Exact Categories And Quillen Monoids

As a generalization of Grothendieck monoids,the concept of Quillen monoids of exact categories is introduced.Some properties of exact categories are studied,it is proven that the algebraic K-group K_n（N）of an exact category N is the group completion of M_n（N）for each n and the Nenashev relation holds for M₁（N）.It is shown that there is a bijection between the set of cofinal subcategories of N which are closed under isomorphic inflation series and the set of cofinal monoids of M₀（N）.Two sufficient and necessary conditions for M₀（N）of a split exact category to be cancellative are given and the concept of Heller monoids of exact categories is introduced.It is shown that K₀（N）is the group completion of Heller monoid H（N）.Moreover,a characterization of M₁（N）of a cancellative split exact category is given and the additivity theorem for the algebraic K-groups is proven to hold for Quillen monoids.