A short course in K-theory by Marc Levine

By Marc Levine

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We can just as well replace Sets with an arbitrary category C, giving the notion of a simplicial object of C. A cosimplicial object of C is a functor T : Ord → C; T ([n]) is the n-cosimplices of T . 1. The morphisms in Ord are generated by the coboundary maps δin : [n] → [n + 1], i = 0, . . , n + 1, δin (j) = j j+1 if j < i if j ≥ i, and the codegeneracy maps σin : [n] → [n − 1], i = 1, . . , n, σin (j) = j j−1 if j < i if j ≥ i. These satisfy certain relations, which we don’t specify here. A simplicial object S is thus often given by defining the n-simplices Sn , the boundary maps ∂in = S(δin−1 ) : Sn → Sn−1 , and the degeneracy maps sni = S(σin+1 ) : Sn → Sn+1 .

Fn )) = (g1 , . . , gm ), where gi : ah(i−1) → ah(i) is the composition of fh(i−1)+1 , . . , fh(i) if h(i − 1) < h(i), and the identity on ah(i−1) if h(i − 1) = h(i). The classifying space of the category C is defined as BC := |N C|. 2. (1) Let X be a set. Let E(X) be the category with objects X, and with a unique morphism x → y for each x, y ∈ X. Clearly N (E(X))n = X n+1 , with ∂in (x0 , x1 , . . , xn ) = (x0 , . . , xi−1 , xi+1 , . . , xn ) We write EX for |E(X)|; it is not hard to see that EX is contractible.

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