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Homological algebra over non-unital rings and algebras, with applications to $(\infty, 1)$-categories

Eric Goubault, Eliot Médioni

Abstract

The article is developing homological algebra in modules over non-unital rings and algebras. The main application is the definition and study of (directed) homology of $(\infty,1)$-categories and of directed spaces, including relative homology and its exact sequence.

Homological algebra over non-unital rings and algebras, with applications to $(\infty, 1)$-categories

Abstract

The article is developing homological algebra in modules over non-unital rings and algebras. The main application is the definition and study of (directed) homology of -categories and of directed spaces, including relative homology and its exact sequence.
Paper Structure (32 sections, 27 theorems, 124 equations)

This paper contains 32 sections, 27 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Main results
  4. Background on modules and (nonunital) algebras
  5. Nonunital algebras
  6. Modules over (nonunital) algebras
  7. Unitalization of algebras and modules, and their properties
  8. On categories of closed, firm and s-unital modules
  9. Nonunital tensor product
  10. Categories of firm and closed modules
  11. Extension of scalar for firm modules
  12. Categories of $\mathbf{s}$-unital modules
  13. A convenient abelian category of bimodules
  14. Application: directed homology of $(\infty,1)$-categories
  15. Defining the algebra of paths
  16. ...and 17 more sections

Key Result

Proposition 1

Theorems & Definitions (94)

  • Definition 1: Rings and rings without unit
  • Definition 2: Algebras and unital algebras
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • Remark 4
  • Definition 3: Unitalization
  • Definition 4
  • Remark 5
  • ...and 84 more