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<U<
             @@     g~~~    @h0@   j0-      Eg00EEE  yyyy
             f@@@__~~~        @ØØ --          EE"" yyy f0

                 Y @@          ?V?C             <Y<
                            j ~    C  00      yyy
                    @@@     ~~       hC     yy
                    f0@@ØØ~           C!! yyy f
                        U _____________==_________________ X
                           ____________________________________________________*
 *_________________________
                             __________________________________________________*
 *______________________________________________________________@
                                f00


Assume that h = gOf, j00=  f0Og00, and (f, f0, f00) and (g, g0, g00) are distin*
 *guished.
If h0 and h00are given such that (h, h0, h00) is distinguished, then there are *
 *maps j
and j0 such that the diagram commutes and (j, j0, j00) is distinguished.  We ca*
 *ll
the diagram a braid of distinguished triangles generated by h = g O f or a braid
cogenerated by j00=  f0O g00.


  We have labeled our axioms (T?), and we will compare them with Verdier's
original axioms (TR?). Our (T1) is Verdier's (TR1) [17], our (T2) is a weak form
of Verdier's (TR2), and our (T3) is Verdier's (TR4).  We have omitted Verdier's
(TR3), since it is exactly the conclusion of the following result.


Lemma 2.2 (TR3). If the rows are distinguished and the left square commutes
in the following diagram, then there is a map k that makes the remaining squares




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES         7


commute.
                         f       g       h
                     X _____//_Y____//_Z____//_Ø X

                     i||      j||     kØØ     ||i
                      fflffl|ffflffl|0fflfflg0fflffl|h0
                     X0 ____//_Y_0__//Z0____// X0

Proof.This is part of the 3x3 lemma, which we state and prove below. The point *
 *is
that the construction of the commutative diagram in that proof requires only (T*
 *1),
(T2), and (T3), not the conclusion of the present lemma; compare [2, 1.1.11].

  Verdier's (TR2) includes the converse, (T20) say, of (T2). That too is a cons*
 *e-
quence of our (T1), (T2), and (T3). A standard argument using only (T1), (T2),
(TR3), and the fact that   is an equivalence of categories shows that, for any *
 *object
A, a distinguished triangle (f, g, h) induces a long exact sequence upon applic*
 *ation
of the functor C (A, -). Here we do not need the converse of (T2) because we are
free to replace A by  -1A. In turn, by the five lemma and the Yoneda lemma, this
implies the following addendum to the previous lemma.

Lemma 2.3. If i and j in (TR3) are isomorphisms, then so is k.

Lemma 2.4 (T20). If (g, h, - f) is distinguished, then so is (f, g, h).

                                                 0      h0
Proof.Choose a distinguished triangle X __f__//Yg__//_Z0__//_ X.By (T2), the
triangles (- f, - g0, - h0) and (- f, - g, - h) are distinguished.  By Lem-
mas 2.2 and 2.3, they are isomorphic. By desuspension, (f, g, h) is isomorphic *
 *to
(f0, g0, h0). By (T1), it is distinguished.

  Similarly, we can derive the converse version, (T30) say, of Verdier's axiom *
 *(T3).

Lemma 2.5 (T30). In the diagram of (T3), if j and j0are given such that (j, j0,*
 * j00)
is distinguished, then there are maps h0 and h00such that the diagram commutes
and (h, h0, h00) is distinguished.

Proof.Desuspend a braid of distinguished triangles generated by j00=  f0Og00.

Lemma 2.6 (The 3 x 3 lemma). Assume that j O f = f0O i and the two top rows
and two left columns are distinguished in the following diagram.

                        f        g       h
                   X ______//_Y_____//_Z_____//Ø X

                   i||       j||      kØØ      |i|
                    fflffl|f0fflffl|g0fflfflh0 fflffl|
                   X0 _____//_Y_0___//Z0____//_Ø X0

                  i0||       j0||     k0ØØ     |i0|
                    fflffl|f0fflffl|0gfflffl00hfflffl|00
                   X00` ` `//Y 00```//Z00```//Ø X00

                  i00||      j00||    k00ØØ    -|i00|
                    fflffl|  fflffl|  fflffl   fflffl|
                   X  __f__// Y__g__// Z-_h_//_ 2X


Then there is an object Z00and there are dotted arrow maps f00, g00, h00, k, k0,
k00such that the diagram is commutative except for its bottom right square, whi*
 *ch
commutes up to the sign -1, and all four rows and columns are distinguished.




8                              J. P. MAY


Proof.The bottom row is isomorphic to the triangle (- f, - g, - h) and is thus
distinguished by (T2); similarly the right column is distinguished. Applying (T*
 *1),
we construct a distinguished triangle



                       X__jOf//_Y_0p//_Vq__// X.


Applying (T3), we obtain braids of distinguished triangles generated by j O f a*
 *nd
f0O i. These give distinguished triangles

                                             00
                       Z__s__//V_t__//Y_00gOj//_ Z


                           0     t0    i0Oh0
                      X00_s__//_V___//Z0___// X00

such that

             p O j = s O g,  t O p = j0,  q O s = h,  j00O t =  f O q


           p O f0 = s0O i0,  t0O p = g0,  q O s0= i00,  h0O t0=  i O q.

Define k = t0O s : Z -! Z0. Then k O g = g0O j and h0O k =  i O h, which already
completes the promised proof of Lemma 2.2. Define f00= t O s0and apply (T1) to
construct a distinguished triangle


                         f00     00g00 00h00    00
                     X00____//_Y____//Z____//_ X.


Applying (T3), we obtain a braid of distinguished triangles generated by f00= t*
 *Os0.
Here we start with the distinguished triangles (s0, t0,  i0O h0) and (t,  g O j*
 *00, - s),
where the second is obtained by use of (T2). This gives a distinguished triangle

                           0     k00   - k
                      Z0 _k__//_Z00__//V___//_ Z`


such that the squares left of and above the bottom right square commute and


                 g00O t = k0O t0 and  -  s O k00=  s0O h00.


The commutativity (and anti-commutativity of the bottom right square) of the
diagram follow immediately.  It also follows immediately that (f00, g00, h00) a*
 *nd
(k0, k00, - k) are distinguished. Lemma 2.4 implies that (k, k0, k00) is distin*
 *guished.



Remark 2.7. Conversely, Verdier's axiom is implied by (T1), (T2), and the 3 x 3
lemma. To see this, apply the 3 x 3 lemma starting with the top left square


                                 h=gOf
                              X _____//Z

                              f||     ||||
                               fflffl|||
                              Y __g__//Z.




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES         9


               3. Weak pushouts and weak pullbacks

  In any category, weak limits and weak colimits satisfy the existence but not
necessarily the uniqueness in the defining universal properties. They need not *
 *be
unique and need not exist. When constructed in particularly sensible ways, they
are called homotopy limits and colimits and are often unique up to non-canonical
isomorphism. As we recall here, there are such homotopy pushouts and pullbacks
in triangulated categories. Homotopy colimits and limits of sequences of maps in
triangulated categories are studied in [3, 13], but a complete theory of homoto*
 *py
limits and colimits in triangulated categories is not yet available.  The mater*
 *ial
in this section is meant to clarify ideas and will not be used in the proofs of*
 * the
additivity theorems. However, it seems to me that there should be better proofs
that do make use of this material, although I have not been able to find them.

Definition 3.1. A homotopy pushout of maps f : X -! Y and g : X -! Z is a
distinguished triangle

                       (f,-g)     (j,k)   i
                     X ____//_Y __Z__//_W____// X.

A homotopy pullback of maps j : Y - ! W and k : Z -! W is a distinguished
triangle
                           -1i   (f,g)    (j,-k)
                     -1W -____//X____//Y __Z___//W.

The sign is conventional and ensures that in the isomorphism of extended triang*
 *les

                    - -1i// (f,-g)//   (j,k)//   i //
               -1W  _____X  _____Y _ Z _____W _____  X
                ||        ||       |         ||      ||
                ||        ||       |(id,- id)||      ||
                ||  - -1i ||(f,g)  fflffl|(j,||-k)i  ||
               -1W  _____//X____//_Y _ Z___//_W____// X,

the top row displays a homotopy pushout if and only if the bottom row displays a
homotopy pullback.

  At this point we introduce a generalization of the distinguished triangles.

Definition 3.2. A triangle (f, g, h) is exact if it induces long exact sequence*
 *s upon
application of the functors C (-, W ) and C (W, -) for every object W of C .

  The following is a standard result in the theory of triangulated categories [*
 *17].

Lemma 3.3. Every distinguished triangle is exact.

  If (f, g, h) is distinguished, then (f, g, -h) is exact but generally not dis*
 *tin-
guished. These exact triangles (f, g, -h) give a second triangulation of C , wh*
 *ich
we call the negative of the original triangulation.

Problem 3.4. The relationship between distinguished and exact triangles has not
been adequately explored in the literature. Can a triangulated category C admit*
 * a
triangulation with a given functor   that differs from both the original triang*
 *ulation
and its negative?  Consideration of automorphisms of objects shows that there
usually are exact triangles in C that are in neither the original triangulation*
 * nor
its negative. Nevertheless, it seems possible that the answer is no.




