How Properads Generalize Operads and Props

This article provides a clear overview of the mathematical relationship between properads, operads, and props. It explains how properads serve as a broader framework that extends the capabilities of operads and incorporates key features of props. Readers will learn the basic definitions of each structure and understand how properads unify these theories to handle complex algebraic operations involving multiple inputs and outputs.

What Are Operads?

To understand properads, it helps to start with operads. In simple terms, an operad is a mathematical structure that describes operations with many inputs but only one output. You can think of this like a function in computer programming or algebra. For example, adding three numbers together takes three inputs and produces one sum. Operads are excellent for studying processes where many things combine to create a single result. However, they are limited because they cannot naturally describe operations that produce multiple results at once.

What Are Props?

Props, which stands for Products and Permutations, are a step up in complexity from operads. A prop describes operations that can have many inputs and many outputs. Imagine a machine with several wires going in and several wires coming out. Props allow mathematicians to study these multi-output systems and how they connect. While props are very powerful, their structure can be quite complex because they allow for disconnected networks of operations. This means some parts of the system might not interact directly with others, which can make calculations difficult.

The Properad Framework

Properads sit between operads and props in terms of structure. The theory of properads generalizes operads by removing the restriction of having only one output. Like props, properads allow for many inputs and many outputs. However, properads impose a specific rule about connectivity. In a properad, the graphs representing the operations must be connected. This means every input and output must be part of a single, unified network without isolated pieces. This requirement simplifies the theory while keeping the flexibility of multiple outputs.

Unifying the Theories

The theory of properads generalizes the theory of operads and props by providing a unified language for algebraic structures. It generalizes operads by expanding them to handle multiple outputs, which operads cannot do. It relates to props by focusing on connected graphs, which captures the essential interactions found in prop theory without the complexity of disconnected components. By using properads, mathematicians can study a wider range of operations than operads allow, while maintaining a more streamlined structure than full props. This makes properads a powerful tool for understanding complex systems in algebraic topology and category theory.