A Gentle Introduction to Algebraic Operads

TL;DR

This paper introduces the theory of algebraic operads, a unifying mathematical framework for various algebraic structures, demonstrating their equivalence to classical algebra categories and exploring their extensions and applications.

Contribution

It provides a comprehensive, self-contained presentation of operads, including classical, partial, and functorial definitions, and proves the correspondence between operads and classical algebraic categories.

Findings

Operads can be fully recovered from categories of representations.

The paper establishes isomorphisms between operad algebras and classical algebra categories.

Operads serve as a higher-level language encoding entire algebraic theories.

Abstract

This text, based on the author's Bachelor's thesis, introduces the theory of Algebraic Operads, a mathematical formalism that provides a unifying framework for modern algebra. We demonstrate how the fundamental theories of associative, commutative, and Lie algebras can be fully recovered as categories of representations of three archetypal operads: $\mathsf{As}$, $\mathsf{Com}$ and $\mathsf{Lie}$ -- the so-called 'three graces' of algebra. Following a deductive and self-contained approach, the notion of an operad is initially presented in its classical form, as a single-object multicategory. Subsequently, alternative definitions -- namely, the partial and functorial definitions -- are provided. This framework allows for the extension of classical algebraic notions, such as free objects and quotients, to the operadic context, thereby enabling operads to be formally presented through generators and relations. The central result of this work is the rigorous proof of the correspondence between operads and algebras. We establish isomorphisms of categories between the algebras over the operads $\mathsf{As}$, $\mathsf{Com}$ and $\mathsf{Lie}$, and their respective classical counterparts. This thesis thus highlights how the theory of operads offers a higher-level language for algebra, encoding entire algebraic theories within single mathematical objects. This formalism not only unifies known structures but also lays the foundation for advanced concepts, such as 'algebras up to homotopy,' with applications in fields like theoretical physics and geometry.