Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2602.04425

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

Augustin Albert·Jérémy Dubut·Eric Goubault
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CTmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Augustin Albert, Jérémy Dubut, Eric Goubault. 2026-02-04. Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems. https://arxiv.org/abs/2602.04425

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The category of nominal sets is locally monoidal closed

The category Nom of nominal sets was proposed by Pitts and Gabbay as a setting for the semantics of abstract syntax with variable bindings. It is well-known that Nom is a topos, also known as the Schanuel topos, and in particular it follows that Nom is locally closed, i.e., every slice category Nom/X is cartesian-closed. In this paper, we show that Nom has a much stronger property: every monoidal (closed) structure on Nom induces a monoidal (closed) structure on all slice categories Nom/X. One monoidal closed structure of particular interest on Nom is the separated product A * B, whose right adjoint A -* B is called the separated function space. In particular, it follows that Nom is locally separatedly closed.

math.CT↗
arXiv2026

Hochschild cohomology of the second kind: Koszul duality and Morita invariance

We define Hochschild cohomology of the second kind for differential graded (dg) or curved algebras as a derived functor in the twisted derived category. The Hochschild cohomology of the second kind of a curved or curved algebra $A$ is then equivalent to the classical Hochschild cohomology of the twisted derived dg category of $A$, which is often geometrically meaningful. Examples include the category of $\infty$-local systems on a topological space, the bounded derived category of a complex manifold and the category of matrix factorizations. We also show that Hochschild cohomology of the second kind is preserved under (nonconilpotent) Koszul duality and weak equivalences of curved algebras. The main technical ingredient is a new bimodule version of Koszul duality.

math.CT↗
arXiv2026

Localization of lax symmetric monoidal categories

In this note, we explain in some detail how one can fiberwise localize a (co)lax symmetric monoidal infinity-category. This construction was tacitly used in Section 5 of our recent paper "On the equivalence of the Lurie's infinity-operads and dendroidal infinity-operads". Version 2: A stronger version of the result is proven. Given a locally cocartesian fibration $f:X\to B$ and a collection of marked arrows $X^\circ\subset f^{-1}(B^{eq})$ in $X$ closed under locally cocartesian liftings, we prove that the localization $\mathcal{L}(X,X^\circ)\to B$ is also a locally cocartesian fibration whose fibers are localizations of the fibers of $f$. This result is applied to the description of localizations of lax symmetric monoidal categories.

math.CT↗