FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS Any errors discovered in the reading should be forwarded to the author who will deal with them promptly. 1. Intuition Fact 1.1. 0 N. 2 Throughout these notes, denote by CGHaus the category of compactly generated Hausdorff spaces, and by Top we denote the category of spaces. Definition 1.2. Let N := 1 N. The set N is naturally an object of Ord,the − {− } [ − category of partially ordered sets: 1 0 1 2 . {− ! ! ! ! ···} For each n N we define the real n-simplex by 2 − n+1 n n (x0,...,xn) R 0 i=0 xi =1 n 0 := 2 ≥ ≥ |4 | (? n = 1 P − topologized by the obvious inclusion n Rn+1. |4 | ⇢ For most n,thesets n 1 n HomCGHaus − , 4 |4 | and n+1 n HomCGHaus , 4 |4 | Figure 1. The real 2-simplex 2 |4 | 1 FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 2 Figure 2. The commuting of di’s. are quite large. In particular, they’ve at least the cardinality of n . There are however |4 | some morphisms in those Hom-sets which are easy to write down. Definition 1.3. For n N and 0 i n,wedefinemorphismsofCGHaus, di and si,by 2 − the following rule. i i n 1 d n s n+1 − / o |4 | |4 ||4 | (x0,x1,...,xn 1) / (x0,x1,...,xi 1, 0,xi,...,xn 1) − − − (x0,x1,...,xi + xi+1,xi+2,...,xn)(o x0,x1,...,xn) Remark 1.4. We consider a pair of diagrams in CGHaus.Observe: 1 (1, 0) d1 8 |4 | d2 8 ✏ & 2 ' 0 2 (1) (1, 0, 0) . |4 | d1 d1 8 |4 | 7 & & / 1 (1, 0) |4 | In a more geometric vein we envision this commutative square as in Figure 2. Observe: 0 (1) 6 ⌘ s0 8 |4 | d0 - 1 & 1 ( (x0,x1) (0, 1) . |4 | d0 s1 8 |4 | ⌘ 6 & ( - 2 (0,x ,x ) |4 | 0 1 Again, in a more geometric vein, we envision this commutative square as in Figure 3. FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 3 Figure 3. The commuting of di’s and sj’s. Atrivialexercise: Exercise 1.5. For each n N ,provideacellulardecompositionforthespace n ,and 2 − |4 | do so in such manner that the maps sj and di are cellular. An exercise to be completed at least once in your life. You may also wait for the second instance of this exercise later on. Exercise 1.6. Verify that for n N and i, j such that the compositions involved are 2 − defined, the morphisms di and sj obey the following relations, called the co-simplicial identities. j i i j 1 d d = d d − if i<j sjdi = disj 1 i<j 8 − if j j j j+1 >s d =1=s d > j i i 1 j <>s d = d − s if i>j+1 sjsi = sisj+1 i j > if > > Definition 1.7. For each n :0,andforeach0 i n,letdi,calledaface map,tobe ≥ the map n n 1 HomCW ( ,X) / HomCW ( − ,X) |4 | |4 | f / f di. ◦ We define d0 1 d0 0 o 1 o Sing (X):= Hom ( − ,X) o Hom ( ,X) d1 Hom ( ,X) o , ( |4 | |4 | o |4 | o ···) and we consider this entity as a diagram in Set calling it the singular set of X.Wedefine the object (N )op S (X) Ob AbGrp − , • 2 ⇣ ⌘ FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 4 Figure 4. Fix X Ob (CGHaus),andfixf : 2 X and consider the 2 |4 | ! procession above, from a diagram in CGHaus to a diagram in Set. by n 1 ( 1)− d n 1 i=0 − i n S (X):= ZHom ( − ,X) o ZHom ( ,X) , • ··· |4 | P |4 | ··· and refer to it as the singular chain complex. Exercise 1.8. Two formal exercises: Prove the constructions Sing (X) and S (X) to be functorial. • • Observe that the co-simplicial identities pass contravariantly to another set of iden- • tities, the simplicial identities, and as a consequence find that the singular chain complex is indeed a chain complex. FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 5 Proposition 1.9. The composition HSing CGHaus • / Ch (AbGrp) • I forget . CGHaus / Ch (AbGrp) H S • • is a reduced homology theory on CGHaus . • Proof. See Hatcher, May, or Goerss & Jardine. ⇤ Were we to restrict ourselves to those spaces which are CW, then we’ve the following corollary. Corollary 1.10. We’ve a natural isomorphism of chains of abelian groups, HSing (X) ⇠ HCW (X) . • ! • What about the data we had assembled right before Sing (X), what kind of thing was it? adiagraminCGHaus or equivalently a subcategory of CGHaus;or • adiagraminCGHaus/X or equivalently a subcategory of CGHaus/X. • Definition 1.11. Let X be the category such that: |4| # Ob ( X)= n X : CGHaus n N |4| # {|4 | ! | 2 −} and n m ✓ n / m |4 | |4 | |4 | |4 | ik i1 j` j1 Hom X 0 , 1 = 8 ✓ = d d s s 9 . |4|# ✏ ✏ } ··· ··· X X <> => B C X @ A Let F : X CGHaus be the> forgetful functor ( n X) n . > |4| # ! : |4 | ! 