Operads

An operad is a symmetric multicategory with one object.

On the other hand, a multicategory could be defined as a colored/typed non-symmetric operad, so the terminology is redundant. (I am assuming that operads are symmetric by default and multicategories are not, a common but not universal convention. Non-symmetric operads are also called planar operads.)

The words “operad” and “multicategory”, however, evoke different ideas. In topology and applied category theory, operads often model hierarchical composition, such as substitution of little disks or wiring diagrams. Multicategories model maps having definite arities. The prototypical example is the multicategory of vector spaces and multilinear maps. For this reason, I have found it convenient to separate the pages for operads and multicategories.

The analogues of operads and multicategories that have multiple inputs and multiple outputs are dioperads and polycategories .

Operads can be enriched over any symmetric monoidal category \((\mathcal{V},\otimes,I)\). In fact, topological operads were the first kind to be defined.

Table 1: Special nomenclature for enriched operads
Base category \(\mathcal{V}\) Operads in \(\mathcal{V}\)
\((\mathbf{Set},\times,1)\) (set-theoretic) operads
\((\mathbf{Top},\times,1)\) topological operads
\((\mathbf{sSet},\times,1)\) simplicial operads
\((\mathbf{Vect}_k,\otimes,k)\) algebraic operads, aka linear operads
\((\mathbf{Mod}_R,\otimes,R)\) operads, for some authors

Expository literature

Blog posts

  • Tai-Danae Bradley, 2017: What is an operad? (1 ,2 )
  • Tim Hosgood, 2017-18: Loop spaces, spectra, and operads (1 ,2 ,3 )

Introductions and surveys

  • Fong & Spivak, 2019: An invitation to applied category theory, Ch. 6: Electric circuits: Hypergraph categories and operads (doi)
  • Vallette, 2014: Algebra + homotopy = operad (pdf, arxiv)
  • Markl, 2008: Operads and PROPS (doi, arxiv), in Hazewinkel, ed., Handbook of Algebra, Vol. 5
  • Stasheff, 2004, in AMS Notices: What is an operad? (pdf)
  • May, 1997: Operads, algebras, and modules (doi, pdf)
    • May, 1997: Definitions: Operads, algebras, and modules (doi, pdf)

Books and monographs (MO )

  • Yau, 2016: Colored operads (doi)
    • Book review by Nick Gurksi (doi)
    • MAA review helpfully suggests starting at Part II or even Part III
  • Mendez, 2015: Set operads in combinatorics and computer science (doi)
  • Loday & Vallette, 2012: Algebraic operads (doi, pdf)
  • Moerdijk, 2010: Lectures on dendroidal sets, Lecture 1: Operads & Lecture 2: Trees as operads
    • Part I in: Moerdijk & Toën, 2010: Simplicial methods for operads and algebraic geometry (doi)
  • Markl, Shnider, Stasheff, 2002: Operads in algebra, topology, and physics (doi)
    • Short book review by John Baez (doi, pdf)
  • Kříž & May, 1995: Operads, algebras, modules and motives (pdf, alternate )

Examples and constructions

Operads of wiring diagrams

Spivak et al on undirected wiring diagrams:

  • Spivak, 2013: The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits (arxiv)
  • Spivak, 2014: Category Theory for the Sciences, Sec. 5.4: Operads, especially Sec. 5.4.2.4: Wiring diagrams

Spivak et al on directed wiring diagrams:

  • Spivak & Rupel, 2013: The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes (arxiv)
  • Vagner, Spivak, Lerman, 2015: Algebras of open dynamical systems on the operad of wiring diagrams (pdf, arxiv)
  • Spivak, 2015: The steady states of coupled dynamical systems compose according to matrix arithmetic (arxiv)
  • Spivak et al, 2016: String diagrams for traced and compact categories are oriented 1-cobordisms (doi, arxiv)
  • Spivak & Tan, 2016: Nesting of dynamical systems and mode-dependent networks (doi, arxiv)
  • Patterson, Spivak, Vagner, 2020: Wiring diagrams as normal forms for computing in symmetric monoidal categories (arxiv)

Other works on operads of wiring diagrams:

  • Yau, 2018: Operads of wiring diagrams (doi, arxiv)
    • Comprehensive and largely self-contained monograph
    • Main objective is to give finite presentations of operads of wiring diagrams (directed and undirected) and also of their algebras
    • Ch. 2 is a readable introduction to operads and wiring diagrams

Operads of combinations

Linear combinations, affine combinations, conical combinations, convex combinations, and other kinds of combinations, involving different coefficients, all correspond to operads. This viewpoint is standard enough to feature in the Wikipedia article on operads, yet I have not been able to find many references in the primary literature.

A construction generating an operad from a monoid, which accounts for linear and conical combinations through the multiplicative monoids \(\mathbb{R}\) and \(\mathbb{R}_+\), appears with a different motivation in the combinatorics research of Samuele Giraudo.

  • Giraudo, 2015: Combinatorial operads from monoids (doi, arxiv)
    • Early version: Giraudo, 2012: Constructing combinatorial operads from monoids (pdf, arxiv)
  • Giraudo, 2017, habilitation thesis: Operads in algebraic combinatorics (arxiv), Ch. 4: From monoids to operads
  • Giraudo, 2018: Nonsymmetric operads in combinatorics (doi)

This construction is mentioned independently by Tom Leinster in a curious line of work connecting entropy and operads. The operad of simplices, corresponding to convex combinations, also appears and plays a larger role.

  • Leinster, 2011, blog post: An operadic introduction to entropy (nCat Cafe )
  • Leinster, 2011: notes: An operadic characterization of Shannon entropy (pdf, nLab , nCat Cafe )
  • Leinster, 2017, talk: The categorical origins of entropy (video, slides, alternative )

Theory of operads

Operads as monoids

Like so many other structures, operads can be defined as monoid objects in a suitable monoidal category, namely the category of species. This was first observed by Kelly and is emphasized in the book (Mendez, 2015) cited above. Other references:

  • Kelly, 1972: On the operads of J.P. May (pdf)
  • Markl, Shnider, Stasheff, 2002: Operads in algebra, topology, and physics, Sec. II.1.8: Operads as monoids
  • Obradovic, 2017: Monoid-like definitions of cyclic operad (pdf, arxiv)
  • Foissy et al, 2020: Families of algebraic structures (arxiv), Sec. 3: Reminders on operads and colored operads in the species formalism

Computing with operads*

See section on operads at computational category theory.