Operads: history

The historical development of the concept of the operad spans over a century, beginning with initial formalizations of universal operations and concluding with modern applications in higher-dimensional category theory and mathematical physics.

Initial Formalizations (1898–1959)

• 1898: A.N. Whitehead describes quadratic operads in the category of vector spaces within his “A Treatise on Universal Algebra”. He defines “invariant equations of condition”, which constitute the relations on free operads generated by binary operations. The fact that Whitehead was implicitly studying operads has been observed by C. Weibel. 🗎
• 1954–1955: Michel Lazard introduces the “analyseur” to axiomatize formal group laws. This structure is functionally equivalent to the modern notion of operad. A “formal group law” is a power series $F(x, y) \in \mathbb Z\llbracket x,y\rrbracket$ that satisfies the axioms of a group structure in a formal neighborhood of the identity: the associativity identity $F(F(x, y), z) = F(x, F(y, z))$ and the unit law $F(x, 0) = x$ and $F(0, y) = y$. 🗎
• Late 1950s: Kolmogoroff and Arnold study the operad of continuous operations $\mathbb{R}^n \to \mathbb{R}$, in connection with Hilbert’s 13th problem. The problem asked to determine that $n$-ary continuous functions ($n \ge 3$) can be represented by the superposition of unary and binary operations, establishing that the operad is generated by its components in arities 1 and 2.

Emergence in Topology and Logic (1963–1970)

• 1963: James Dillon Stasheff introduces $A_\infty$ spaces and the associahedron (Stasheff polytope) to characterize based loop spaces. 🗎
• 1964: Murray Gerstenhaber initiated the deformation theory of associative rings and algebras through the analysis of the Hochschild cochain complex. Gerstenhaber found that the Hochschild cochain complex $C^*(A, A)$ of an associative algebra $A$ inherits a Lie bracket, making the cohomology ring a certain kind of graded Lie algebra. Operadic deformation theory generalizes these results by demonstrating that for an algebra $A$ over an operad $P$, the set of $P$-algebra structures is naturally isomorphic to a subset (of “Maurer–Cartan elements”) in the convolution dg Lie algebra $\mathfrak{g} = Hom_{\mathbb{S}}(P^{\text{!}}, End_A)$. Given a $P$-algebra structure $\phi$, the associated deformation complex is the cochain complex $(Hom_{\mathbb{S}}(P^{\text{!}}, End_A), \partial_{\phi})$, where the cohomology groups are the André–Quillen cohomology of the $P$-algebra. Gerstenhaber found the theorem for $P=E_2$ (the “little 2-diks” operad).
• 1965: Saunders Mac Lane defines “props” (PROducts and Permutation category) to describe structures with multiple inputs and multiple outputs, such as bialgebras. 🗎
• 1966–1967: J. Michael Boardman and Rainer M. Vogt develop “categories of operators in standard form”, utilizing props to study homotopy-invariant algebraic structures on topological spaces. 🗎
• 1969: Joachim Lambek introduces “multicategories” in the context of logic and linguistics, defining what are now termed colored non-symmetric operads.

Formal Definition and Naming (1972–1977)

• 1972: J. Peter May publishes “The Geometry of Iterated Loop Spaces”, providing the formal definition of the “operad”. The term is a portmanteau of “operation” and “monad”. 🗎
• 1972: G. Maxwell Kelly introduces “clubs” to address coherence problems in categories, a structure closely related to operads. 🗎
• 1974: Boardman and Vogt utilize the “little $n$-cubes” operad $C_n$ to characterize $n$-fold loop spaces. 🗎
• 1977: Dennis Sullivan develops infinitesimal computations in topology, utilizing quasi-free commutative algebras which relate to operadic structures.

Link with Enumerative Combinatorics (1981–1986)

• 1981: André Joyal establishes the combinatorial theory of species. It was recognized in 1972 by Max Kelly that operads in the sense of May are monoids in what we now call the category of species, equipped with the “plethystic substitution” monoidal structure. 🗎
• 1986: Joyal identifies that set-theoretic operads are equivalent to analytic monads on the category of sets. He demonstrates that the category of analytic functors and weakly cartesian transformations is equivalent to the category of symmetric sequences. Incidentally, Lawvere theories can be described as particular operads.

The Operadic Renaissance (1990s)

• 1990s: The theory undergoes a “renaissance”, shifting focus from topology toward algebra and mathematical physics. (The theory of algebras up to homotopy defined by operad action is a subject whose time has come. –Peter May) 🗎
• 1994: Victor Ginzburg and Mikhail Kapranov establish the theory of Koszul duality for operads. They define the quadratic dual operad and demonstrate that operads such as $Lie$ and $Com$ are dual to each other. 🗎
• 1994: Ezra Getzler and John D.S. Jones define the Batalin–Vilkovisky operad and its applications to topological conformal field theories. 🗎
• 1995: Getzler and Kapranov introduce cyclic operads, where the output is exchangeable with the inputs. 🗎
• 1998: Getzler and Kapranov define modular operads to include compositions along graphs of any genus. 🗎
• 1999: Maxim Kontsevich proves the formality theorem for Poisson manifolds using $L_\infty$-algebras and operadic structures.

Modern Developments (2000s–Present)

• 2003: Clemens Berger and Ieke Moerdijk establish an axiomatic homotopy theory for operads by defining model category structures on operads in symmetric monoidal categories. 🗎
• 2004: Tom Leinster provides a systematic treatment of generalized operads and higher-dimensional category theory in “Higher Operads, Higher Categories”. 🗎
• 2007: Jacob Lurie develops the theory of $\infty$-operads as functors to the category of finite pointed sets satisfying specific fiber conditions. 🗎
• 2011: Denis-Charles Cisinski and Moerdijk introduce dendroidal sets as a model for $\infty$-operads, generalizing the relationship between simplicial sets and categories. 🗎
• 2017: David Gepner, Rune Haugseng, and Joachim Kock characterize $\infty$-operads as analytic monads on the $\infty$-category of spaces. 🗎
• 2017: Benoit Fresse identifies that the group of rational homotopy automorphism classes of $E_2$-operads is isomorphic to the pro-unipotent Grothendieck–Teichmüller group.