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Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems

Augustin Albert, Jérémy Dubut, Eric Goubault

TL;DR

The paper develops a general homological framework for directed structures by introducing abelian framed bicategories, where each local hom-category is abelian and horizontal composition is additive. It defines chain and cochain complexes inside these local categories, and proves homology and cohomology theories, including long exact sequences for relative pairs and a Mayer–Vietoris sequence, as well as a Künneth theorem under suitable monoidal/closed hypotheses. It also proves embedding theorems showing that module-like abelian framed bicategories embed, in a coherent, pointwise manner, into bimodule categories over End$(U_I)$-algebras, connecting directed homology to classical module theory via Gabriel/Freyd–Mitchell-type results. The framework unifies and extends directed homology notions from precubical settings, enabling robust algebraic tools (Künneth, Eilenberg–Zilber, Mayer–Vietoris) to be applied to directed topology and related directed-space constructions. This has potential to provide new computational and conceptual avenues for directed homology in areas like trace spaces and directed patches of spaces.

Abstract

We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular that similar exact sequences as the relative homology and Mayer-Vietoris long exact sequences can be shown to hold. Also, for closed monoidal abelian framed bicategories, Künneth theorem holds as well. Finally, we prove embedding theorems similar to the Gabriel and Freyd-Mitchell theorems, for particular abelian framed bicategories, allowing to see those as bicategories of bimodules over algebras. This naturally links to the original motivation of this work, which was to generalize directed homology developed in the abelian framed bicategory of bimodules over (path) algebras.

Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems

TL;DR

The paper develops a general homological framework for directed structures by introducing abelian framed bicategories, where each local hom-category is abelian and horizontal composition is additive. It defines chain and cochain complexes inside these local categories, and proves homology and cohomology theories, including long exact sequences for relative pairs and a Mayer–Vietoris sequence, as well as a Künneth theorem under suitable monoidal/closed hypotheses. It also proves embedding theorems showing that module-like abelian framed bicategories embed, in a coherent, pointwise manner, into bimodule categories over End-algebras, connecting directed homology to classical module theory via Gabriel/Freyd–Mitchell-type results. The framework unifies and extends directed homology notions from precubical settings, enabling robust algebraic tools (Künneth, Eilenberg–Zilber, Mayer–Vietoris) to be applied to directed topology and related directed-space constructions. This has potential to provide new computational and conceptual avenues for directed homology in areas like trace spaces and directed patches of spaces.

Abstract

We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular that similar exact sequences as the relative homology and Mayer-Vietoris long exact sequences can be shown to hold. Also, for closed monoidal abelian framed bicategories, Künneth theorem holds as well. Finally, we prove embedding theorems similar to the Gabriel and Freyd-Mitchell theorems, for particular abelian framed bicategories, allowing to see those as bicategories of bimodules over algebras. This naturally links to the original motivation of this work, which was to generalize directed homology developed in the abelian framed bicategory of bimodules over (path) algebras.
Paper Structure (26 sections, 50 theorems, 81 equations, 3 figures)

This paper contains 26 sections, 50 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contents of the paper and main results
  3. Related work
  4. Framed bicategories
  5. Double categories
  6. A framed bicategory
  7. Abelian framed bicategories
  8. Homology and cohomology in abelian framed bicategories
  9. Chain complexes in abelian framed bicategories
  10. Homology
  11. Cohomology
  12. Exact sequences through relative pairs
  13. Relative pairs
  14. Long exact sequences for relative homology
  15. Mayer–Vietoris sequence
  16. ...and 11 more sections

Key Result

Lemma 2.2

For all objects $A$ in a double category $\mathbb{D}$, the two isomorphisms $\mathfrak l_{U_A}, \mathfrak r_{U_A}: U_A \odot U_A \to U_A$ coincide.

Figures (3)

  • Figure 1: Non-cancellative behavior in Fahrenberg's matchbox fahrenberg2004directed. The blue and green directed paths are not dihomotopic because any homotopy would have to go through the upper face and so through undirected paths. However, they become dihomotopic after extending them with the red directed path.
  • Figure 2: Proof of $\beta \circ_T \alpha = m \circ (\alpha \odot \beta) \circ \mathfrak r^{-1}_{U_A}$.
  • Figure :

Theorems & Definitions (157)

  • Definition 2.1: ehresmann1963categories
  • Lemma 2.2
  • Example 2.3
  • Example 2.4: shulman2007framed,ponto2014linearity
  • Example 2.5
  • Definition 2.6: shulman2007framed
  • Definition 2.7
  • Definition 2.8: shulman2007framed
  • Definition 2.9: shulman2007framed
  • Example 2.10
  • ...and 147 more