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Cochain complexes over a functor

Germán Benitez, Pedro Rizzo

Abstract

In this paper we propose unifying the categories of cochain complexes $\text{Ch}(\mathcal{C})$ and modules $\widehat{A}\text{-mod}$ over a repetitive algebra $\widehat{A}$. Motivated by their striking similarities and importance, we introduce a novel category encompassing both. Our analysis explores key properties of this unified category, highlighting its parallels and divergences from the original structures. We study whether it preserves crucial aspects like limits, colimits, products, coproducts, and abelianity. Besides, we establish a family of projective and injective indecomposable objects within this framework. Moving beyond theoretical foundations, we examine the influence and interaction over these novel categories of the category of endofunctors and its monoidal structure. Finally, we explore the implications of our constructions over representation theory of algebras and algebraic geometry.

Cochain complexes over a functor

Abstract

In this paper we propose unifying the categories of cochain complexes and modules over a repetitive algebra . Motivated by their striking similarities and importance, we introduce a novel category encompassing both. Our analysis explores key properties of this unified category, highlighting its parallels and divergences from the original structures. We study whether it preserves crucial aspects like limits, colimits, products, coproducts, and abelianity. Besides, we establish a family of projective and injective indecomposable objects within this framework. Moving beyond theoretical foundations, we examine the influence and interaction over these novel categories of the category of endofunctors and its monoidal structure. Finally, we explore the implications of our constructions over representation theory of algebras and algebraic geometry.
Paper Structure (13 sections, 30 theorems, 49 equations)

This paper contains 13 sections, 30 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Roadmap and overview of the results
  3. Cochain complexes over a functor
  4. Some properties of LF
  5. (Co-)Completeness
  6. Projective and injective objects
  7. The (monoidal) category eq:End(C), and the categories eq:LF(C) and eq:RF(C)
  8. The contravariant functors eq:L and eq:R
  9. Monoidal categories and its relations with eq:LFC and eq:RFC
  10. Some applications
  11. (Locally) Finite categories
  12. Generalized Repetitive algebras or Cochain complexes over representable functors
  13. Some geometric applications

Key Result

Theorem 2.5

Let $\mathcal{C}$ be an additive category and let $F:\mathcal{C}\longrightarrow\mathcal{C}$ be a covariant functor. Under the above notations, the categories $\mathfrak{L}_F(\mathcal{C})$ and $\mathfrak{gr}_F(\mathcal{C})$ are isomorphics.

Theorems & Definitions (74)

  • Example 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • ...and 64 more