Operads and multicategories

Operads and colored operads, aka multicategories , are like categories, except that the morphisms can have multiple inputs (and a single output). As one might expect, operads are related to monoidal categories. They are useful for modeling hierarchical composition, such as substitution in wiring diagrams. Meanwhile, dioperads and colored dioperads, better known as polycategories , have multiple inputs and multiple outputs.

Operads and multicategories can be defined in (enriched over) any symmetric monoidal category \((\mathcal{V},\otimes,I)\).

Table 1: Special nomenclature for operads in symmetric monoidal categories
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

General theory

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) and Definitions: Operads, algebras, and modules (doi, pdf)

Books and monographs (MO )

  • Yau, 2016: Colored operads (doi)
    • MAA review helpfully suggests starting at Part II or even Part III
    • Book review by Nick Gurksi (doi)
  • 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)
  • Leinster, 2004: Higher operads, higher categories (doi, arxiv), Ch. 2: Classical operads and multicategories
  • 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 )

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 (arxiv, pdf)
  • 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)

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 )

Other topics in 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 on operads

See computational category theory.