10                             J. P. MAY


  The fact that the triangles in Definition 3.1 give rise to weak pushouts and *
 *weak
pullbacks depends only on the fact that they are exact, not on the assumption t*
 *hat
they are distinguished. This motivates the following definition.

Definition 3.5. For exact triangles of the form displayed in Definition 3.1, we*
 * say
that the following commutative diagram, which displays both a weak pushout and
a weak pullback, is a pushpull square.

                                  g
                              X _____//Z

                             f ||     |k|
                               fflffl|fflffl|
                              Y __j__//W


Lemma 3.6. The central squares in any braid of distinguished triangles generated
by h = g O f are pushpull squares. More precisely, with the notations of (T3), *
 *the
following triangles are exact.

                   (f0,g)         (j,h0)     -g00Oj0
              Y _________//_U __Z_______//_V________// Y

                     0       (j0,h00)        (g00,- f)
             Y ___jOf___//_V________//W _  X_________//_ Y

Proof.Although rather lengthy, this is an elementary diagram chase.


Remark 3.7. We would like to conclude that the triangles displayed in the lemma
are distinguished and not just exact. Examples in [12] imply that this is not t*
 *rue
for all choices of j and j0. The braid in (T3) gives rise to a braid of disting*
 *uished
triangles that is cogenerated by -g00O j0or, equivalently, generated by  -1(j00*
 *O g0).
Here  -1(j00O g0) = 0 since j00=  f0O g00. This implies that the central term in
the braid splits as U _ Z. Application of (T3) gives a distinguished triangle

                                               00Oj0
              Y ____ff___//_U __Z__fi___//_V_-g_____// Y.
                                                 __0                 __0
inspecting the relevant_braid,_we see that ff = (f , g) and fi = (j, h). Howeve*
 *r, we

cannot always replace f0and h0by f0 and h0and still have a distinguished triang*
 *le.
  This leaves open the possibility that the triangles displayed in Lemma 3.6 are
distinguished for some choices of j and j0. It was stated without proof in [2, *
 *1.1.13]
that j and j0 can be so chosen in the main examples, and we shall explain why
that is true in x5.  It was suggested in [2, 1.1.13] that this conclusion shoul*
 *d be
incorporated in Verdier's axiom if the conclusion were needed in applications. *
 *This
course was taken in [10], and we believe it to be a sensible one. However, rath*
 *er than
try to change established terminology, we offer the following modified definiti*
 *on.

Definition 3.8. A triangulation of C is strong if the maps j and j0 asserted to
exist in (T3) can be so chosen that the two exact triangles displayed in Lemma *
 *3.6
are distinguished.

Remark 3.9. Neeman has given an alternative definition of a triangulated catego*
 *ry
that is closely related to our notion of a strong triangulated category; compar*
 *e [12,
1.8] and [13, x1.4]. It is based on the existence of particularly good choices *
 *of the
map k in (TR3).




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        11


                     4.The compatibility axioms

  In the rest of the paper, we return to our standing hypothesis that C is a cl*
 *osed
symmetric monoidal category with a öc mpatible" triangulation. In this section,
we state and explain the compatibility axioms. Let Sn =  nS for any integer n,
where  n is the n-fold iterate of   if n is positive or the (-n)-fold interate *
 *of  -1
if n is negative. Of course, S-n is isomorphic to DSn [11, 2.9].

Definition 4.1. The triangulation on C is compatible with its closed symmetric
monoidal structure if axioms (TC1)-(TC5) are satisfied.

Axiom TC 1. There is a natural isomorphism ff : X ^ S1 -!  X such that the
composite
                         -1         fl        ff
              S2 =  S1 ff--!S1 ^ S1 -!S1 ^ S1 -!  S1 = S2

is multiplication by -1.


Axiom TC 2. For a distinguished triangle X __f__//Y_g_//_Zh__// X
and an object W , each of the following triangles is distinguished.


      X ^ W  ___f^id__//_Y ^_W__g^id__//Z ^ W__h^id___// (X ^ W )


              id^f             id^g              id^h
    W ^ X  __________//_W ^ Y_________//_W ^ Z__________// (W ^ X)

             F(id,f)           F(id,g)            F(id,h)
  F (W, X)__________//_F (W, Y_)_______//_F (W,_Z)________// F (W, X)


              -F(h,id)            F(g,id)           F(f,id)
 -1F (X, W )__________//_F (Z, W_)_______//_F (Y, W_)_______//_F (X, W ).


Remark 4.2. In (TC2) and in later axioms, we implicitly use isomorphisms such as

( X) ^ Y ~= (X ^ Y ) ~=X ^ ( Y )  and F ( -1X, Y ) ~= F (X, Y ) ~=F (X,  Y )

that are implied by (TC1). We often write  DX where the canonically isomorphic
object D( -1X) might seem more natural.  We can deduce from (TC1) that " :
D( -1X) ^  -1X - ! S agrees with " : DX ^ X - ! S under the canonical
isomorphism of sources, and similarly for j when X is dualizable. The first tri*
 *angle
displayed in (TC2) is isomorphic to the second, by application of fl to all ter*
 *ms, so
that the second one is redundant.

Remark 4.3. Our (TC1) and (TC2) are equivalent to the compatibility conditions
specified by Hovey, Palmieri, and Strickland [6, A.2].  Indeed, using associati*
 *vity
isomorphisms implicitly, we have the composite natural isomorphism

                   -1^id         id^fl          ff
         ( X) ^ Yff___//X ^ S1 ^_Y___//X ^ Y ^ S1__//_ (X ^ Y ).

Calling this map eX,Y, we see that the conditions prescribed in [6, A.2] are sa*
 *tisfied.
Conversely, isomorphisms eX,Y as prescribed there are determined by the eS,Y via
the diagram on [6, p. 105], and the eS,Y determine and are determined by the ma*
 *ps


               ffY : Y ^ S1 fl-!S1 ^ Y -eS,Y--! (S ^ Y ) ~= Y.

These maps give a natural isomorphism ff that satisfies (TC1) and (TC2).




12                             J. P. MAY


  The need for the axioms (TC1) and (TC2) is clear, and we view them as ana-
logues of the elementary axioms (T1) and (T2) for a triangulated category. The
new axioms (TC3)-(TC5) encode information about the ^-product of distinguished
triangles that holds in the examples but is not implied by (TC1) and (TC2). The
reader may recoil in horror at first sight of the diagram in the following axio*
 *m but,
as we shall explain shortly, it is really quite natural.

Axiom TC 3 (The braid axiom for products of triangles). Suppose given distin-
guished triangles

                       X __f__//Yg__//_Zh__// X

and
                          f0     0g0   0h0     0
                      X0_____//Y____//Z____// X .

Then there are distinguished triangles

                        p1    j1        f0^h0         0
                Y ^ X0_____//V___//_X ^ Z___//_ (Y ^ X )


                            p2    j2        -g0^g0     0
                -1(Z ^ Z0)_____//V___//_Y ^_Y___//Z ^ Z

                        p3    j3        h0^f0          0
                X ^ Y 0____//V___//_Z ^ X___//_ (X ^ Y )

such that the following diagram commutes.



        -1(Y ^ Z0)              X ^ X0K              -1(Z ^ Y 0)
           ____KK__________________sssKKK           0ss   ______________________
          _______KKK_____________________f^idsssid^fKKKsss ____________________*
 *________
          _________KKK___________________ssKKK  sss         ___________________*
 *_________
  -1(id^h0)___________KKK___________________ssssKKKsss      __-1(h^id)_________*
 *___________________
          ______________KKKK-1(g^id)K_______________________ssssKKKKs-1(id^g0)s*
 *ss_____________________________________
          __________________KKK__________________sssKKKssss____________________*
 *________________
           _oeoe_______________K%%K____yysssyyss     KK%% ____________________
          Y ^_X0LL:___________ -1(Z ^:Z0)             Xr^ Y_0___________
           ___:::LLL_____________________________________|:::__________________*
 *______________________rrrr
          ______:::LLLp1LL___________________________|::p3r____________________*
 *_____________rrr
          ________::___LL_____________________p2|::rrr      ___________________*
 *_____________
         __________::____LLL_______________|::rrr           ___________________*
 *__________
         ___________0::____LLL_____________|::rrr           ___________________*
 *_________
         _______id^f__::______LLL______________|::rrrf^id    __________________*
 *___________
     g^id______________:::______LL%%__________________fflffl|:::id^g0__________*
 *____________________________yyrr
         ________________::_____________________-10-1:::VrrrL__________________*
 *___________________LLL
         _________________::___________|rr(id^hr)::L(h^id)LL___________________*
 *_________
         __________________::__________|rrrLLL    ::        ___________________*
 *_________
          __________________::__________|rrrLLL    ::       ___________________*
 *_________
          ____________________::________j2|rrr LLL  ::     ____________________*
 *________
          _____________________::_______|j3rrrj1 LLL  ::   ____________________*
 *________
           _____________________::_______|rrr       LLL:: _____________________*
 *_______
            _______________xxrr  :ÆÆfflffl|           L&ÆÆ_______________&
          Z_^_X0___________L    Y ^ YL0               X ^ Z0_
          __________________________LLLrrLL          r0r  _____________________*
 *_____
          __________________________h^idLLrrrLLL- id^hrrr  ____________________*
 *_____
          _______________________LLLLg^idrrrrid^g0LLLLrrrr  ___________________*
 *_________________
     id^f0________________________________________________LLLLrrrrLLLLf^id_____*
 *______________________________rrrr
          __________________________LLLrrrrrr   LLL         ___________________*
 *_______
          ______________________LLLLrrrrrrr        LLL     ____________________*
 *_______________
           _øø____________________%%Lyyrrryyrr       LL%% _____________________
          Z ^ Y 0              (X ^ X0)               Y ^ Z0




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        13


  There are several ways to understand (TC3). In concrete terms, one can pretend
that X and X0 are subobjects of Y and Y 0with quotient objects Z and Z0 and
that V is the pushout

(4.4)                V = (Y ^ X0) [X^X0 (X ^ Y 0).

