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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2408.03818 |
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| _version_ | 1866913706396352512 |
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| author | Kaipel, Maximilian |
| author_facet | Kaipel, Maximilian |
| contents | We take a novel lattice-theoretic approach to the $τ$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: \mathfrak{T}(A) \to \mathfrak{T}(A/I)$. If $\mathrm{tors } A$ is finite, $F_I$ is a regular epimorphism in the category of small categories and we characterise when $F_I$ is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the $τ$-cluster morphism category up to equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $τ$-cluster morphism categories of factor algebras Kaipel, Maximilian Representation Theory We take a novel lattice-theoretic approach to the $τ$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: \mathfrak{T}(A) \to \mathfrak{T}(A/I)$. If $\mathrm{tors } A$ is finite, $F_I$ is a regular epimorphism in the category of small categories and we characterise when $F_I$ is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the $τ$-cluster morphism category up to equivalence. |
| title | $τ$-cluster morphism categories of factor algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2408.03818 |