7! |4 | ; Remark 1.12. Observe that F ( X) is a diagram in CGHaus,andmore,sinceforeach |4| # n appearing in that diagram we’ve maps n X which commute with the morphisms |4 | |4 | ! of the diagram, we get a map lim n X. |4 | ! F ( ! X) |4|# It must also be noted that if X is pointed, then this colimit is naturally pointed by the 0-simplex corresponding to the point 0 = X. |4 | • ! Lemma 1.13. The induced map lim n X |4 | ! F ( ! X) |4|# is a weak homotopy equivalence. Proof. It suffices to prove that for each n N,andallX Ob (CGHaus) that the induced 2 2 morphism of groups n ⇡n lim ⇡n (X) 0 |4 |1 ! F ( ! X) |4|# @ A FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 6 is an isomorphism, and for that lesser criterion a proof of bijectivity will suffice. Injectivity: suppose Sn lim n ! |4 | F ( ! X) |4|# to generate a homotopy class of spheres in that colimit which passes to the trivial class in X. We may thus assemble the following commutative diagram. n S / lim • |4 | 7 F ( ! X) |4|# n+1 7 |4 | ⇠ Dn+1 6 ⇠ n ✏ " ✏ S I+ /, X ^ nullhomotopy The first isomorphism is from a common characterization of a nullhomotopy, the second isomorphism is evident; the indicated lift is the canonical inclusion of n+1 into the colimit |4 | corresponding to the morphism n+1 X. Then since n+1 is contractible, so too the |4 | ! |4 | image of Sn in the colimit. Indeed [Sn X]=[ ] and the induced morphism of homotopy ! • groups is injective. Surjectivity: suppose Sn X to be some morphism of CGHaus. Then we may assemble ! the following diagram. lim • |4 | 7 F ( ! X) |4|# @ n+1 6 |4 | ⇠ @Dn+1 7 ⇠ # ✏ Sn /, X In this diagram the isomorphisms are evident and the lift is gotten by the universal property of the colimit on the sub-diagram of F ( X) corresponding to the n +2faces @ n+1 |4| # |4 | in X. Thus the induced morphism of homotopy groups is surjective. ⇤ 2. Formalism We have already observed that S (X) may be thought of either as a diagram in AbGrp or op • (N )op as a functor (N ) AbGrp,i.e.anobjectofthefunctorcategoryAbGrp − .Untilthis − ! point we’ve thought of Sing (X) only as a diagram in Set, but as with S we may envision • this diagram as a functor, with as of yet unnamed source. FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 7 Definition 2.1. Let ,calledtheaugmented simplex category,bethefullsubcategory 4− of Ord subtended by the objects 0 < 1 < <n n 0 Ob ( ):= [n]:= { ··· } ≥ . 4− ? n = 1 ( ( − ) Lemma 2.2. The category is generated by two types of morphisms: 4− di si [n 1] / [n][o n +1] − 0 / 00o 1 / 11o . i 1 / i 1 o i 1 − − − i ⇡ iilo . , . i +1k ⇡ i +1 . n 1 . ⌫ i +2 − ⌫ . + n l . ⇡ n +1 and these morphism satisfy the co-simplicial identities. Proof. Exercise. ⇤ Definition 2.3. There are two important variations on .Let be the full subcategory of 4− 4 subtended by the objects [n] for n N,andcallthisthesimplex category.Denoteby 4− 2 +the wide subcategory of wherein morphisms are generated only by the di,andcallitthe 4 4 + semi-simplex category. Denote likewise by ( ) the wide subcategory of wherein 4− 4− morphisms are generated by only by the di,andcallthistheaugmented semi-simplex category. Afunctor ( )op Set 4− ! is called an augmented simplicial set.Afunctor op Set 4 ! is called a simplicial set.Afunctor op + Set 4 ! is called a semi-simplicial set.Lastly,afunctor op + Set 4− ! is called an augmented semi-simplicial set. Remark 2.4. The profusion of categories here may seem unnecessary, but there are certain advantages. Our implicit use of until now, as oppose to is precisely why we get 4− 4 a reduced homology theory, as opposed to a homology theory. Similarly, it will be seen that generalization of the construction lim • ,didnotdependonkeepingtrackof |4 | F ( ! X) |4|# FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 8 those maps sj between simplices, so the categories +and + will do just as well for that 4− 4 construction in some ways. This is not to say that the maps sj are irrelevant in general, in fact they are essential for products as will be seen later. We’ve already seen one instance of such an object, Sing (X),butthereareofcourse others. Perhaps the most important for our purposes amongst these simplicial sets are the representable pre-sheaves. n := Hom (_, [n]) 4 Exercise 2.5.