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By Marc Levine

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An−1 −→ an . → a1 − a0 − 32 A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 If h : [m] → [n] is order-preserving define h((f1 , . . , 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 , .

Algebraic cycles and vector bundles over affine three-folds, Ann. of Math. (2) 116 (1982), no. 3, 579–591. , Algebraic cycles and vector bundles over affine varieties, Algebra and its applications (New Delhi, 1981), 25–28, Lecture Notes in Pure and Appl. , 91, Dekker, New York, 1984. [28] Mumford, D. Rational equivalence of zero-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195-204. , Zero cycles and projective modules, Ann. of Math. (2) 140 (1994), no. 2, 405–434. 56 A SHORT COURSE IN K-THEORY MEXICO CITY MAY, 2002 [30] Quillen, Daniel.

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 . The fundamental example of a cosimplicial space (C = Top) is ∆ : Ord → Top. ∆([n]) is the standard topological n-simplex: n ∆([n]) = ∆n := {(t0 , .

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