By Hvedri Inassaridze

This publication exposes equipment of non-abelian homological algebra, corresponding to the speculation of satellites in summary different types with admire to presheaves of different types and the speculation of non-abelian derived functors of staff valued functors. purposes to K-theory, bivariant K-theory and non-abelian homology of teams are given. The cohomology of algebraic theories and monoids also are investigated. The paintings is predicated at the contemporary paintings of the researchers on the A. Razmadze Mathematical Institute in Tbilisi, Georgia. *Audience:* This quantity may be of curiosity to graduate scholars and researchers whose paintings includes class concept, homological algebra, algebraic K-theory, associative earrings and algebras; algebraic topology, and algebraic geometry.

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A map I : G* ---+ G~ of pseudosimplicial groups is called simplicial if it satifies the condition In+lsf = sfln for n 2: 0, 0 ~ i ~ n. 10. A map I : G* ---+ G~ of pseudosimplicial groups is called a fibration if 1) I is simplicial, 2) for every n 2: 0 the homomorphism In : Gn ---+ G~ is surjective. 11. If I : G* ---+ G~ is a fibration then the sequence of homotopy groups . . ---+ 7fn +l (G~) ---+ 7f n(K er J) ---+ 7fn ( G*) ---+ 7fn ( G~) ---+ ... is exact . Proof. Clearly K er I is a pseudosimplicial group with respect to the boundery and pseudodegeneracy operators induced by the pseudosimplicial group G*.

Let (X*, a, A) be a contractible P-resolution and let T E Ob CS. Then there is a spectral sequence Proof. 32, Theorem 2. 34 and Quillen's spectral sequence. 35. 36. Let T : C --+ Gr be a cosheaf and let (X*, a, A) be a contractible P -resolution such that L~T(Xm) = 0 for all n > 0, m 2:: 0 and L5T(Xm) = T(Xm) , m 2:: O. Then ·there is an isomorphism L~T(A) ~ 1l"nT(X*), n 2:: O. Proof. 35. 37. If the projective class P is induced by a cotriple F then there is an isomorphism for any T : C --+ Gr.

With no essential change we will follow Quillen's proof for a bisimplicial group [84]. Let G* = {G n, 8i, sf} be a pseudosimplicial group. Define a pseudosimplicial group EG* by (EG*)n = {x E Gn+1 : 8~85 .. (x) = I} . -with boundary and degeneracy operators induced from G*. Further let C1ro(G*) be a constant pseudosimplicial group with (C1rOG*)n 1ro(G*) and identity boundary and degeneracy operators. There is a sequence of pseudosimplicial groups . 1 - t nG* ~ EG* {) -t . 3) where {)n : (EG*)n - t Gn is induced by the homomorphism 8~ti : Gn+!