Notes on presheaf representations of strategies and cohomological refinements of $k$-consistency and $k$-equivalence

TL;DR

This paper explores how positional strategies in $k$-pebble games can be represented as presheaves, linking them to sheaf-theoretic models of contextuality and analyzing cohomological $k$-consistency.

Contribution

It introduces a presheaf-based representation of strategies in $k$-pebble games and connects this to sheaf-theoretic models, providing new insights into cohomological $k$-consistency.

Findings

Positional strategies correspond to certain presheaves.

Presheaf representations align with sheaf-theoretic models of contextuality.

Cohomological $k$-consistency can be studied through this framework.

Abstract

In this note, we show how positional strategies for $k$-pebble games have a natural representation as certain presheaves. These representations correspond exactly to the sheaf-theoretic models of contextuality introduced by Abramsky-Brandenburger. We study the notion of cohomological $k$-consistency recently introduced by Adam O' Conghaile from this perspective.