# Effect algebras as presheaves on finite Boolean algebras

## TL;DR

This paper explores the categorical structure of effect algebras via presheaves on finite Boolean algebras, characterizing properties and tensor products through categorical and functorial methods.

## Contribution

It introduces a presheaf-based framework for effect algebras, characterizes key properties categorically, and describes tensor products using Kan extensions and Day convolution.

## Key findings

- Properties like being an orthoalgebra are characterized categorically.
- Tensor products of effect algebras are described as Kan extensions.
- Tensor product can be expressed via Day convolution of presheaves.

## Abstract

For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra $A$ can be characterized by properties of the category of elements of the presheaf $R(A)$. We prove that the tensor product of of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. As a consequence, the tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.