Then the map j2 : V  -! Y ^ Y 0corresponds to the evident inclusion, while
j1 : V - ! X ^ Z0 and j3 : V - ! Z ^ X0 correspond to the maps obtained by
quotienting out Y ^ X0 and X ^ Y 0. The diagram then corresponds to a folklore
diagram in classical algebraic topology.  Starting from this idea, we shall exp*
 *lain
how to use standard cofiber sequences to verify the axiom in x6.
  In axiomatic terms, it is more instructive to explain (TC3) in terms of Verdi*
 *er's
axiom. Consider the canonical 3 x 3 diagram



(4.5)    X ^ X0 __f^id__//Y ^ X0_g^id__//Z ^ X0h^id_//_ (X ^ X0)

       id^f0||             |id^f0|        |id^f0|        ||(id^f0)
            fflffl|f^id    fflffl|g^id    fflffl|h^id    fflffl|
         X ^ Y 0________//_Y ^ Y_0_____//_Z ^_Y_0___// (X ^ Y 0)

        id^g0||            |id^g0|        |id^g0|        ||(id^g0)
            fflffl|f^id    fflffl|g^id    fflffl|h^id    fflffl|
         X ^ Z0 ________//_Y ^ Z0______//_Z ^_Z0_____// (X ^ Z0)

       id^h0||             |id^h0|        |id^h0|        |-|(id^h0)
            fflffl| (f^id) fflffl| (g^id) fflffl|- (h^id)fflffl|
         (X ^ X0)_____// (Y ^ X0)____// (Z ^ X0)____// 2(X ^ X0).

All squares except the bottom right one commute, and that square commutes up
to the sign -1. To see this, observe that we have implicitly used the identific*
 *ation

          ( X) ^ Z0~=(X ^ S1) ^ Z0~=(X ^ Z0) ^ S1 ~= (X ^ Z0).

It is the anticommutativity of this square that forces the sign in the diagram *
 *of
(TC3). By (TC2) and (T2), the rows and columns are distinguished. The signs are
inserted in the bottom right square to ensure this. Each square gives two compo*
 *sites
to which Verdier's axiom can be applied.
  The diagram in (TC3) arranges in a single picture parts of braids generated by
desuspending the composites

               (g ^ id) O (id^g0) = g ^ g0= (id^g0) O (g ^ id)

               (id^h0) O (f ^ id) = f ^ h0= (f ^ id) O (id^h0)

               (id^f0) O (h ^ id) = h ^ f0 = (h ^ id) O (id^f0).

By expanding the relevant diagrams (T3) slightly, we see that pairs of these six
composites appear in each of three distinct Verdier braids.  To avoid expanding
an already complicated diagram, we have omitted from the diagram in (TC3) the
generating and cogenerating triangles from the three relevant braids as display*
 *ed
in (T3), thus including only those subdiagrams from (T3) that involve at least *
 *one
dotted arrow.
  The point of (TC3) is that the cited three braids are duplicative. If we star*
 *t with
a given distinguished triangle (p2, j2, -g^g0), then applications of Verdier's *
 *axiom to
the two composite descriptions of the desuspension of g ^g0construct distinguis*
 *hed
triangles (p1, j1, f ^ h0) and (p3, j3, h ^ f0).  On the other hand, applicatio*
 *n of




14                             J. P. MAY


Verdier's axiom to the desuspension of the composite (f ^id)O(- id^h0) construc*
 *ts
(p3, j3, h ^ f0) from (p1, j1, f ^ h0). The axiom (TC3) says that we can use th*
 *e same
maps in these a priori different ways of generating braids with the same object*
 *s.
This discussion leads to the following addendum. Compare Definition 3.8.

Lemma 4.6. In the diagram of Axiom TC3, the six squares that have V as a vertex
are pushpull squares.


Proof.Three of the squares have side arrows (p1, p2), (p1, p3), (p2, p3) with t*
 *arget
V , three have side arrows (j1, j2), (j1, j3), (j2, j3) with source V .  These *
 *squares
pair up as the central squares in the three braids cited in the paragraph above*
 *, and
the conclusion is immediate from Lemma 3.6.

  Applying (TC3) to the distinguished triangles (- -1h, f, g) and (- -1h0, f0, *
 *g0),
we obtain the following equivalent form of that axiom.

Lemma 4.7 (TC30). For the distinguished triangles (f, g, h) and (f0, g0, h0) di*
 *s-
played in (TC3), there are distinguished triangles

                               q1       h^0g0         0
                X ^ Z0_k1__//W____//Z ^ Y___//_ (X ^ Z )

                          q2           0- (f^f0)           0
            Y ^ Y 0k2_//_W___//_ (X ^ X_)________// (Y ^ Y )

                               q3       g0^h0         0
                Z ^ X0_k3__//W____//Y ^ Z___//_ (Z ^ X )

such that the following diagram commutes.

          X_^_YK0K___________ -1(Z ^ Z0)K            Ys^ X0_____________
          _______KKK_________________________________s-1(h^id)ssss-K-1(id^h0)KK*
 *KKssss________________________________________
          _________KKK____________________sssKKKsss       _____________________*
 *________
     id^g0____________KKK___________________sssKKKsss      g^id________________*
 *_______________
          ______________KKKf^idKK_________________________ssssKKKKid^f0ssss____*
 *___________________________________
          __________________KKK____________________ssssKKKKssss________________*
 *______________________
           _oeoe______________yK%%Kyssyyss           K%% _______________
          X ^_Z0LL9____________Y ^ Y90               Zr^_X0____________
           ____99LLL9__________________________________|999____________________*
 *__________________rrrr
           _____99_LLLk1L_______________________|99k3rrr ______________________*
 *_______
          ________99__LL__________________k|99rrr         _____________________*
 *_______
          _________99___LLL_________________2|99rrr       _____________________*
 *________
          __________99____LLL______________|99rrr          ____________________*
 *________
         _______id^h099______LLL________________|99rrrh^id ____________________*
 *_________
         _____________999______LL%%___________________fflffl|9990______________*
 *________________________yyrr
      f^id______________99________________________99WssKKK _id^f_______________*
 *______________________
         ________________99____________id^g0|ss9g^id9sKKK  ____________________*
 *________
          _________________99____________|sssKKK99         ____________________*
 *_________
          __________________99__________|sssKKK  99       _____________________*
 *_______
          ___________________99__________q2|sssKKK99      _____________________*
 *________
           ___________________99_________|q3sssq1KKK99   ______________________*
 *______
           ____________________99_________|sss    KKK99  ______________________*
 *_______
            ____________________9øøfflffl|yyss       %9øø____________________%K
          Y_^_Z0____________K  (X ^ X0)              Z ^ Y_0_
           __________________________KKssKKK        s0s  ______________________*
 *____
          __________________________g^idKKKsssKKKid^gsss  _____________________*
 *____
          ________________________KKKKs(f^id)sssK(id^f0)KKK____________________*
 *_________________ssss
     id^h0________________________________________________KKKKsssKKKh^id_______*
 *_____________________________ssss
          __________________________KKKssssss  KKK        _____________________*
 *_____
          ___________________KKKssss   sss       KKKK     _____________________*
 *__________
           _oeoe__________________%%KKyyssyysss     K%%  ___________________
          (Y ^ X0)              Z ^ Z0              (X ^ Y 0)




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        15


  For intuition, pretending that we have inclusions of X in Y and X0 in Y 0with
quotient objects Z and Z0, we can think of W as the quotient object

(4.8)                   W = (Y ^ Y 0)=(X ^ X0),

with k2 being the quotient map and k1 and k3 being the inclusions of

       X ^ Z0~=(X ^ Y 0)=(X ^ X0)  and Z ^ X0~= (Y ^ X0)=(X ^ X0).

Lemma 4.9. In the diagram of Axiom TC30, the six squares that have W as a
vertex are pushpull squares.

  The following obvious remark is quite useful.

Remark 4.10. We can reverse the order of our given_triangles_(f, g, h) and (f0,*
 * g0, h0)
and apply (TC3) and (TC30). We agree to write V, W , and similarly for maps_in
such resulting_diagrams. By (TR3), we can obtain_equivalences_fl : V - ! V and
fl : W -! W  such that _p2O fl = fl O p2, j2fl = flj2,_k2O_fl = fl O k2, and _q*
 *2fl = flq2.
We can then redefine the remaining maps _r(_r= _pi, ji, ki, _qi, i = 1 and 3) by
taking _r= fl Or Ofl-1. It follows from the axioms that the new diagrams still *
 *satisfy
the properties specified in (TC3) and (TC30). We say that the new diagrams are
involutions of the original diagrams for (f, g, h) and (f0, g0, h0).

