Categorical Background (Lecture 2)

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1 Cateorical Backround (Lecture 2) February 2, 2011 In the last lecture, we stated the main theorem o simply-connected surery (at least or maniolds o dimension 4m), which hihlihts the importance o the sinature σ as an invariant o an (oriented) Poincare complex. Let us bein with a ew remarks about how this invariant is deined. Let V be a inite dimensional vector space over the real numbers and let q : V R be a nondeenerate quadratic orm on V, with associated bilinear orm (, ) : V V R. We can always choose an orthoonal basis {x 1,..., x a, y 1,..., y b } or V satisyin (x i, x i ) = 1 (y i, y i ) = 1. The sum a + b is the dimension o the vector space V, and the dierence a b is called the sinature o q, and denoted σ(q). Theorem 1 (Sylvester). Let V be a inite dimensional vector space over R equipped with a nondeenerate quadratic orm q : V R. Then the sinature σ(q) is well-deined: that is, it does not depend on the choice o orthoonal basis or V. Moreover, i V is another inite dimensional vector space over R with a quadratic orm q : V R, then there exists an isometry (V, q) (V, q ) i and only i dim V = dim V and σ(q) = σ(q ). The notion o a vector space V with a quadratic orm makes sense over an arbitrary ield k. We have emphasized the case k = R or two reasons: irst, the classiication o quadratic orms over R is particularly simple (because o Theorem 1). Second, inormation about the intersection orm on the middle cohomoloy H 2m (; R) o a simply connected Poincare complex o dimension 4m plays an important role in determinin whether or not is homotopy equivalent to a maniold. However, quadratic orms over other ields are also o eometric interest. For example, i is a simply connected Poincare complex o dimension 4m + 2 > 4, then the problem o indin a maniold in the homotopy type o turns out to depend on more subtle properties o quadratic orms over the ield F 2. We would ultimately like to treat maniolds o all dimensions in a uniorm way, which will require us to somehow interpolate between the ields k = R and k = F 2. We can do this by considerin quadratic orms over the inteers, or over more eneral rins. The theory o quadratic orms over an arbitrary rin R can be very complicated. For example, the classiication o quadratic orms over the ield Q o rational numbers is a nontrivial achievement in number theory (the Hasse-Minkowski theorem). In eneral, we cannot detect whether two quadratic orms are isomorphic by means o simple inteer invariants, as in Theorem 1. However, there are more elaborate theories that are desined to eneralize the dimension and sinature to other contexts: 1

2 K-theory L-theory input: projective module module with quadratic orm or R-vector spaces dimension sinature classical version Grothendieck roup K 0 Witt roup invariant o maniolds Euler characteristic sinature local-lobal principle Gauss-Bonnet theorem Hirzebruch sinature ormula We are ultimately interested in understandin the riht hand column o this table. But let us irst spend some time discussin the let column, which is perhaps more amiliar. We bein by recallin the deinition o the Grothendieck roup K 0 (A) o an associative rin A. The collection o isomorphism classes o initely enerated projective A-modules orms a commutative monoid under the ormation o direct sums. The roup completion o this monoid is denoted by K 0 (A), and called the (0th) K-roup o A. Put another way, the K-roup K 0 (A) is the abelian roup enerated by symbols [P ], where P ranes over all projective let A-modules o inite rank, subject to the relations iven by [P P ] = [P ] + [P ]. Remark 2. I A is a ield, then all initely enerated A-modules are automatically projective, and are determined up to isomorphism by their rank (in other words, by their dimension as A-vector spaces). Consequently, there is a canonical isomorphism K 0 (A) Z, which assins to each A-module P its dimension dim A (P ). Consequently, we can think o the K-roup K 0 (A) as a device which allows us to eneralize the notion o dimension to the case o modules over arbitrary rins. Let us now consider the ollowin question: Question 3. What is K-theory an invariant o? We ive several answers, beinnin with the obvious. (a) K-theory is an invariant o rins. However, we can immediately improve on (a). Note that the Grothendieck roup K 0 (A) is deined purely in terms o the cateory o initely enerated projective modules over A. In particular, Morita equivalent rins (such as matrix rins M n (A)) have the same K-theory as A. We can thereore improve on our irst answer: (b) K-theory is an invariant o additive cateories. For our purposes, it will be convenient to ive a variation on this answer. Recall that we are ultimately interested in assinin eometric invariants to maniolds and other Poincare complexes. We can obtain some by considerin alebraic invariants to vector spaces. Let k be a ield. For any inite CW complex, we can consider the Betti numbers b i = dim k H i (; k). In eneral, these invariants depend on the ield k. However, the Euler characteristic χ() = i ( 1) i b i = i ( 1) i dim k H i (; k) 2

