在阿贝尔框架双范畴中的同调代数：精确序列与嵌入定理
Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems
AI总结：
本文提出阿贝尔框架双范畴，用于开发有向结构的同调与上同调理论，并证明了与经典定理类似的嵌入定理，拓展了有向同调的应用范围。
摘要：
我们引入了阿贝尔框架双范畴，这些是局部阿贝尔的特殊框架双范畴，并展示了它们适合开发有向结构的同调和上同调理论。这特别意味着类似于相对同调和Mayer-Vietoris长精确序列的类似精确序列可以被证明成立。此外，对于闭合单子阿贝尔框架双范畴，Künneth定理也成立。最后，我们证明了类似于Gabriel和Freyd-Mitchell定理的嵌入定理，对于特定的阿贝尔框架双范畴，允许将它们视为代数的双模双范畴。这自然地联系到本工作的原始动机，即推广在阿贝尔框架双范畴中开发的有向同调，该双范畴是双模 over (路径) 代数。
英文摘要：
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.
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