Iterated magnitude homology
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913758200201216 |
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| author | Roff, Emily |
| author_facet | Roff, Emily |
| contents | Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology -- the homology of the classifying space of a small category. The classifying space can also be generalized in a different direction: it extends from categories to bicategories as the geometric realization of the geometric nerve. This paper introduces a hybrid of the two ideas: an iterated magnitude homology theory for categories with a second- or higher-order enrichment. This encompasses, for example, groups equipped with extra structure such as a partial ordering or a bi-invariant metric. In the case of a strict 2-category, iterated magnitude homology recovers the homology of the classifying space; we investigate its content and behaviour when interpreted for partially ordered groups, normed groups, and strict $n$-categories for $n > 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00577 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Iterated magnitude homology Roff, Emily Algebraic Topology Category Theory Group Theory Metric Geometry 18D20, 18N10, 18G90, 20J99, 57T99 Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology -- the homology of the classifying space of a small category. The classifying space can also be generalized in a different direction: it extends from categories to bicategories as the geometric realization of the geometric nerve. This paper introduces a hybrid of the two ideas: an iterated magnitude homology theory for categories with a second- or higher-order enrichment. This encompasses, for example, groups equipped with extra structure such as a partial ordering or a bi-invariant metric. In the case of a strict 2-category, iterated magnitude homology recovers the homology of the classifying space; we investigate its content and behaviour when interpreted for partially ordered groups, normed groups, and strict $n$-categories for $n > 2$. |
| title | Iterated magnitude homology |
| topic | Algebraic Topology Category Theory Group Theory Metric Geometry 18D20, 18N10, 18G90, 20J99, 57T99 |
| url | https://arxiv.org/abs/2309.00577 |