Homology and cohomology Spring 2009

To work effectively with simplicial sets, which provide useful algebraic or combinatorial models of topological spaces, we need to get serious about geometric realization, colimits, limits, and adjunctions. A first step is to recall the Yoneda Lemma. Given any pair of functors F,G : D−→C, denote by Nat(F,G) := {φ| φ : F−→G} the collection of all natural transformations from F to G. Denote by sSet the category of simplicial sets and their maps, and recall from Series 5 that sSet = Set op . Hence, a simplicial set Y is the same as a ∆op-shaped diagram Y : ∆op−→Set, and sSet(X,Y ) = Nat(X,Y ). Proposition 1 (The Yoneda Lemma). Let D be a category and consider any r, s ∈ D. (a) If X : D−→Set is a functor, then there exists an isomorphism Nat (