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Problems

April 12, 2012

1. Describe explicitly (co)products and (co)equalizers in Set, Top, Mo dA , SSet and DGMo dA . Explain how to compute a (co)limit of an arbitrary small diagram in all these categories. 2. For X SSet denote by skn (X ) the minimal simplicial subset of X containing all simplices of X of degree n and less (put sk
-1

(X ) = ).
n X )N

· Prove that skn (X ) is isomorphic to the colimit of the following functor F : ( Here (n )N X is the subcategory of the category of elements
X

SSet.

of X (see lectures) containing all

non-degenerate simplices of degree n and less, and F sends a non-degenerate simplex k X to k SSet. · Prove that for any n N there is a pushout diagram
n

- skn-1 (X )

?

n

? - skn (X )

What is the set over which the coproduct on the left is taken? · Prove that the realisation | n | is isomorphic to S n in Top. · Prove that |X | is a CW-complex. 3. In , define1 i : [n - 1] [n] to be the unique injective monotone map not containing i in its image. Denote also by i : [n + 1] [n] the unique surjective monotone map which maps i and i + 1 in [n + 1] to i in [n]. For a simplicial set X , denote by di = X (i ) : X (n) X (n - 1) and si = X (i ) : X (n) X (n + 1). These are called, respectively, i-th face and degeneracy maps. · Prove the identities di dj = dj di sj = dj
-1 di -1 si

(i < j ), (i < j ),
+1 sj

dj sj = 1 = dj di sj = sj di si sj = s
1

-1

(i > j + 1) (i j )

j +1 si

[n] is omitted from the notation

1


· Prove that any injective (surjective) map in can be written as a composition of i (i ). Consequently, prove that any map in can be written as a composition of i followed by i . · Prove that n is isomorphic to the colimit of
0i
0in



n-1

(what are the two maps in this diagram?) · Let A[-] : Set Mo dA denote the free A-module functor. For a simplicial set X , define A[X ]-i = A[X (i)] and d-i : A[X (i)] A[X (i - 1)] X to be the sum
j

(-1)j dj . Prove that this gives a functor from SSet to DGMo d0 . A

4. A groupoid is a category C such that any morphism in Mor C is an isomorphism · Prove that for any category D and X in SSet, a map f : X N (D) is determined by f0 , f1 and f2 (fn : X (n) N (D)(n)). · Prove that for a groupoid C, the nerve N (C) is fibrant in the standard model structure on SSet. 5. Let M be a model category. Prove Whitehead's theorem A morphism f : X Y between fibrantcofibrant ob jects is a weak equivalence if and only if it is an isomorphism in Mcf . (Hint: for 'only if ' part, it is enough to prove it only for trivial (co)fibrations. For 'if ' part, it might be useful to factor f = p i, so that i is a trivial cofibration and p is a fibration, and then try to show that p is a weak equivalence.) 6. Prove Ken Brown's lemma : let M be a model category. If a functor F : M D takes trivial cofibrations between cofibrant ob jects to isomorphisms in D, then F takes all weak equivalences between cofibrant ob jects to isomorphisms in D and the left derived functor LF exists. (Hint: if f : A B is a weak equivalence between cofibrant ob jects, factor the induced map A
B B - B as

f id

a cofibration followed by trivial fibration. Also, remember that cofibrations are stable under pushout.) 7. Let M F
-

G

N

be a Quillen adjunction. Prove that the following are equivalent · The pair LF Ho (M) - Ho (N) RG are inverse to each other up to natural isomorphism (that is, (LF, RG) is an adjoint equivalence of categories). · For any cofibrant ob ject X of M and any fibrant ob ject Y of N a morphism F (X ) Y is a weak equivalence in N if and only if the adjoint morphism X G(Y ) is a weak equivalence in M.

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