  The heart of our work concerns the interplay between (TC3) and (TC30); We
retain their notations.

Axiom TC 4 (The additivity axiom). The maps ji and ki can be so chosen that
the following diagram is a pushpull square.

                                     j2
                            V ____________//_Y ^ Y 0

                        (j1,j3)||             k2||
                             fflffl|          fflffl|
                    (X ^ Z0) _ (Z ^ X0)(k1,k3)//_W


In particular, k2 O j2 = k1 O j1 + k3 O j3.

  We will show how to use equations (4.4) and (4.8) to derive this axiom in x6.*
 * In
fact, we will see that, in practice, the square comes from a distinguished tria*
 *ngle;
compare Remark 3.7 and Definition 3.8. This suggests the following strengthened
alternative to the concept of compatibility that we are in the process of defin*
 *ing.

Definition 4.11. A strong triangulation of C is strongly compatible with its sy*
 *m-
metric monoidal structure if the maps of (TC3) and (TC30) can be so chosen that
the pushpull squares of Lemmas 4.6 and 4.9 and of (TC4) all arise from distin-
guished triangles.

  To see the plausibility of (TC4), observe that there is yet another Verdier b*
 *raid
in sight, coming from the relation

                   (j1, j3) O p2 = ( -1h ^ id,  -1 id^h0).

We have a given distinguished triangle

                           p2     j2       -g0^g0      0
                -1(Z ^ Z0)____//_V___//Y ^ Y____//Z ^ Z .

If the triangulation is strong, we also have distinguished triangles




16                             J. P. MAY
                                          0,h^id)   (p3O(id^f0))
     V ___(j1,j3)//_(X ^ Z0) _ (Z(^iX0)d^h//_ (X ^ X0)_______// V

           ( -1h^id, -1id^h0)    0          0(k1,k3)   (g^id)q3       0
 -1(Z ^ Z0)______________//(X ^ Z ) _ (Z ^_X_)_____//W_________//Z ^ Z .

Taking these three triangles as input in Verdier's axiom and noting that

                    j2 O  (p3 O (id^f0)) = - (f ^ f0),

(TC4) states that the maps asserted to exist by Verdier's axiom can be taken to*
 * be
the maps k2 and q2 of the given distinguished triangle

                          q2           0- (f^f0)           0
           Y ^ Y 0_k2_//W____//_ (X ^ X_)_______//_ (Y ^ Y ).

  Finally, we need an axiom that relates duality to ^-products of distinguished
triangles.  Keeping the original distinguished triangle (f, g, h), we specializ*
 *e the
distinguished triangle (f0, g0, h0) to

                                       D( -1h)
                 DZ __Dg_//DY_Df_//_DX_________// DZ.

We can construct diagrams as in (TC3) and (TC30) for both this pair of distin-
guished triangles and for the same pair of distinguished triangles in the rever*
 *se
order. We adopt the notations of Remark 4.10 for the relevant objects and maps.
Observe that we have a natural map æ : X -! DDX and a natural composite


(4.12)       , : X ^ DX_æ^id//_DDX ^ DX __^_//_D(DX ^ X),

both of which are isomorphisms when X is dualizable.  We have the following
pleasant observation. The duals of the diagrams in (TC3) and (TC30), if flipped
over (or read from bottom to top), give further diagrams of the same shape. This
is not an accident.

Lemma 4.13. Let X, Y , and therefore Z be dualizable. Then, using isomorphisms
,, the dual of a diagram as in (TC3) for the triangles (f, g, h) and (Dg, Df, D*
 * -1h)
is a diagram as in (TC30) for the same triangles in the reverse order.
                  ___
  That is, taking W  = DV with V as in (TC3) and taking
        __  __  __                          _   _   _
       (k1, k2, k3) = (Dj3, Dj2, Dj1)  and (q1, q2, q3) = (Dp3, Dp2, Dp1),

we obtain a diagram as in (TC30) for (Dg, Df, D -1h) and (f, g, h). We can now
formulate our last compatibility axiom. Despite considerable effort, I have not*
 * been
able to deduce it from the others. Recall Remark 4.10.

Axiom TC 5 (The braid duality axiom). There is a diagram as in (TC30) for the
triangles (Dg, Df, D -1h) and (f, g, h) which satisfies the following propertie*
 *s.
                          ___
   (a) There is a map _": W - ! S such that the following diagram commutes.

                               (_k1,_k3)___    _k2
         (DZ ^ Z) _ (DXT^ X) __________//_Woo________DY ^ Y
                        TTTT             _|       nnnnn
                            TTTTT        "|   nnn"n
                             (",")TTTTT**TTTfflffl|vvnnnnn
                                        S




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        17


   (b) If X, Y , and Z are dualizable, then the chosen diagram as in (TC30) is
       isomorphic to the dual of a diagram as in (TC3) for the triangles (f, g,*
 * h)
       and (Dg, Df, D -1h) and satisfies the additivity axiom (TC4) with respect
       to an involution of the latter diagram.

  Here (b) ensures that the dual of (a) also holds.  A diagram chase [8, III.1.*
 *4]
shows that the coevaluation map j : S -! X ^ DX of a dualizable object X is


                D" : S ~=DS_____//D(DX ^ X) ~=X ^ DX.

Therefore, via isomorphisms æ and ,, the dual of the commutative diagram in (a)
gives the commutative diagram in the following result.

Lemma 4.14 (TC5a0). With the diagram (TC3) for (f, g, h) and (Dg, Df, D -1h)
taken as in (TC5b), there is a map __j: S -! V such that the following diagram
commutes.

                                     jjjS PPP
                             (j,j)jjjjjj _| PPPP j
                            jjjj         j|     PPPPP
                       ttjjjjj  (j3,j1)  fflffl|j2  PP((
         (Z ^ DZ) _ (X ^ DX) oo_________V __________//_Y ^ DY

  This reduces the proof of Theorem 0.1 to quotation of the axioms.

The proof of Theorem 0.1.We are assuming Axioms (TC1)-(TC5), and we have
the following commutative diagram.  The desired formula Ø(Y ) = Ø(X) + Ø(Z)
follows by traversing its outer edge.

                                     jjjS PPP
                             (j,j)jjjjjj _| PPPP j
                            jjjj         j|     PPPPP
                       ttjjjjj  (j3,j1)  fflffl|j2  PP((
         (Z ^ DZ) _ (X ^ DX) oo_________V ___________//Y ^ DY
                  |                                      |
                  |                      fl|             |
                  |                      |               |
                  |                     _fflffl|_        |
              (fl,fl)|                j V P              fl|
                  |        _  _   jjjjj    PPPP _        |
                  |       (j1,j3)jjjj         PPj2P      |
                  |       jjjjj                  PPPP    |
                  fflffl|ttjjj (_k1,_k3)___    _k2  PP'' fflffl|
         (DZ ^ Z) _ (DXT^ X) __________//_Woo________DY ^ Y
                        TTTT             _|       nnnnn
                            TTTTT        "|   nnn"n
                             (",")TTTTT**TTTfflffl|vvnnnnn
                                        S

Here the top and bottom pairs of triangles are given by (TC5a) and (TC5a0),
the trapezoids involving maps fl are given by (TC5b) and Remark 4.10, and the
remaining trapezoid is given by (TC5b) and (TC4).


                  5. How to prove Verdier's axiom

  To prepare for the proofs of the compatibility axioms, we first recall the st*
 *andard
procedure for proving Verdier's axiom (T3).
  We assume that our given category C is the "derived categoryö  r öh motopy
categoryö  btained from some Quillen model category B.  One can give general
formal proofs of our axioms that apply to the homotopy categories associated to




18                             J. P. MAY


"simplicial", öt pological", or öh mological" model categories that are enriched
over based simplicial sets, based spaces, or chain complexes, respectively. We *
 *shall
be informal, but we shall give arguments in forms that should make it apparent
that they apply equally well to any of these contexts. An essential point is to*
 * be
careful about the passage from arguments in the point-set level model category
B, which is complete and cocomplete, to conclusions in its homotopy category C ,
which generally does not have limits and colimits.
  We assume that B is tensored and cotensored over the category in which it
is enriched.  We then have canonical cylinders, cones, and suspensions, together
with their Eckmann-Hilton duals. The duals of cylinders are usually called äp th
objects" in the model theoretic literature (although in based contexts that term
might more sensibly be reserved for the duals of cones). When we speak of homo-
topies, we are thinking in terms of the canonical cylinder X   I or path object*
 * Y I,
and we need not concern ourselves with left versus right homotopies in view of *
 *the
adjunction
                       B(X   I, Y ) -! B(X, Y I).

