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Universal Homotopy Theories and Associated Homological Algebras

Ahmad Rouintan

Abstract

Let $\mathscr{C}$ be a small category. For every commutative ring $R$ with unity, we associate an $R\mathrm{-linear}$ abelian category with the universal homotopy category of $\mathscr{C}$, where we can do the corresponding homological algebra.

Universal Homotopy Theories and Associated Homological Algebras

Abstract

Let be a small category. For every commutative ring with unity, we associate an abelian category with the universal homotopy category of , where we can do the corresponding homological algebra.
Paper Structure (2 sections, 8 theorems, 2 equations)

This paper contains 2 sections, 8 theorems, 2 equations.

Table of Contents

  1. Associated homological algebra with coefficients in $\mathbb{Z}$
  2. Associated homological algebra with coefficients in an arbitrary ring $R$

Key Result

Proposition 1.1

An object $g\in \mathscr{C}$ is a group object in $\mathscr{C}$ if and only if $\mathrm{Mor}_\mathscr{C}(-, g)$ is a group object in the functor category $\mathbf{Set}^{\mathscr{C}^\mathrm{op}}$.

Theorems & Definitions (14)

  • Proposition 1.1: MacLane1998, Section 3.6, Proposition 1
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Theorem 1.4: Quillen, Quillen1967, II, § 2, Theorem 2.4
  • Corollary 1.5: BousfieldKan1972, Page 314
  • Proposition 1.6: Dugger2001, Proposition 2.3
  • Corollary 1.7
  • proof
  • Proposition 2.1
  • ...and 4 more