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Structures on the Category of N-Complexes

Felix Küng

Abstract

The theory of $N$-complexes is a generalization of both ordinary chain complexes and graded objects. Hence it yields deeper insight in the structure of these and offers a broader range of applications. This work generalizes the tensor product of chain complexes and graded objects to the case of $N$-complexes using the structures of $q$-binomial coefficients. We then study different approaches to realize the derived category of $N$-complexes. In particular we realize it as the Verdier quotient of the homotopy category of $N$-complexes, as the $\mathrm{h}$-projective objects and as the homotopy category of a category admitting a Quillen model structure.

Structures on the Category of N-Complexes

Abstract

The theory of -complexes is a generalization of both ordinary chain complexes and graded objects. Hence it yields deeper insight in the structure of these and offers a broader range of applications. This work generalizes the tensor product of chain complexes and graded objects to the case of -complexes using the structures of -binomial coefficients. We then study different approaches to realize the derived category of -complexes. In particular we realize it as the Verdier quotient of the homotopy category of -complexes, as the -projective objects and as the homotopy category of a category admitting a Quillen model structure.
Paper Structure (15 sections, 67 theorems, 124 equations)

This paper contains 15 sections, 67 theorems, 124 equations.

Table of Contents

  1. The category $\mathrm{C}_N(\mathcal{A})$
  2. Interlude on combinatorics of $q$-binomial coefficients
  3. Background
  4. Vanishing at roots of unity
  5. Lifting functors
  6. Monoidal structure
  7. Background on distributive monoidal structures
  8. Closed monoidal structure
  9. Frobenius exact structure on $\mathrm{C}_N(\mathcal{A})$
  10. Properties of the homotopy category of $N$-complexes
  11. Two approaches to the derived category of $N$-complexes
  12. Verdier quotient
  13. The category of h-projectives
  14. Semifree resolutions
  15. A model structure on $\mathrm{C}_N(\mathrm{Mod}_R)$

Key Result

Lemma 1.2

The collection of $\infty$-complexes in $\mathcal{C}$ together with morphisms of $\infty$-complexes in $\mathcal{C}$ defines a category, which we will refer to as $\mathrm{C}_\infty (\mathcal{C})$.

Theorems & Definitions (188)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • proof
  • Definition 1.4
  • Definition 1.6
  • Example 1.7
  • Proposition 1.8
  • proof
  • ...and 178 more