nLab sequential multicategory

Idea

The 2-category $Cat$ of categories embeds fully faithfully into the 2-category $Multicat$ 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 $C_\blacktriangleright$ induced by a category $C$ by
   $$C_\blacktriangleright(X_1, \ldots, X_n; Y) := C(X_1, Y) \times \cdots \times C(X_n, Y)$$

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

Properties

• Every sequential multicategory is symmetric.

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

• A category $C$ is preadditive if and only if its corresponding sequential multicategory $C_\blacktriangleright$ 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 $(-)_\blacktriangleright : Cat \to Multicat$ is left adjoint to the category of monoids construction $Mon : Multicat \to Cat$, and this restricts to cartesian multicategories. In fact, this is a consequence of the more general fact that $Multicat$ is copowered over $Cat$: see Pisani 2013 and Pisani 2014.

References

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:

• Claudio Pisani, Some remarks on multicategories and additive categories, arXiv:1304.3033 (2013).

• Claudio Pisani, Sequential multicategories, Theory and Applications of Categories 29.19 (2014), arXiv:1402.0253

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.