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

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.

💡 Research Summary

The paper introduces a new categorical framework called abelian framed bicategories, which are framed bicategories whose horizontal hom‑categories D(A,B) are abelian and whose horizontal composition ⊙ is additive. This structure is designed to support homology and cohomology theories for directed (non‑reversible) objects such as directed spaces, trace spaces, and precubical sets.

After recalling the basic notions of double categories and framed bicategories (following Shulman), the authors define chain complexes inside each local abelian hom‑category and describe morphisms between complexes via vertical arrows together with compatible 2‑cells. The homology of an object X is then obtained as the homology of a chosen chain complex M*(X) in D(L(X),R(X)).

The main technical contributions are three families of classical exact‑sequence results, now proved in the setting of abelian framed bicategories:

1. Relative homology long exact sequence (Theorem 5.6). For any relative pair (X,Y) defined within the bicategory, there is a long exact sequence
   … → Hₙ₊₁(X,Y) → Hₙ(X) → Hₙ(Y) → Hₙ(X,Y) → …
   mirroring the classical relative homology sequence. The proof relies on the exactness of restriction and extension functors and the additive nature of ⊙.

2. Mayer–Vietoris long exact sequence (Theorem 5.10). For a “good cover” X = X₁ ∪ X₂ satisfying suitable pullback conditions, the authors construct a long exact sequence that relates the homology of X, X₁, X₂, and their intersection. This result shows that the bicategorical framework can handle gluing constructions in a way analogous to classical algebraic topology.

3. Künneth theorem (Theorem 6.1). In a closed monoidal abelian framed bicategory where the tensor product functor ⊗ satisfies the usual algebraic Künneth hypotheses (flatness, Tor‑vanishing, etc.), a short exact sequence
   0 → Tor₁(H*(X),H*(Y)) → H*(X⊗Y) → H*(X)⊗H*(Y) → 0
   holds. The authors adapt the standard homological algebra proof, using the fact that ⊗ is a bifunctor between abelian hom‑categories and that derived functors can be computed objectwise.

Beyond these homological results, the paper develops an embedding theory for a special class of abelian framed bicategories called module‑like (Definition 7.26). A module‑like bicategory is locally cocomplete, closed, possesses an initial coefficient I, has faithful restriction functors, and U I is a compact projective generator of D(I,I). For any such bicategory A, Theorem 7.47 constructs a framed lax functor
F : A → End(U I)Mod
that is fully faithful and locally an equivalence of categories. This is a bicategorical analogue of Gabriel’s theorem and the Freyd–Mitchell embedding theorem, showing that under the stated hypotheses every module‑like abelian framed bicategory can be realized concretely as a bicategory of bimodules over a ring (or algebra).

The paper illustrates the theory with three families of examples:

• Bimodules over algebras (Example 2.3), the prototypical case where objects are algebras, 1‑cells are bimodules, and 2‑cells are bilinear maps. Horizontal composition is given by the usual tensor product of bimodules.

• V‑distributors (profunctors) (Example 2.4), where V is a bicomplete closed symmetric monoidal category (e.g., Set, R‑Mod). Here objects are small categories, 1‑cells are V‑valued profunctors, and composition is via coends.

• Bimodules over absorption monoids (Example 2.5), motivated by directed homotopy theory. Absorption monoids model trace monoids of directed paths; bimodules over them capture the action of concatenation, absorption, and neutral elements.

These examples demonstrate that the abstract framework indeed captures a wide range of concrete settings previously used for directed homology, such as the homology of precubical sets developed in earlier work.

The authors discuss the significance of their results: by embedding the homological algebra of directed structures into an abelian bicategorical context, they obtain a uniform language for relative homology, Mayer–Vietoris, and Künneth phenomena, while also providing a representation theorem that guarantees a concrete algebraic model (bimodules over a ring) whenever the module‑like conditions hold. This bridges the gap between abstract higher‑categorical constructions and computable algebraic models.

Potential limitations are noted. The “good cover” condition required for Mayer–Vietoris may be restrictive in practice, and the monoidal hypotheses for Künneth are fairly strong (e.g., flatness of the tensor product). Moreover, the module‑like axioms (existence of an initial coefficient, compact projective generator, faithful restrictions) may not be satisfied in many naturally occurring directed settings, limiting the immediate applicability of the embedding theorem.

Future work suggested includes relaxing these hypotheses, developing explicit computational tools for concrete directed systems (e.g., concurrent processes, non‑reversible dynamical systems), and exploring connections with other higher‑categorical homology theories such as derivators or ∞‑categories.

In summary, the paper establishes abelian framed bicategories as a robust setting for directed homological algebra, proves analogues of the fundamental exact‑sequence theorems, and provides a Gabriel‑type embedding into bimodule bicategories, thereby offering both theoretical depth and a pathway to concrete algebraic models.