Let X and Y be finite CW complexes equipped with base points x P X and y P Y. One of the primary aims of algebraic topology is to describe the set rX, Ys of homotopy classes of pointed maps from X to Y. This is generally quite difficult, even in the case whereX andY are relatively simple spaces (such as spheres). A more reasonable (but still difficult) problem is to determine the setrX, Yssof stablehomotopy classes of maps fromX toY, which is defined as the direct limit lim
ÝÑrΣnX,ΣnYs; here ΣnX and ΣnY denote the n-fold suspensions ofX and Y, respectively. To study these invariants systematically, it is convenient to introduce the following definition:
Definition 0.2.3.1. The Spanier-Whitehead categorySW is defined as follows:
• An object of the categorySW consists of a pair pX, mq, whereX is a pointed finite CW complex andmPZis an integer.
• Given a pair of objectspX, mq,pY, nq PSW, the set of morphisms frompX, mqtopY, nq is given by the direct limit lim
ÝÑkrΣm`kX,Σn`kYs(note that the set rΣm`kX,Σn`kYs is well-defined as soon asm`k and n`k are both nonnegative).
Given a pointed finite CW complex X and an integermPZ, one should think of the object pX, mq PSW as playing the role of the suspension ΣmX. Note thatm is allowed to be negative: the Spanier-Whitehead category enlarges the homotopy category of pointed finite CW complexes by allowing “formal desuspensions.”
Example 0.2.3.2. For each integer n P Z, we let Sn denote the object of the Spanier-Whitehead category given by pS0, nq, where S0 is the 0-sphere. We will refer to Sn as the n-sphere. Note that for ně0, the objectSn can be identified with the pair pSn,0q. Remark 0.2.3.3. Let X andY be pointed finite CW complexes, and let us abuse notation by identifyingX andY with the objectspX,0q,pY,0q PSW. Then we have HomSWpX, Yq is the setrX, Yss of stable homotopy classes of maps fromX toY.
Remark 0.2.3.4. For any pair of objectspX, mq,pY, nq PSW, it follows from the Freuden-thal suspension theorem that the diagram of sets trΣm`kX,Σn`kYsu is eventually constant:
that is, the natural map
rΣm`kX,Σn`kYs Ñ rΣm`k`1X,Σn`k`1Ys is bijective for k"0.
Let H denote the category whose objects are pointed finite CW complexes and whose morphisms are homotopy classes of pointed maps. Then the construction X ÞÑ ΣX
determines a functor Σ from the category H to itself. Unwinding the definitions, the Spanier-Whitehead category can be described as the direct limit of the sequence of categories
¨ ¨ ¨ ÑHÝÑΣ HÝÑΣ HÝÑΣ HÝÑΣ HÑ ¨ ¨ ¨.
The category H arises naturally as the homotopy category of an8-category: namely, the 8-category Sfin˚ whose objects are pointed Kan complexes X for which the geometric realization|X|has the homotopy type of a finite CW complex. Moreover, the suspension functor Σ :HÑH is obtained from a functor fromSfin˚ to itself, which we will also denote by Σ. It follows that the Spanier-Whitehead category SW can also be described as the homotopy category of an 8-category: namely, the direct limit of the sequence
¨ ¨ ¨ ÑSfin˚ ÝÑΣ Sfin˚ ÝÑΣ Sfin˚ ÝÑΣ Sfin˚ ÝÑΣ Sfin˚ Ñ ¨ ¨ ¨.
We will denote this direct limit by Spfin and refer to it as the8-category of finite spectra.
The set rX, Yss of stable homotopy classes of maps fromX toY is generally easier to compute than the set rX, Ys. This is in part because the problem is more structured: for example, the set rX, Yss has the structure of an abelian group. In fact, one can say much more: the Spanier-Whitehead categorySW is an example of atriangulated categoryin the sense of Verdier (see [?]). The next definition axiomatizes those features of the8-category Spfin that are responsible for this phenomenon:
Definition 0.2.3.5. LetC be an8-category. We will say thatC isstable if it satisfies the following axioms:
paq The 8-categoryC admits finite colimits.
pbq The 8-category C has an object which is both initial and final (we will refer to such an object as azero object of C and denote it by 0PC).
pcq The suspension functor Σ :CÑC(given by the formula ΣX“0>X0) is an equivalence of 8-categories.
Example 0.2.3.6. The 8-category Sfin˚ satisfies axioms paq and pbq of Definition 0.2.3.5:
axiompbqfollows from the fact that we are working withpointedspaces (so that the one-point space is both initial and final), and axiompaq follows from from the observation that the pointed finite spaces are precisely those that can be built from the 0-sphere S0 by means of finite colimits. However, the8-categorySfin˚ does not satisfy pcq: for example, the 0-sphere S0 cannot be obtained as the suspension of another space. The8-category Spfin of finite spectra can be regarded as remedy for the fact that Sfin˚ does not satisfy pcq: it satisfies propertypcqby construction, and inherits properties paqand pbq fromSfin˚ . Consequently, Spfin is a stable 8-category.
