Algebraic K-Theory by V. Srinivas (auth.)

By V. Srinivas (auth.)

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Hence the diagram from X to T. -- Z x y V >------+ T defines an arrow in One checks that the isomorphism class of this diagram depends only on the isomorphism classes of X <-<-- Z >---+ Y and Y - v >---+1; so that we have a well defined composition rule for morphisms; next, one verifies that composition is associative. Hence i f ~ is a small category, then is well defined, and Let 0£0b~ is a small category. 0): (where K0 d~) Q~ There is a natural isomorphism is as defined above). This motivates Quillen's second definition of higher K-theory.

Denotes the group operation in [ = rM' 1 . ''] commute, so the homomorphism iM = ioiM' , We note that from the sequence (E) above, and the facts that q~ 1 = q~ 1 , o q in diagram ~;here 1:; , giving notice 2 more arrows 0 2-simplices in ---;. I I i! oqir• q·. i u : 0 __,. ~~ in Q y. M in cc B Q~. == q~qMo in Q'( , ~;e have a From the diagram, we also I i! 0 q~l ' and namely By definition the composite M''! corresponding to the diagram I q~1 i~l! o iM'! and I the shaded triangles commute: The shaded triangles give that in fact I I qM' .

Math. S. (1971)). is the reduced suspension and Q On o"" Sm and the symmetric group, We note that for any (X, x) if I:X X the loop space of loops based at natural base points, then there is a natural continuous map applying this to I: rx n subgroup, so we can form the space as: , aOO awlyibg the functor rP we I X- Qm l:m X. ,. whose direct limit is called Q l: x, with ... The Plus Construction 26 so that we have a direct system of groups Let l'tlw ith stable homotopy group of be the 1!. 1 stable homotopy group of X.

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