  Hovey [5] gives an exposition of much of the relevant background material on
simplicial model categories. Discussions of topological model categories appear*
 * in
[4] and [9]. Homological model categories appear implicitly in [5] and [7, IIIx*
 *1]. Of
course, we must assume that the functor   on B induces a self-equivalence of C .
This is enough for the verification of most of the axioms but, to verify parts *
 *of (TC2)
and (TC5), we assume more precisely that the adjunction between   : B -! B
and its right adjoint   is a Quillen equivalence of model categories. (See e.g.*
 * [5,
1.3.3] for a discussion of this notion).
  The distinguished triangles in C are the triangles that are isomorphic in C t*
 *o a
canonical distinguished triangle of the form

                                i(f)    p(f)
(5.1)                 X __f__//Y____//Cf___//_ X

in B. Here Cf = Y [f CX, where CX is the cone on X, and i(f) and p(f) are the
evident canonical maps. Then (T1) is clear and (T2) is a standard argument with
cofiber sequences. One uses formal comparison arguments (as in [17, II.1.3.2]) *
 *to
reduce the verification of (T3) in C to consideration of canonical cofiber sequ*
 *ences
in B. In B, one writes down the following version of the braid in (T3).

                __h____________________________________________________________*
 *_______________________i(g)___________________________________@
              _________________________________________________________________*
 *______________________________________________________________@
            _____________!!____________________________________________________*
 *____""___________________________________________________##___@
(5.2)     X              Z C              Cg                Cf
            @@         "">>  Ci(h)    0  ==z EEE          x<<x
             @@@     g"""      C     j z       Ep(g)EE  xxx
             f @@ØØ@"""         C!! z z          E""Exxx  i(f)

                 Y AA           -Ch=D=            <Y<z

                     AAA     j -      Dp(h)D   zzzz
                   i(f)A__AA--           D"" zzz f
                         Cf _____________ << X
                            ___________________________________________________*
 *________________________________________________________
                              _________________________________________________*
 *______________________________________________________________@
                                 p(f)

Here h = g O f, j and j0 are evident canonical induced maps, j00=  i(f) O p(g),
and the diagram commutes in B. One proves (T3) by writing down explicit inverse




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        19


homotopy equivalences

                   , : Cg -! Cj  and    : Cj -! Cg

such that j0 =   O i(j) and j00= p(j) O ,. Details of the algebraic argument ar*
 *e in
[17, pp. 75-77], and the analogous topological argument is an illuminating exer*
 *cise.
  We could go on to use these Verdier braids to prove (TC3) and (TC4), but there
are simpler proofs that give more information. To see this, we need a reformula*
 *tion
of the original triangulation.
  Assuming, as can be arranged by cofibrant approximation, that f is a cofibrat*
 *ion
between cofibrant objects, the quotient Y=X is cofibrant. Let Mf be the mapping
cylinder of f. Passage to pushouts from the evident commutative diagram

                          *oo___X  ____//_Mf

                          ||     ||       '|
                          ||     ||       |
                          ||     ||       fflffl|
                          *oo___X  __f__//_Y


gives a quotient map q(f) : Cf - ! Y=X.  By [5, 5.2.6], we have the following
standard result. It is central to our way of thinking about triangulated catego*
 *ries.

Lemma 5.3. Let f : X -! Y be a cofibration between cofibrant objects. Then the
quotient map q(f) : Cf -! Y=X is a weak equivalence.

  Now define
                          ffi(f) : Y=X -!  X

to be the map in C represented by the formal öc nnecting map"

                              q(f)   p(f)
(5.4)                    Y=X oo___Cf ____//_ X

in B.  Observe that (5.4) gives a functor from cofibrations in B to diagrams in
B. The composite q(f) O i(f) : Y -! Y=X is the evident quotient map, which we
denote by ß(f). Therefore, when we pass to C , our canonical distinguished tria*
 *ngle
(5.1) is isomorphic to the triangle represented by the diagram

                               ß(f)     ffi(f)
(5.5)                 X __f_//_Y___//Y=X____// X

in B, and our triangulation consists of all triangles in C that are isomorphic *
 *to one
of this alternative canonical form. This reformulation has distinct advantages.
  Returning to Verdier's axiom, we can replace the given maps f, g, and thus
h = g O f by cofibrations between cofibrant objects, and then the quotient obje*
 *cts
Y=X, Z=X and Z=Y  are cofibrant.  The point of Verdier's axiom now reduces
to just the observation that Z=Y is canonically isomorphic in B to (Z=X)=(Y=X).
Using our new canonical cofibrations (5.5) starting from f, g, h, and the cofib*
 *ration
j : Y=X -! Z=X, we obtain the following braid.




20                             J. P. MAY
              __h______________________________________________________________*
 *_____________________ß________________________________________@
           ____________________________________________________________________*
 *______________________________________________________________@
         ______________!!______________________________________________________*
 *__________$$__________________________________________________@
(5.6)  X               Z=F=               Z=Y                Y=X
         >>           __  FFF           xx<<  DDD           x;;x
           >>>     g___     FßFF      ßxxx      DffiDD   xxx
          f >>OEOE>____        F""F xxx           DD""Dxxxxß

               Y BB             Z=X<F<             z<Y<
                  BB        jxxxx    FFffiFF     zzz
                  ß BB!!BB xxx         FFF     zzzf
                          x              F"" zz
                      Y=X _________       :: X
                           ____________________________________________________*
 *______________________________________________________________@
                             __________________________________________________*
 *______________________________________________________________@
                                  ffi

Expanding the arrows ffi as in (5.4), we find that this braid in C is represent*
 *ed by an
actual commutative diagram in B, but of course with some wrong way arrows. With
this proof of Verdier's axiom, there is no need to introduce the explicit homot*
 *opies
, and   of our first proof. Modulo equivalences, the two central braids in (5.6*
 *) are
as follows. Here and later, we generally write C(Y, X) instead of Cf for a given
cofibration f : X -! Y .

                  Y _______//_Z  C(Z, X) ____//_C(Z, Y )
                  |         |        |           |
                  |         |        |           |
                  fflffl|   fflffl|  fflffl|     fflffl|
                Y=X  _____//Z=X     X  ________//_ Y

These are both pushouts in which the horizontal arrows are cofibrations and all
objects are cofibrant. By the following lemma, this implies that, in C , these *
 *two
squares give pushpull diagrams that arise from distinguished triangles. We conc*
 *lude
that C is strongly triangulated in the sense of Definition 3.8.

Lemma 5.7. Suppose given a pushout diagram in B,

                                  f
                              X _____//_Y

                             g ||     |j|
                               fflffl|fflffl|
                              Z __k__//W,

in which f and therefore k are cofibrations and all objects are cofibrant. Then*
 * there
is a distinguished triangle

                    (-f,g)          (j,k)
                X _________//_Y __Z_______//_W____// X.

in C . Thus the original square gives rise to a pushpull square in C .

Proof.Standard topological arguments work model theoretically to give a weak
pushout (double mapping cylinder) M(f, g) in B which fits into a canonical tria*
 *ngle

                     (j0,k0)         ß       ffi
             X _ Y _________//_M(f,_g)__// X____// X _  Y

as in (5.5).  It is easy to check that ffi = (f, -g) in C and that there is a w*
 *eak
equivalence M(f, g) -! W under Y _ Z in B. The conclusion follows.




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        21


          6. How to prove the braid and additivity axioms

  We now consider axioms (TC1)-(TC4). Here, in addition to the assumptions of
the previous section, we assume that the closed symmetric monoidal structure on
the homotopy category C is induced from a closed symmetric monoidal structure
on B. There are model theoretic axioms, specified in [15] and [5], that codify *
 *the
relationship between products ^ and cofibrations in reasonable monoidal model
categories, and we assume that such standard properties hold in B.  They are
known to hold in the usual examples. The most important axiom, which is called
the pushout-product axiom in [15], asserts that, for cofibrations f : X -! Y and
f0 : X0- ! Y 0, the evident induced map

                 V = (X ^ Y 0) [X^X0 (Y ^ X0) -! Y ^ Y 0

is a cofibration which is acyclic if either f or g is acyclic. It follows that,*
 * for any
object T , the dual induced map

      F (Y ^ Y 0, T ) -! F (V, T ) ~=F (X ^ Y 0, T ) xF(X^X0,T)F (Y ^ X0, T )

is a fibration. Some equivalent conditions are given in [5, 4.2.2].
  The verification of (TC1) is trivial, and the verification of (TC2) is standa*
 *rd; see
Hovey [5, 6.41., 6.6.3]. For the cofiber sequences of (TC2) that involve ^, one*
 * uses
the triangulation by cofibrations. For the cofiber sequences of (TC2) that invo*
 *lve
the internal hom functor F , one verifies that the negative of the triangulation
by cofiber sequences is the triangulation given by fiber sequences in B, which *
 *are
Eckmann-Hilton dual to cofiber sequences and whose development is word-for-word
dual to that described in x5. For a given map f with fiber F f and cofiber Cf, *
 *there
is a map j : F f -!  Cf that is suitably related to the unit and counit of the
( ,  ) adjunction. This can be used in a direct verification of (TC2), as in [8*
 *, IIIx2];
see also [4, II.6.4]. The argument in the homological context is easier.
  We consider the new axiom (TC3). We may assume without loss of generality
that the given distinguished triangles are of the form (5.5). We write them as

                                          0     g0     h0
       X _f__//_Yg__//Zh__//_ X and  X0__f__//Y_0__//Z0___// X0.

