Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914305461452800 |
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| author | Albert, Augustin Dubut, Jérémy Goubault, Eric |
| author_facet | Albert, Augustin Dubut, Jérémy Goubault, Eric |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04425 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems Albert, Augustin Dubut, Jérémy Goubault, Eric Category Theory Algebraic Topology 18G90 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. |
| title | Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems |
| topic | Category Theory Algebraic Topology 18G90 |
| url | https://arxiv.org/abs/2602.04425 |