nLab sequential multicategory



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Category theory



The 2-category Cat Cat of categories embeds fully faithfully into the 2-category MulticatMulticat of multicategories in two ways:

  1. As the unary multicategories (i.e. in which every multimorphism has unary domain).
  2. Via the “discrete cocones” construction, in which we define the multimorphisms of the multicategory C C_\blacktriangleright induced by a category CC by
    C (X 1,,X n;Y):=C(X 1,Y)××C(X n,Y) C_\blacktriangleright(X_1, \ldots, X_n; Y) := C(X_1, Y) \times \cdots \times C(X_n, Y)

A multicategory isomorphic to one induced by the discrete cocones construction is called a sequential multicategory (since its multimorphisms are sequences of unary morphisms).


  • Every sequential multicategory is symmetric.

  • A sequential multicategory C C_\blacktriangleright is representable if and only if CC has coproducts (Example 8.8(2) of Hermida 2000).

  • A category CC is preadditive if and only if its corresponding sequential multicategory C C_\blacktriangleright is cartesian (see Pisani 2013 and Pisani 2014); and is hence additive if and only if its corresponding multicategory is cartesian and representable.

  • The discrete cocones construction () :CatMulticat(-)_\blacktriangleright : Cat \to Multicat is left adjoint to the category of monoids construction Mon:MulticatCatMon : Multicat \to Cat, and this restricts to cartesian multicategories. In fact, this is a consequence of the more general fact that MulticatMulticat is copowered over CatCat: see Pisani 2013 and Pisani 2014.


The discrete cocones construction was introduced in Example 2.2(2) of:

  • Claudio Hermida, Representable multicategories, Adv. Math. 151 (2000), no. 2, 164-225 (pdf)

and studied extensively in:

The discrete cocones construction is also discussed in Example 2.1.6 of Higher Operads, Higher Categories.

The sequential multicategory structure was rediscovered in the one-object setting as the “T construction” of:

  • Samuele Giraudo, Combinatorial operads from monoids, Journal of Algebraic Combinatorics 41 (2015): 493-538.

A generalisation of the concept to generalised multicategories is given in:

  • Alex Cebrian, Plethysms and operads, Collectanea Mathematica 75.1 (2024): 247-303.

Last revised on April 28, 2024 at 14:35:54. See the history of this page for a list of all contributions to it.