Thus all objects are cofibrant, f and f0 are cofibrations, Z = Y=X and Z0= Y 0=*
 *X0,
g and g0 are quotient maps, and h and h0are connecting maps ffi. We are thinking
of these as diagrams in B, the arrows h and h0being shorthand for pairs of arro*
 *ws
as displayed in (5.4).
  As in (4.4) and (4.8), we set

        V = (Y ^ X0) [X^X0 (X ^ Y 0)  and W = (Y ^ Y 0)=(X ^ X0).

We have many canonical isomorphisms of quotients, such as

     X ^ Z0~=(X ^ Y 0)=(X ^ X0),  X ^ C(Y 0, X0) ~=C(X ^ Y 0, X ^ X0),

         V=(X ^ X0) ~=(Z ^ X0) _ (X ^ Z0),  (Y ^ Y 0)=V ~=Z ^ Z0.

These are used heavily in verifying the claims that we are about to make.
  By the cited axioms for a monoidal model category, or standard verifications *
 *in
the usual examples, we have the following canonical triangles as in (5.5). Thus*
 *, in
each case, the first map is a cofibration, the second map is a quotient map, and
the third map is a connecting map as in (5.4).  Note that j2 is a cofibration by
the pushout-product axiom. The maps p2 and q2 are defined in terms of displayed




22                             J. P. MAY


connecting maps, and the symbol ' indicates an identification of the class of a*
 * given
map ffi in C ; each such identification can be verified by an elementary diagra*
 *m chase.

                     p1     j1        0ffi'f^h0         0
             Y ^ X0 ____//_V___//X ^ Z_________// (Y ^ X )

                                  0       ffi  p2
                 V__j2_//Y ^ Yg0^g//_Z ^_Z0______//_ V

                     p3     j3        0ffi'h^f0          0
             X ^ Y 0____//_V___//Z ^ X_________// (X ^ Y )

                            q1         0ffi'h^g0        0
             X ^ Z0__k1_//W____//Z ^ Y_________// (X ^ Z )

                    f^f0        0k2     ffi q2          0
             X ^ X0_____//Y ^ Y____//W_________// (X ^ X )

                            q3        0ffi'g^h0         0
             Z ^ X0__k3_//W____//Y ^ Z_________// (Z ^ X )

Note that there are no signs here. The sign inserted in one of the corresponding
distinguished triangles listed in each of (TC3) and (TC3') is dictated by (T2).
Straightforward diagram chases, using commutative diagrams in B, show that the
diagrams displayed in (TC3) and (TC30) commute in C . Each of these diagrams
has one arrow whose label is given with a minus sign. Without the sign, the squ*
 *are
of which the arrow is one side would anticommute for the same reason that the
bottom right square of (4.5) anticommutes.
  This completes the verification of (TC3) and (TC30). While (TC30) also follows
formally from (TC3), its present proof makes (TC4) obvious. Indeed, the followi*
 *ng
square is a pushout in B to which Lemma 5.7 applies.

                                     j2
                            V ____________//_Y ^ Y 0

                        (j1,j3)||             k2||
                             fflffl|          fflffl|
                    (X ^ Z0) _ (Z ^ X0)(k1,k3)//_W


Reinterpreting the squares of (TC3) and (TC30) in terms of equivalent Verdier
braids, we conclude that the strong form of the axioms (TC3) and (TC4) specified
in Definition 4.11 holds.

              7. How to prove the braid duality axiom

  We must still verify (TC5).  We retain the assumptions of the previous two
sections. We will use the following elementary observation.

Lemma 7.1. Suppose given a commutative diagram

                                 wA GG
                             fwww    GGGg
                             ww         GGG
                           --wwwg0     f0 G##
                        X  ______//_Y_____________Zoo_
                                    _________________
                        p||      k2||__"2_______________________________q||
                         fflffl|   fflffl|_____________________________________*
 *___fflffl|
                            ____//_  o_______________________o_
                       X=A F k1  Y=AØ _______________________Z=Ak3
                            FF        _______________________xx
                            "FFFF_"Ø _____________________"xxxx
                             1  F##F______________________fflfflØ3__xxx
                                  T




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        23


in B in which the maps f, g, f0, and g0 are cofibrations between cofibrant obje*
 *cts,
the maps p, k2, and q are quotient maps, and the maps k1 and k3 are induced by
g0 and f0. Let _": Y=A -! T be the map induced by passage to quotients from "2.
Then _Ö k1 = "1 and _Ö k3 = "3.

  As in x4, replace (f0, g0, h0) by the distinguished triangle

                                       D( -1h)
(7.2)            DZ __Dg_//DY_Df_//_DX_________// DZ.

A priori, this lies in C . As we discuss in more detail shortly, we can represe*
 *nt it in
B by a canonical triangle of the form displayed in (5.5). We can then define

      V = (Y ^ DZ) [X^DZ  (X ^ DY )  and W = (Y ^ DY )=(X ^ DZ)

as in x6, with canonical maps ji, pi, qi, and ki. We can also define
      __                                 ___
      V = (DY ^ X) [DZ^X  (DZ ^ Y )  and W  = (DY ^ Y )=(DZ ^ X)
                    _   _   _       __
with canonical maps ji, pi, qi,_and_ki. The commutativity_isomorphism fl for ^ *
 *in B
induces isomorphisms fl : V -! V and fl : W -! W  under which fl O riO fl-1 = _*
 *ri
for r = j, p, q, k, as in Remark 4.10.  Thus, with these choices, the involution
condition of (TC5b) is immediate. We must verify (TC5a) and the rest of (TC5b).
For (TC5a), the idea is to apply Lemma 7.1 to the diagram

(7.3)                         DZ ^ XM
                        id^fqqqqq    MMMDg^idMM
                          qq             MMMM
                       xxqqqDg^id      id^f M&&
                  DZ ^ Y _____//DY ^_Y____________DYo^oX_
                                    __________________
                 id^g||         _k2||_"__________________________________Df^id||
                     fflffl|     _fflffl|______________________________________*
 *___fflffl|_
                  DZ ^ Z _______//_oo_________________________DX_^_X
                       MM   k1   WØ  ________________________k3qq
                         MMM     _Ø ________________________qqq
                           "MMMM "Ø _________________"qqq
                               M&&M_________________fflfflxxqqqq
                                 S.

The problem is that, a priori, the solid arrow part of the diagram only commutes
in C . We must show that we can arrange representative objects and maps in B so
that the diagram is already defined and commutative there.
  We have internal hom objects F (X, Y ) in B.  When X is cofibrant and Y  is
fibrant, F (X, Y ) is fibrant. We need the following small observation.

Remark 7.4. In topological examples, S is fibrant, but in cases where that is n*
 *ot so
we must use a fibrant approximation ~ : S -! T . Here ~ is an acyclic cofibrati*
 *on.
Since a pushout of an acyclic cofibration is an acyclic cofibration, it follows*
 * from
the pushout-product axiom that we have a composite of acyclic cofibrations

             T ~=S ^ T_____//(T ^ S) [S^S (S ^_T_)//_T ^ T.

Since T is fibrant, there is a retraction r : T ^ T -! T , and r is clearly a w*
 *eak
equivalence.

  For a cofibrant object X, a cofibrant approximation OE : DX -! F (X, T ) gives
a fibrant and cofibrant representative DX in B for the dual of X in C . Moreove*
 *r,
the composite

                    DX ^ X _OE^id//_F (X, T_)"^_X//T




24                             J. P. MAY


in B represents the evaluation map " : DX ^ X -! S in C .
  We have the cofibration f : X -! Y and the quotient map g : Y - ! Z. The
composite

                          F(g,id)     F(f,id)
                    F (Z, T_)__//F (Y,_T_)//_F (X, T )

is the trivial map, and F (f, id) is a fibration.  Choose a cofibrant approxima*
 *tion
_ : DZ -! F (Z, T ) as above and factor F (g, id)O_ as the composite of a cofib*
 *ration
Dg : DZ -! DY and an acyclic fibration Ø : DY - ! F (Y, T ).  Define DX =
DY=DZ and let Df : DY  -! DX be the quotient map.  Since the composite
F (f, id) O Ø : DY -! F (X, T ) is trivial when restricted to DZ, it factors as*
 * OE O Df
for a map OE : DX = DY=DZ -! F (X, T ). Clearly Ø is a cofibrant approximation,
and it is implicit in the verification of (TC2) by use of fiber sequences that *
 *OE is a
weak equivalence and thus a cofibrant approximation. Setting

                      D( -1h) = ffi : DX -!  DZ,

we have the required canonical triangle (7.2).  Moreover, we have the following
commutative diagram in B. It represents_the_diagram (7.3) in C and allows us to
apply Lemma 7.1 to construct a map _": W - ! T in B that represents the map _"
in C that is required to verify (TC5a).


