Monoidal categories

Monoidal categories are categories whose morphisms can have multiple inputs and multiple outputs. They are a fundamental concept in category theory and have become especially popular in applied category theory due to their graphical languages.



  • Baez & Stay, 2010: Physics, topology, logic, and computation: a Rosetta stone (doi, arxiv)
    • Lovely introduction to monoidal categories and string diagrams
    • Describes a speculative graphical language for closed categories (not necessarily compact)
    • Sec 2.3.3, p. 145 has references on decision procedures for the word problem in free monoidal categories
  • Coecke & Paquette, 2010: Categories for the practicing physicist (doi, arxiv)
    • Another nice introduction at a level similar to Baez & Stay
    • Detailed coverage of the category of relations
    • Related papers at this level:
      • Abramsky & Coecke, 2008: Categorical quantum mechanics (arxiv)
      • Abramsky & Coecke, 2004: A categorical semantics of quantum protocols (arxiv)
  • Street, 2012: Monoidal categories in, and linking, geometry and algebra (doi, arxiv)
  • Savage, 2019: String diagrams and categorification (arxiv)
    • Sec 2: Short but clear introduction to strict linear monoidal categories and string diagrams, including the Eckmann-Hilton argument
    • Sec 3: Presenting monoidal categories by generators and relations, mainly by example: group algebras of the symmetric and braid groups, Hecke algebras, and wreath product algebras
    • Sec 4: Autonomous and pivotal monoidal categories and isotopy of string diagrams
    • Sec 5: The Grothendieck ring and trace as two forms of decategorification
    • Sec 6: Final, more technical section on current research directions

Graphical languages: string diagrams

  • Selinger, 2010: A survey of graphical languages for monoidal categories (doi, arxiv)
    • An excellent reference, more encyclopedic than Baez & Stay
  • Mellies, 2006: Functorial boxes in string diagrams (doi, pdf)
  • Bartlett, 2014: Quasistrict symmetric monoidal 2-categories via wire diagrams (arxiv)
  • Miatto, 2019: Graphical calculus for products and convolutions (arxiv)
    • String diagrams for matrix algebra, encompassing the dot, tensor, Kronecker, Hadamard, Kathri-Rao and Tracy-Singh products
  • Kim, Oh, Kim, 2021: Boosting vector calculus with the graphical notation (doi, arxiv)

Graphical languages: proof nets

  • Selinger, 2010, Sec. 9: Beyond a single tensor product

For further references on proof nets, see Selinger’s survey and my page on linear logic.

Type theories

  • Jay, 1989: Languages for monoidal categories (doi, pdf)
  • Shulman, 2019: A practical type theory for symmetric monoidal categories (arxiv, nCat Cafe )

For cartesian monoidal categories, see algebraic theories, and for cartesian closed categories, see lambda calculus and type theory.

Completeness theorems, in the sense of model theory

  • Hasegawa, Hofmann, Plotkin, 2008: Finite dimensional vector space are complete for traced symmetric monoidal categories (doi, pdf)
    • Cf. correspondence between traces and parametrized fixed point operators on cartesian monoidal categories
  • Selinger, 2011: Finite dimensional Hilbert spaces are complete for dagger compact closed categories (doi, arxiv)
  • Kissinger, 2015: Finite matrices are complete for (dagger-)hypergraph categories (arxiv)
  • Bartha, 2017: On the completeness of the traced monoidal category axioms in (Rel,+) (doi, pdf)