0.2. PREREQUISITES 49 Remark 0.2.3.7. IfC is a stable8-category, then its homotopy category hC inherits the structure of a triangulated category. Moreover, essentially all of the triangulated categories which arise naturally can be described as the homotopy category of a stable 8-category.
Remark 0.2.3.8. Suppose we are given a commutative diagramσ : X //
Y
W //Z
in a stable8-category C. Then σ is a pullback square if and only if it is a pushout square.
Remark 0.2.3.9. Let Cbe an8-category with a zero object 0, and suppose we are given a commutative diagramσ :
inC. Ifσ is a pullback square, we abuse terminology by saying that the diagram XÝÑf Y ÝÑg Z
is afiber sequence(here we are implicitly referring to the entire diagramσ, which we can think of as supplying the morphismsf andg together with a nullhomotopy of the composition g˝f). Similarly, ifσ is a pushout square, then we abuse terminology by saying that the diagram
XÝÑf Y ÝÑg Z
is a cofiber sequence. If the 8-category C is stable, then the fiber sequence and cofiber sequences inC are the same.
For many purposes, the 8-category Spfin of finite spectra is too small: it admits finite limits and colimits, but does not admit many other categorical constructions such as infinite products. One can remedy this by passing to a larger 8-category.
Construction 0.2.3.10. LetCbe a small8-category. Then one can form a new8-category IndpCq, called the8-category ofInd-objects of C. This8-category admits two closely related descriptions:
paq It is obtained fromCby formally adjoining filtered colimits. In particular, every object of IndpCq can be written as the colimit lim
ÝÑCα of some filtered diagram tCαuin C, and the mapping spaces in IndpCqcan be described informally by the formula
MapIndpCqplim
pbq If C admits finite colimits, then IndpCq can be described as the full subcategory of FunpCop,Sq spanned by those functors which preserve finite limits.
We let Sp denote the 8-category IndpSpfinq. We will refer to Sp as the 8-category of spectra. A spectrumis an object of the8-category Sp.
Remark 0.2.3.11. If C is a stable 8-category, then the 8-category IndpCq is also stable.
In particular, the 8-category Sp is stable, so the homotopy category hC is triangulated.
Definition 0.2.3.12. For eachnPZ, let SnPCW be defined as in Example 0.2.3.2 and regardSn as an object of the 8-category Sp. In the special case n“0, we will denote Sn simply byS and refer to it as thesphere spectrum.
LetE be an arbitrary spectrum and letnPZbe an integer. Since the homotopy category hSp is additive, the set HomhSppSn, Eq “π0MapSppSn, Eq has the structure of an abelian group. We will denote this group byπnE and refer to it as the nth homotopy group of E.
We say that a spectrum E isconnective if the homotopy groupsπnE vanish for nă0. We let Spcn denote the full subcategory of Sp spanned by the connective spectra.
There are many different ways of looking at the notion of a spectrum (most of which lead to alternative definitions of the 8-category Sp). Let us summarize a few of the most useful:
Spectra are infinite loop spaces: Let E be a spectrum. For each integernPZ, we let Ω8´nE denote the mapping space MapSppS´n, Eq. We refer to Ω8´nE as the nth space of E. Note that each Ω8´nE can be identified with the loop space of Ω8´n´1E.
Consequently, the construction E ÞÑ tΩ8´nEunPZ determines a functor from the 8-category Sp to the inverse limit of the tower of 8-categories
¨ ¨ ¨ÝÑΩ S˚ ÝÑΩ S˚ ÝÑΩ S˚ÝÑΩ S˚ÝÑ ¨ ¨ ¨Ω .
One can show that this functor is an equivalence of8-categories. In other words, the data of a spectrum E is equivalent to the data of an infinite loop space: that is, a sequence of pointed spacestEpnqunPZ which are equipped with homotopy equivalences Epnq »ΩEpn`1q.
Spectra are cohomology theories: LetE be a spectrum. For every spaceX, letEnpXq denote the setπ0MapSpX,Ω8´nEqof homotopy classes of (unpointed) maps from X into thenth space ofE. We will refer toEnpXqas thenth cohomology group ofX with coefficients inE. One can show that the constructionX ÞÑ tEnpXqunPZ(which extends in a canonical way to an invariant of pairs of spacesAĎX) is ageneralized cohomology theory: that is, it satisfies all of the Eilenberg-Steenrod axioms characterizing singular cohomology, with the exception of the dimension axiom. Moreover, the converse is true
0.2. PREREQUISITES 51