(7.5)
                           gggDZ ^ X XXXX
                 id^fgggggggg            XXXXXXDg^idXXX
               ggggg                             XXXXXXXX
         ssgggggggDg^id                        id^f      XX++X
  DZ ^|YN_____________________//_NDY ^oYo__________________ DYo^|X
     |   NN_^idNNN                |                   Ø^idooooo|
     |       NNN                  Ø^id|              ooo       |
     |         N''     F(g,id)^id fflffl|id^f       wwoo       |
 id^g|        F(Z, T) ^ Y___//_F(Y, T) ^oYo_F(Y, T) ^ X        |Df^id
     |                                                         |
     |             |              |              |             |
     |             |              |              |             |
     fflffl|       |              |              |             fflffl|
  DZ ^ ZNN         id^g||         "|      F(f,id)^id||      DXo^ X
         NNNN      |              ||             |       oooo
         _^idNNNN  |              |              |   oooOE^ido
               N'' fflffl|"       fflffl|"       fflffl|wwoo
              F(Z, T) ^ Z________//Too______F(X, T) ^ X


  To complete the verification_of (TC5b), it remains to show that the diagram
of (TC30) centered around W  is isomorphic in C to the dual of the diagram of
(TC3) centered around V . We need a standard observation that can be verified by
comparing cofiber and fiber sequences as in [5, 6.3] or [8, III.2.3].

Remark 7.6. Let p : E -! B be a fibration in B with fiber i : F -! E, so that F
is the pullback of p along * -! B. We have a fiber sequence in canonical form


                        B __ffi//_Fi_//_Ep__//B.

This is Eckmann-Hilton dual to (5.5).  Shifting to the right, it gives rise to a
distinguished triangle


(7.7)                  F __i__//Ep__//_B_ffi//_ F




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        25


in C .  We have an explicit comparison of this triangle with the canonical dist*
 *in-
guished triangle


                      F __i_//_E___//C(i)___// F.

In fact, the canonical composite q : Ci -! E=F -! B in B is a weak equivalence
that restricts to p on E and makes the following diagram commute in C :


                            C(i)_____// F

                             q||       ||||
                              fflffl|  ||
                             B ___ffi//_ F.


  We use isomorphisms , : DX ^_Y_- ! D(X ^ DY ) in C of (4.12) to identify
all entries other than DV and W  in our diagrams (TC30) and (TC3), but we must
again distinguish between B and C . We use the duals and cofibrant approximatio*
 *ns
in B discussed above. In B, we have a map


                          æ : X -! F (DX, T ),


namely the adjoint of the composite


                    DX ^ X _,^id//_F (X, T )_^"X//_T.


Using this and the map r : T ^ T -! T of Remark 7.4, we obtain a map

                                       F(id,r)O^
         DZ ^ X __^æ//_F (Z, T ) ^ F (DX,_T_)___//F (Z ^ DX, T ).


We write , for this map and for other similarly defined maps. They are represen*
 *ta-
tives in B for maps , as in (4.12), hence they are weak equivalences. Observe t*
 *hat
this depends on arguments in C that are based on the assumption that X, Y , and
Z are dualizable.                                                  ___
  Let us write D0(-) for the functor F (-, T ) on B. We define , : W  -! D0(V )
as follows. The composite of the map


          (id^g, Df ^ id) : DY ^ Y -! (DY ^ Z) xDX^Z  (DX ^ Y )


and the cofibration Dg ^ f  : DZ ^ X - ! DY ^ Y  is the trivial map, hence
(id^g, Df ^ id) factors through a map
                   ___
                   W  -! (DY ^ Z) xDX^Z  (DX ^ Y ).


The maps , for DY ^ Z, DX ^ Z, and DX ^ Y are compatible, hence they induce
a map


    (DY ^ Z) xDX^Z  (DX ^ Y ) -! D0(Y ^ DZ) xD0(X^DZ) D0(X ^ DY ).


Since the functor D0 converts pushouts to pullbacks, the target here is isomorp*
 *hic
to D0(V ). The composite of the last two maps is the desired map
                               ___    0
                           , : W - ! D (V ).




26                             J. P. MAY


Immediate diagram chases from the definitions give that the following diagrams
commute in B.
                            _k   ___   _q
                 DZ ^ Z  ____1___//_W___1___//_DX ^ Y

                   , ||           |,|           |,|
                     fflffl|      fflffl|       fflffl|
                D0(Z ^ DZ) D0(j//_D0(V_)0_//_D0(X ^ DY )
                               3)     D (p3)

                            _k    ___   _q
                 DX ^ X  ____3___//_W___3___//_DY ^ Z

                    ,||            ,||          |,|
                     fflffl|       fflffl|      fflffl|
               D0(X ^ DX)  D0(j_//D0(V_)0__//D0(Y ^ DZ)
                               1)     D (p1)

Now consider the following diagram

                  Dg^f             _k2   ___   _q2
        DZ ^ X  ________//_DY ^ Y_______//_W________// DZ ^ X

          , ||             ,||            ,||           ||,
            fflffl|         fflffl|       fflffl|       fflffl|
      D0(Z ^ DX)D0(g^Df)//_D0(Y ^ DYD)0(j//_D0(V0)//_ D0(Z ^ DX).
                                         2)  D (p2)

The left square clearly commutes in B, and another immediate diagram chase from
the definitions_shows that the middle square commutes in B. We must prove that
, : W  -! D0(V ) is a weak equivalence and that the right hand square, which is
the only one in sight that involves connecting maps, commutes in C . This will_*
 *give
that our cofibrant version in B of the diagram of (TC30) centering around W  is
essentially a cofibrant approximation of a fibrant version in B of the dual of *
 *the
diagram of (TC3) centering around V .
  In the bottom row of the last diagram, D0(j2) is a fibration, D0(g ^ Df) is i*
 *ts
fiber, and, by inspection of duality, D0(p2) can be identified in C with a map *
 *of the
form  ffi as in (7.7). Since the left square commutes in B and its vertical arr*
 *ows are
weak equivalences, there results a weak equivalence , : C(Dg ^f) -! CD0(g ^Df)
that fits into a comparison of canonical distinguished triangles

               Dg^f
     DZ ^ X  ________//_DY ^ Y_______//C(Dg ^ f)_______// DZ ^ X

       , ||             ,||               ,||              ||,
         fflffl|         fflffl|          fflffl|          fflffl|
   D0(Z ^ DX)D0(g^Df)//_D0(Y ^ DY_)_//CD0(g ^ Df)____// D0(Z ^ DX).


Moreover, the following diagram commutes in B, where the bottom arrow q is as
in Remark 7.6:

                        C(Dg ^ f) ___q___//__W

                           , ||            |,|
                             fflffl|       fflffl|
                       CD0(g ^ Df) __q__//D0(V ).
                                                                      ___
Since both maps q and the left map , are weak equivalences, so is , : W - ! D0(*
 *V ).
Moreover, a diagram chase from the two diagrams above and the diagram in Remark




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        27


7.6 shows that the right hand square in the third diagram above commutes in C .
This completes the proof of (TC5b).


         8. The proof of the additivity theorem for traces

  We adopt the methods of the previous section to prove Theorem 1.9. We retain
the assumptions and notations there.  The idea is to construct a commutative
diagram as follows in C .


(8.1)
                                      hhhhS SSS
                             (j,j)hhhhhhhh|_  SSSSSSj
                          hhhhh           |j        SSSSS
                     sshhhhhh  (j3,j1)    fflffl| j2    SSS))
     (Z ^ DZ) _ (X ^ DX) oo_______________V ______________//_Y ^ DY
               |                                              |
               |                          |fl                 |
               |                          |                   |
               |                          fflffl|_            |
           (fl,fl)|                    hh V SS                |fl
               |           _  _   hhhhhh      SSSS _          |
               |          (j1,j3)hhhhhh          SSj2SS       |
               |       hhhhhh                        SSSSS    |
               fflffl|sshhh   (_k1,_k3)  ___      _k2    SS)) fflffl|
     (DZ ^ Z) _ (DX ^ X) ________________//WØoo____________DY ^ Y

     (id^!,id^OE)||                       Ø                   |id^_|
               fflffl|        (_k1,_k3)  _fflfflØ__k2         fflffl|
     (DZ ^ Z) _ (DX ^ X) ________________//WØoo____________DY ^ Y

     (id^ ,id^ )||                        Ø                   |id^|
               fflffl|       (_k1^id,_k3^ifflfflØd)__k2^id    fflffl|
  (DZ ^ Z ^ C) _ (DX ^VX ^ C)__________//_W^ Coo_________DYk^ Y ^ C
                     VVVVVV               |____      kkkkkk
                           VVVVVV         |"^id  kkkk
                          ("^id,"^id)VVVV++VVVVVVfflffl|"^iduukkkkkk
                                          C

Traversing the outer edge of (8.1), we read off the additivity relation of Theo*
 *rem
1.9. In view of the diagram in the proof of Theorem 0.1 at the end of x4, it re*
 *mains
only to construct dotted arrows that make (8.1) commute.
  We first concentrate on the upper dotted arrow. We are given the solid arrow
portion of the following diagram in C .

