Monoids are one of the most fundamental building blocks of mathematics and therefore also of physics and nature. They are everywhere! A monoid is a collection of objects together with an operation that fuses two of those objects into a new object. Mathematicians often express this with the following diagram.

You already know two monoidal structures: The addition or multiplication of numbers. In both operations, two numbers are combined to form a new number.

But what is a comonoid?

A comonoid is obtained by turning a monoid upside down, literally. Or in other words, if you have an operation that does not make one out of two objects, but divides one object into two new ones.

This may seem somewhat counterintuitive at first glance, but especially in quantum physics and computer science you can find many comonoids. In particular, they play a major role in quantum computing.