# The fundamental group of binoid varieties

## TL;DR

This paper computes the topological fundamental group of binoid schemes over the complex and real numbers, introducing a 2-category theory approach and simplifying the process for Stanley Reisner Rings.

## Contribution

It introduces a method to calculate the fundamental group of binoid schemes, including explicit computations over the reals and complexes, and simplifies the process for Stanley Reisner Rings.

## Key findings

- Calculated the fundamental group of binoid schemes over b6 and b7.
- Provided an explicit method using 2-category theory for the real case.
- Simplified the fundamental group computation for Stanley Reisner Rings.

## Abstract

Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let $X$ be a binoid scheme. The aim of this paper is to calculate the topological fundamental group of $KX$, where $K=\mathbb{C}$ or $\mathbb{R}$. For the latter, we will give an explicit way of calculating the fundamental group using methods from 2-category theory. Indeed, we will calculate the more general fundamental groupoid. As a specialisation, we will also look at the Stanley Reisner Rings. Our method simplifies in this case, allowing us to describe the fundamental groupoid in terms of the simplicial complex directly.