                          f      g      h
                      X _____//Y____//Z____//_Ø X

                     OE||     |_|    Ø!      ||OE
                       fflffl|fflffl|fflfflØ fflffl|
                      X _f^id//_Yg^id//_Zh^id//_ X


As in the previous section, we may take this to be a diagram in B, where f is a
cofibration between cofibrant objects, Z = Y=X, g is the quotient map, and h is
the canonical connecting map of (5.4). We may as well assume further that X and
Y are fibrant, although Z need not be.
  Since maps in C between fibrant and cofibrant objects are homotopy classes
of maps, the left square is homotopy commutative. We may apply the homotopy
extension property [14, p. 1.7] to a homotopy _ O f ' f O OE to obtain a homoto*
 *py




28                             J. P. MAY


from _ to a map _0 such that _0O f = f O OE. Replacing _ by _0, we may as well
assume that the left square commutes.  It then induces a map ! : Z -! Z by
passage to quotients. With this choice of !, the middle square commutes and the
right square induces a commutative diagram in C . Now the solid arrow portion of
the following diagram is easily checked to commute, and Lemma 7.1 applies to gi*
 *ve
the required dotted arrow.

                          kDZ ^ X  VVV
                   id^fkkkkkk        VVVVVDg^idVV
                   kkk                      VVVVVVV
               uukkkkDg^id                id^f    VV++
         DZ ^ Y __________//_DY ^;Y;oo_______________DY ^ X

        id^g||              _k2||;;;;                   |Df^id|
            fflffl|          _|fflfflid^_;;_            fflffl|
         DZ ^ Z _____________//Woo___;;_____________DX ^ X
            |         k1       Ø      ;;   k3           |
            |                  Ø       ;;               |
            |                  Ø        ;;              |
            |                  Ø          ÆÆ;           |
        id^!||                 Ø        DYq^ Y          |id^OE|

            |                  Ø     qqqq               |
            |                  Ø   qq_q                 |
            fflffl|          __fflfflk2xxqqq_           fflffl|
         DZ ^ Z _______k_____//Woo__________________DX ^ X
                      1                    k3

In the case of Lefschetz constants of maps, where C = S, this completes the pro*
 *of
of Theorem 1.9.
  Now consider the lower dotted arrow in (8.1).  We are given the solid arrow
portion of the following diagram, which we take as above as a diagram in B.


(8.2)          X  ___f____//_Y__g____//_Z___h____//_Ø X

                ||         ||         Ø            ||
                fflffl|    fflffl|    fflfflØ      fflffl|
              X ^ C f^id//_Y ^ Cg^id//_Z ^hC^id//_ (X ^ C).


We may as well assume that C is fibrant and cofibrant. However, there is a slig*
 *ht
catch to applying the argument just given to arrange that the left square commu*
 *tes
on the nose rather than just up to homotopy.

Remark 8.3. The object Y ^ C need not be fibrant, hence p0 : (Y ^ C)I -! Y ^ C
need not be a fibration and the model theoretic version of the homotopy extensi*
 *on
property may not apply; see [14, p. 1.6, 1.7]. In topological situations, all o*
 *bjects
are fibrant and the problem disappears. Moreover, in the applications to natural
diagonal maps that I have in mind, C = Y and   for X is the composite of id^f
and the diagonal X -! X^X. In such cases, the left square does commute in B. It
seems that a fairly elaborate diagram chase using functorial fibrant approximat*
 *ion
can circumvent this problem, but I will leave the details to the interested rea*
 *der.

  Once we have that the left square commutes in (8.2), we can define   on Z by
passage to quotients. Then the solid arrow portion of the following diagram is *
 *easily
checked to commute, and Lemma 7.1 applies to give the required dotted arrow.




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        29

                         DZj^ X  XXXX
                id^fjjjjjj           XXXXXXDg^idXXX
                 jjj                         XXXXXXX
            uujjjjjDg^id                   id^f    XXXXX++
     DZ ^ Y  ____________//_DY?^ Yoo____________________ DY ^ X
                               ??                           |
     id^g||                _k2||???                         |Df^id
         fflffl|            _fflffl|id^???_                 fflffl|
     DZ ^ Z  ______________//_Woo___??__________________DX_^ X
         |         k1        Ø        ??    k3              |
         |                   Ø         ??                   |
         |                   Ø          ??                  |
         |                   Ø            ØØ?               |
    id^ 0||                  Ø        DYo^ Y ^ C            |id^|
         |                   Ø      ooooo                   |
         |                   Ø    oo_o                      |
         fflffl|          ___fflfflwk2^idwoo                fflffl|
    DZ ^ Z ^ C_____k______//W^ Coo____________________ DX ^ X ^ C
                   1^id                    k3^id


               9. Homology and cohomology theories

  When C is the stable homotopy category, one can give a general treatment of t*
 *he
products in homology and cohomology theories that is based solely on the struct*
 *ure
of C as a symmetric monoidal category with a compatible triangulation. There are
four basic products here, two of which are called "slant products". A systematic
exposition is given by Adams [1, IIIx9] and followed by Switzer [16, pp. 270-28*
 *4].
We warn the reader that the treatment of slant products in the literature is ch*
 *aotic.
No other two sources seem to give the same signs, and some standard references
actually confuse the slant product \ with a product that differs only by a sign*
 * from
the slant product =. We run through a version of Adams' definitions and pinpoint
the role played by the new axioms. If we were starting from scratch, our prefer*
 *red
version of slant products would differ by signs from those below, but the logic*
 *al
advantage of writing variables in their most natural order is outweighed by the
need for consistency in the literature. Adams and Switzer make no use of functi*
 *on
spectra F (X, Y ), which were only obtainable by use of Brown's representability
theorem at the time they were writing, and this obscures the formal nature of t*
 *heir
definitions of the products.
  For an object X of C and an integer n, define

                          ßn(X) = C (Sn, X).

When C is the stable homotopy category, ßn(X) is the nth homotopy group of the
spectrum X. When C is the derived category of chain complexes over a commuta-
tive ring R, Sn is the trivial chain complex given by R in degree n and ßn(X) is
the nth homology group of the chain complex X. Applying the product ^ (  in
algebraic settings), we obtain a natural pairing

(9.1)               ßm (X)   ßn(Y ) -! ßm+n (X ^ Y ).

  For objects X and E, algebraic topologists define

            En(X) = ßn(E ^ X)   and  En(X) = ß-n F (X, E).

Equivalently, En(X) ~=C (X,  nE). The four products referred to above are

(9.2)           ^ : Dp(X)   Eq(Y ) -! (D ^ E)p+q(X ^ Y ),




30                             J. P. MAY


(9.3)           [ : Dp(X)   Eq(Y ) -! (D ^ E)p+q(X ^ Y ),


(9.4)           = : Dp(X ^ Y )   Eq(Y ) -! (D ^ E)p-q(X),


(9.5)           \ : Dp(X)   Eq(X ^ Y ) -! (D ^ E)q-p(Y ).

The naturality of slant products is better seen by rewriting them in adjoint fo*
 *rm

(9.6)         = : Dp(X ^ Y ) -! Hom (Eq(X), (D ^ E)p-q(X)),


(9.7)         \ : Eq(X ^ Y ) -! Hom (Dp(X), (D ^ E)q-p(Y )).

  The four products are obtained by passing to ß* and applying the pairing (9.1)
and functoriality, starting from formally defined canonical maps

(9.8)              D ^ X ^ E ^ Y -! D ^ E ^ X ^ Y,


(9.9)            F (X, D) ^ F (Y, E) -! F (X ^ Y, D ^ E),


(9.10)            F (X ^ Y, D) ^ E ^ Y -! F (X, D ^ E),


(9.11)             F (X, D) ^ E ^ X ^ Y -! D ^ E ^ Y.

Here (9.10) is obtained by permuting E and Y and using the natural isomorphism

                      F (X ^ Y, D) ~=F (Y, F (X, D)),

the evaluation map " : F (Y, F (X, D)) ^ Y -! F (X, D), and the natural map

                      : F (X, D) ^ E -! F (X, D ^ E),

while (9.11) is obtained by permuting E and X and using the evaluation map
" : F (X, D) ^ X -! D.
  Of course, when D = E is a monoid in C (ring spectrum in the algebraic topolo*
 *gy
setting), we can compose the given external products with maps induced by the
product E ^ E -! E to obtain internal products.  Similarly, when X = Y  has
a coproduct X - ! X ^ X or product X ^ X - ! X, we can obtain internal
products by composition. In topology, we are thinking of reduced cohomology and
the diagonal map A+ -! (A x A)+ ~=A+ ^ A+ on spaces A. The internalization
of the product \ is the cap product.
  There are many unit, associativity, and commutativity relations relating the *
 *four
products, and these are catalogued in [1] and [16]. Without exception, these fo*
 *rmu-
las are direct consequences of our axioms for a symmetric monoidal category with
a compatible triangulation. In particular, Adams [1, pp. 235-244] and Switzer [*
 *16,
pp. 276-283] catalogue many formulas and commutative diagrams that relate the
four products to the connecting homomorphisms in the homology and cohomology
of pairs (X, A) and (Y, B), the crucial point being the correct handling of sig*
 *ns.
Modulo change of notation, they are considering the behavior of smash products
and function spectra with respect to pairs of distinguished triangles in the st*
 *able
homotopy category.  Our compatibility axioms give what is needed to make the
derivations of these formulas and diagrams formal consequences of the axioms.




          THE ADDITIVITY OF TRACES IN TRIANGULATED CATEGORIES        31


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  Department of Mathematics, The University of Chicago, Chicago, IL 60637
  E-mail address: may@math.uchicago.edu