# Algebraic K-Theory by E. M. Friedlander, M. R. Stein

By E. M. Friedlander, M. R. Stein

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**Example text**

J. 2) we obtain the category nat of natural systems: Objects are pairs (C, D) where £> is a natural system of the small category C. ' is a natural transformation. Here (t>*D : FC -^ Ab is given by {(t>*D)f = D^j and a* = (a)*, /3* = 0(^)*. A natural transformation T : D —^ D yields as well the natural transformation (l>*t:(l)*D-^(j>*D. ^^,^oV'V). 13 below. One has the full inclusion of categories mod C nat which carries (TT,M) to {C,D) where C = TT is the category given by the group TT and where D is the natural system on C with Df = M for f e n and a* = identity and P^{x) = x^ for X G M, ^ E TT.

On the other hand the groups [M{A,n), M{B,m)\ are not at all understood; for A = B = Z these are the homotopy groups of spheres. 7. 13. PROPOSITION. The homotopy decomposition of X can be chosen in Top to be functorial in X. The homology decomposition of X cannot be chosen to be functorial neither in Top nor in the homotopy category Top/ ~. 14) which are called the cohomology group of X, resp. the homotopy group of X in dimension n with coefficients in the abelian group A. 7 yield elements k'^X^k'^£T^n-x{HnX,Xn-x).

On the one hand Loday [63] defines algebraic systems called 'cat"-groups' (see also Porter [75] and Bullejos, Cegarra and Duskin [20]), on the REMARK. Section 4 Homotopy types 21 other hand Conduch6 [26] considers *crossed modules of length 2' representing 3-types which were generalized by Carrasco [21] and Carrasco and Cegarra [22] for n-types; this approach of Conduch6 and Carrasco describes additional structure for the Moore chain complex N{G) which is sufficient to determine the simplicial group G.