Introducing pixelation with applications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912983758667776 |
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| author | Rock, J. Daisie |
| author_facet | Rock, J. Daisie |
| contents | Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25432 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Introducing pixelation with applications Rock, J. Daisie Category Theory Algebraic Geometry Representation Theory Primary: 18E35, 18A25, 18F20. Secondary: 18G15, 16G20, 16G99 Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$. |
| title | Introducing pixelation with applications |
| topic | Category Theory Algebraic Geometry Representation Theory Primary: 18E35, 18A25, 18F20. Secondary: 18G15, 16G20, 16G99 |
| url | https://arxiv.org/abs/2603.25432 |