3 does not depend on k. This invariant χ() is not enerally not the dimension o a vector space (or example, it can be neative). Instead, we should think o it as an invariant o a chain complex o vector spaces: the sinular chain complex C (; k). More enerally, we would like to say that or any rin A, the chain complex C (; A) determines a class in the Grothendieck roup K 0 (A). (O course, we know what this class should be: namely, χ()[a] K 0 (A). Inore this or the moment.) We enerally cannot deine this class to be the alternatin sum ( 1) i [H i (; A)], i because the individual A-modules H i (; A) need not be projective. To show that C (; A) determines a well-deined class in the roup K 0 (A), it is convenient to describe K 0 (A) in a dierent way. Rather than representin K-theory classes by projective modules over A, we take as representatives chain complexes o modules over A. O course, we do not want to allow arbitrary chain complexes. In order to obtain reasonable invariants, we should restrict our attention to chain complexes which are perect: that is, which are quasi-isomorphic to bounded chain complexes o inite projective modules. I is a inite CW complex, then the sinular chain complex C (; A) is always perect: or example, it is quasi-isomorphic to the chain complex which computes the cellular homoloy o. Let Per denote the cateory whose objects are bounded chain complexes o inite projective modules. For every perect complex M, we can ind a quasi-isomorphism P M, where P Per. The chain complex P need not be unique. However, it is unique up to chain homotopy equivalence. We can obtain a stroner uniqueness result by passin to the homotopy cateory. Let hper be the cateory with the same objects as Per, but whose morphisms are iven by chain homotopy classes o chain maps. The cateory hper is called the perect derived cateory o A. Every perect complex o A-modules determines an object o hper, which is well-deined up to isomorphism. Moreover, hper is an example o a trianulated cateory: that is, there is a notion o distinuished trianle P P P P [1] in hper. Let K 0(A) denote the abelian roup enerated by symbols [P ], where P is an object o hper, with relations [P ] = [P ] + [P ] or every distinuished trianle P P P P [1]. Every initely enerated projective A-module can be rearded as a perect complex which is concentrated in deree zero, and this construction determines a map o abelian roups K 0 (A) K 0(A). One can show that this map is an isomorphism. In other words, one can deine the Grothendieck roup usin (perect) chain complexes o modules, rather than individual modules. With this deinition, it is easy to see that C (; A) determines a well-deined K-theory class, or every inite CW complex. This suests a dierent answer to Question 3: (c) K-theory is an invariant o trianulated cateories. For purposes o this course, answer (c) is a little bit misleadin. The passae rom the cateory Per to its homotopy cateory hper has upsides and downsides. It has the virtue o allowin us to treat quasiisomorphic chain complexes as i they are actually isomorphic. Unortunately, it also loses a lot o inormation, because it has the eect o identiyin chain homotopic morphisms without rememberin any inormation about the chain homotopy. For our purposes, we will need to work with an intermediate object D per (A), which will allow us to treat quasi-isomorphisms between chain complexes as i they were isomorphisms while retainin inormation about chain homotopies. The ine print is that unlike Per and hper, D per (A) is not a cateory: rather it is a more eneral object called an -cateory. We are now ready to ive our inal answer to Question 3: (d) K-theory is an invariant o stable -cateories. 3

