A Gentle Introduction to Algebraic Operads
Felicia Ferraioli

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: , and -- 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
