangles). Let X, Y , and therefore Z
be dualizable in the distinguished triangle (0.2). Assume given maps OE : X -! X
and _ : Y -! Y and maps X : X -! X ^ C and Y : Y -! Y ^ C such that
the left squares commute in the following two diagrams:
f g h
X _____//Y____//Z____//_ X
OE|| |_| |!| ||OE
fflffl|fflffl|fflffl| fflffl|
X __f__//Y_g__//Z_h__//_ X
f g h
X ________//_Y_______//_Z________//_ X
X || |Y| ||Z || X
fflffl| fflffl| |fflffl fflffl|
X ^ C f^id//_Y ^ Cg^id//_Z ^hC^id//_ (X ^ C).
Then there are maps ! : Z -! Z and Z : Z -! Z ^ C such that these diagrams
commute and the additivity relation ø(_) = ø(!) + ø(OE) holds.
A result like this was first formulated in [8, III.7.6], in the context of eq*
*uivari-
ant stable homotopy theory. It has important calculational consequences in that
subject, and it should be of comparable significance in other areas.
Remark 1.10. I do not know whether or not the conclusion holds for every choice
of ! and Z that make the displayed diagrams commute, but I would expect not.
This was claimed to hold in [8, III.7.6], but even in that special context the *
*proof is
incomplete. The question is related to Neeman's work in [12], where it is empha*
*sized
that some fill-ins in diagrams such as these are better than others. The theorem
has a slight caveat in the generality of traces, as opposed to Lefschetz consta*
*nts;
see Remark 8.3.
6 J. P. MAY
2.Triangulated categories
We recall the definition of a triangulated category from [17]; see also [2, 6*
*, 10].
Actually, one of the axioms in all of these treatments is redundant, namely the*
* one
used to construct the maps ! and on Z in the additivity theorem just stated.
The most fundamental axiom is called Verdier's axiom, or the octahedral axiom
after one of its possible diagrammatic shapes. However, the shape that I find m*
*ost
convenient, a braid, does not appear in the literature of triangulated categori*
*es. It
does appear in Adams [1, p. 212], who used the term "sine wave diagram" for it.
We call a diagram (0.2) a "triangleä nd use the notation (f, g, h) for it.
Definition 2.1. A triangulation on an additive category C is an additive self-
equivalence : C -! C together with a collection of triangles, called the dist*
*in-
guished triangles, such that the following axioms hold.
Axiom T 1. Let X be any object and f : X -! Y be any map in C .
(a) The triangle X id-!X -! * -! X is distinguished.
(b) The map f : X -! Y is part of a distinguished triangle (f, g, h).
(c) Any triangle isomorphic to a distinguished triangle is distinguished.
Axiom T 2. If (f, g, h) is distinguished, then so is (g, h, - f).
Axiom T 3 (Verdier's axiom). Consider the following diagram.
__h_____________________________________________________________*
*______________________g0______________________________________@
__________________________________________________________________*
*______________________________________________________________@
_________________________________________________________!!________*
*________________________________________________""____________@
X @@ ~Z?@? -=W=EE y__> Ci(h) 0 ==z EEE x<> __ FFF xx<< DDD x;;x
>>> g___ FßFF ßxxx DffiDD xxx
f >>OEOE>____ F""F xxx DD""Dxxxxß
Y BB Z=X__