4 Our oal in this lecture and the next is to explain the meanin o this statement. To do so will require a lon diression. Deinition 4. Let C be a cateory. The nerve o C is a simplicial set N(C) whose n-simplices are iven by composable sequences o morphisms C 0 C 1 C n in C. I C is a cateory, then C can be recovered (up to canonical isomorphism) rom its nerve N(C). The objects o C are in bijection with the 0-simplices o N(C). I and are objects in C, then Map C (, ) can be identiied with the set o 1-simplices o N(C) joinin with. Given a pair o composable morphisms Hom C (, ) and Hom C (, Z), there is a unique 2-simplex o N(C) havin 0th ace and 2nd ace, and the composition is the 1st ace o this 2-simplex: Z. We can summarize this discussion as ollows: the construction C N(C) determines a ully aithul embeddin rom the cateory o (small) cateories into the cateory o simplicial sets. The essential imae is described by the ollowin claim: Fact 5. Let S be a simplicial set. Then S is isomorphic to the nerve o a cateory i and only i the ollowin condition is satisied: ( ) For every pair o inteers 0 < i < n, every map 0 : Λ n i S extends uniquely to an n-simplex : n S. Here Λ n i denotes the ith horn: the simplicial subset o n obtained by removin the interior and the ace opposite the ith vertex. Example 6. When i = 1 and n = 2, condition ( ) says that every pair o composable edes and determine a unique 2-simplex Recall that a simplicial set S is a Kan complex i it satisies the ollowin variant o ( ): ( ) For every pair o inteers 0 i n, every map 0 : Λ n i S extends to an n-simplex : n S. Conditions ( ) and ( ) look similar, but neither implies the other. Condition ( ) requires that we can uniquely ill any inner horn (that is, a horn Λ n i with 0 < i < n), but says nothin about the extremal cases i = 0 and i = n. Condition ( ) requires that we can ill every horn Λ n i, but does not require the iller to be unique. However, these two conditions admit a common eneralization: Deinition 7. An -cateory is a simplicial set S satisyin the ollowin condition: ( ) For every pair o inteers 0 < i < n, every map 0 : Λ n i S extends to an n-simplex : n S. Remark 8. In the literature, -cateories are oten reerred to as quasi-cateories or weak Kan complexes. Example 9. I C is a cateory, then its nerve N(C) is an -cateory. Since passae to the nerve loses no inormation about a cateory, this construction allows us to view the usual deinition o a cateory as a special case o the notion o -cateory. Z. 4

5 Example 10. Any Kan complex is an -cateory. In particular, i is a topoloical space, then the sinular complex Sin() (whose n-simplices are iven by continuous maps rom a topoloical n-simplex into ) is an -cateory. Since the sinular complex Sin() determines up to weak homotopy equivalence, not much inormation is lost by the construction Sin(). Consequently, or many purposes, we can think o -cateories as a eneralization o topoloical spaces. We will typically use the symbol C to denote an -cateory. We will reer to the 0-simplices o C as its objects and the 1-simplices o C as its morphisms. In the simplest case (i = 1 and n = 2), the horn-illin condition ( ) asserts that or every pair o composable morphisms : and : Z, we can ind a 2-simplex σ : h in C. Here we can think o h as a composition o and, and we will write h =. A word o warnin is in order: condition ( ) does not require that σ is unique, so there may be several choices or the composition h. However, one can show that h is unique up to a suitable notion o homotopy. This turns out to be ood enouh or many purposes: Deinition 7 provides a robust eneralization o classical cateory theory. Many o the useul concepts rom classical cateory theory (limits and colimits, adjoint unctors, etcetera) can be eneralized to the settin o -cateories. Z 5