Representation Cohomology of a Small Category
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910157427965952 |
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| author | Klemetti, Markus Levi, Ran Riihimaki, Henri Solch, Daniel |
| author_facet | Klemetti, Markus Levi, Ran Riihimaki, Henri Solch, Daniel |
| contents | Let $C_\bullet$ be a simplicial object in the category $Cat$ of small categories. For a field $k$, taking the Grothendieck groups of isomorphism classes of $kC_n$-modules gives rise to a cochain complex, whose cohomology, which we refer to as representation cohomology, is the object studied in this article. In particular, to any small category $C$, we associate a simplicial object in $Cat$, where for each $n\ge 0$ the objects of the level $n$ category are the simplices of the nerve of $C$. The basic properties of the resulting representation cohomology of these simplicial objects and certain subobjects are then studied in detail. We present some general theoretical computations in favourable cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_20527 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Representation Cohomology of a Small Category Klemetti, Markus Levi, Ran Riihimaki, Henri Solch, Daniel Representation Theory 18G90, 18E99, 18G31, 19G35, 16E20 Let $C_\bullet$ be a simplicial object in the category $Cat$ of small categories. For a field $k$, taking the Grothendieck groups of isomorphism classes of $kC_n$-modules gives rise to a cochain complex, whose cohomology, which we refer to as representation cohomology, is the object studied in this article. In particular, to any small category $C$, we associate a simplicial object in $Cat$, where for each $n\ge 0$ the objects of the level $n$ category are the simplices of the nerve of $C$. The basic properties of the resulting representation cohomology of these simplicial objects and certain subobjects are then studied in detail. We present some general theoretical computations in favourable cases. |
| title | Representation Cohomology of a Small Category |
| topic | Representation Theory 18G90, 18E99, 18G31, 19G35, 16E20 |
| url | https://arxiv.org/abs/2604.20527 |