Introducing pixelation with applications

Abstract

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 $A$.

Figures (5)

• Figure 1.1: Schematic for the proof of Lemma \ref{['lem:pixels are directed']}.
• Figure 2.1: In the proof of Lemma \ref{['lem:pre-dead 0 morphism trick']}: showing $\gamma$ is $\mathfrak{P}$-equivalent to some $\gamma'$, where $[\gamma']=0$ in ${\overline{\mathcal{A}}}^{\mathfrak{P}}$.
• Figure 2.2: The first schematic used in the proof of Proposition \ref{['prop:Init']}. The red boxes represent pixels in $\mathfrak{P}'$. The blue boxes represent pixels in $\mathfrak{P}$. The labels are the names of the pixels used in the proof of Proposition \ref{['prop:Init']}. Points are labeled and paths are labeled near the arrows indicating their directions.
• Figure 2.3: The second schematic used in the proof of Proposition \ref{['prop:Init']}. The red boxes represent pixels in $\mathfrak{P}'$ and the blue boxes represent pixels in $\mathfrak{P}$. The labels are the labels used in the proof of Proposition \ref{['prop:Init']}. Points are labeled and paths are labeled near the arrows indicating their directions.
• Figure 5.1: On the left, example of a screen that pixelates a finite sum of projective indecomposables. Let $\bar{x}_{(1)}=(0,\frac{1}{6})$, $\bar{x}_{(2)}=(\frac{1}{6},\frac{1}{3})$, $\bar{x}_{(3)}=(\frac{1}{3},\frac{1}{2})$, $\bar{x}_{(4)}=(\frac{1}{2},\frac{2}{3})$, and $\bar{x}_{(5)}=(\frac{2}{3},\frac{5}{6})$. In the figure, on the left, is the screen $\mathfrak{P}=\mathfrak{P}_{\bar{x}_{(1)}}\sqcap\mathfrak{P}_{\bar{x}_{(2)}}\sqcap\mathfrak{P}_{\bar{x}_{(3)}}\sqcap\mathfrak{P}_{\bar{x}_{(4)}}\sqcap\mathfrak{P}_{\bar{x}_{(5)}}$. All the pixels that are not completely shaded are dead pixels. Thus, only the blue pixels are not dead pixels. This means $\text{\m@th$\mathcal{A}^{(2)}_{\mathbb{R}}$}^{\mathfrak{P}}$ is equivalent to the path algebra from the quiver on the right, with the usual mesh relations (the diagonal pseudo arrows exit but are superfluous).

Theorems & Definitions (176)

• Theorem A: Theorem \ref{['thm:KCP is a path category']}
• Theorem B: Theorem \ref{['thm:downward closed subsets in V give abelian categories']} and Corollary \ref{['cor:exact restrictions']}
• Theorem C: Corollary \ref{['cor:pixelated lattice']}
• Theorem D: Theorem \ref{['thm:higher auslander categories']}
• Definition 1.1: $\Gamma$
• Definition 1.2: $\sim$
• Remark 1.3
• Example 1.5: running example
• Definition 1.7: $\mathbf{X}$
• Proposition 1.8