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 is 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 < 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. Prove that the assignment

_ : / CGHaus | | 4 n / n 4 |4 | is functorial. We constructed the pre-sheaves Sing (X) from diagrams in CGHaus, and what’s more, we proved that the colimits over those diagrams are weakly equivalent to the original spaces X. We will now discover how, in fact, every simplicial set corresponds to a diagram in a similar way, and how those diagrams can be used to beget spaces in such a way that we recover the construction performed with Sing (X). Towards the extraction from an arbitrary pre-sheaf the correct diagram the following famous lemma suggests the way. Lemma 2.6 (the Yoneda Lemma). Let C be a small category. Letting, for each X Ob (C ), C op 2 hX := Hom (_,X),thenforeachY Ob Set ,we’veanisomorphismofsets C 2 Hom (hX,Y) ⇠ Y (X) C ! which is moreover natural in both X andb Y . C op Corollary 2.7 (the Yoneda Embedding). The functor C Set : X hX is a full ! 7! and faithful embedding. Definition 2.8. Given a pre-sheaf X : op Set,definethecategory X to be the 4 ! 4# category given by setting n op Ob ( X):= X : Set4 n N , 4# 4 ! 2 and n m ✓ n / m 4 4 4 4 Hom X 0 , 1 := 8 9 , 4# ✏ ✏  X X <> => B C X @ A with the obvious composition law. :> ;> Exercise 2.9. Prove the assignment

op Set4 / Cat X / X 4# to be functorial. FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 9

Just as we did earlier, we may choose to let F be the obvious functor ( n X) n, op 4 ! 7! 4 whence F ( X) is a diagram in Set4 .What’smorehowever,itisnotmerelyadiagram op 4# in Set4 but in fact a diagram therein contained in the image of under the Yoneda 4 embedding.

op Corollary 2.10. For any X Ob Set4 we’ve an isomorphism lim • ⇠ X which is 2 4 ! F (!X) 4# natural in X. Definition 2.11. We define the geometric realization of a simplicial set by

_ : / CGHaus | | 4

X b / X := lim • . | | |4 | F (!X) 4# where the colimit lim • is the colimit over the diagram in CGHaus gotten by applying |4 | F (!X) 4# n n op _ : to the diagram F ( X) in Set4 . | | 4 7! |4 | 4# Remark 2.12. We use the same notation as we did for n n as this new functor 4 7! |4 | extends the prior use in a way which admits formalization1. More, it must be noted that the way in which we extend this functor preserves colimits. In particular note that for any op X Ob Set4 we’ve 2 lim • ⇠ X = lim • . 4 | | |4 | F (!X) F (!X) 4# 4# Exercise 2.13. Suppose to be a small category and suppose I / Top I i / Xi to be a diagram in Top. Prove that the space

i Ob( ) Xi/ 2 I ⇠ ` where Xi x y Xj if and only if there exists a chain 3 ⇠ 2 Xi X1 Xj ^ ? _ @

xX0 x1 y _ > ` ··· ? > > x 0 ··· of objects X0,X1,...,morphismsinthediagram,andelementsasabove,enjoystheuniversal property of limXi. ! 1In fact it is a Kan extension along the Yoneda embedding. FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 10

Remark 2.14. You should convince yourself that Sing (X) got this way is the same space | | as lim • which we’ve already proved weakly equivalent to X. |4 | !X |4|# Using what you proved in the previous exercise, observe that for a simplicial set X,the space X is isomorphic to the space | |

0 1 2 X(0) X(1) X(2) / |4 | |4 | |4 |··· ⇠ ` ` ` where the relation is generated by ⇠ n 1 i n x d (x) . 4 3 ⇠ 2 |4 |

Exercise 2.15. Convince yourself that what we’ve proven in this exercise is enough to verify that geometric realization is concerned only with the morphisms coming from + or +, op 4 4 i.e. prove that given any X Ob Set4 ,that 2 ⇣ ⌘ lim • ⇠ lim • ⇠ lim • ⇠ lim • . |4 | ! |4 | ! |4 | ! |4 | F ( !+ X) !+ F (!X) F ( ! X) F ( X) 4 # 4# 4# 4 # Remark 2.16. Degeneracies however are extremely important for agreement of our geometric notion of the product on spaces, with the categorical notion on simplicial sets. See homework 4problem4.

As is suggested by our development of them in parallel, there is a formal relationship between the functors _ and Sing. | | Proposition 2.17. The functors

Sing

op u Set4 CGHaus . ? 9

_ | | comprise an adjunction.

Indeed, the proof of the adjunction will now seem trivial. In fact, the motivation for the tack we have taken through this material was precisely to make this so 2.

2in all truthfulness, it has also been to suggest a way that the Yoneda lemma, all too often seen merely as abstract formalism, admits a natural geometric interpretation. FROM SINGULAR HOMOLOGY TO SIMPLICIAL SETS 11

op Proof. Let X Ob Set4 and let Y Ob (CGHaus). Then observe that 2 2 HomCGHaus ( X ,Y) ⇠ HomCGHaus lim • ,Y | | ! 0 |4 | 1 F (!X) 4# @ A ⇠ lim HomCGHaus ( • ,Y) ! |4 | F ( X) 4# = lim HomCGHaus ( •, Sing (Y )) 4 F ( X) 4#

⇠ Hom op lim •, Sing (Y ) ! Set4 0 4 1 F (!X) 4# where the first isomorphism was observed after the@ definition of the geometricA realization, the second is the universal property, the equality is by definition, and the last isomorphism is again by universal property. ⇤