With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
The 2-category $Cat$ of categories embeds fully faithfully into the 2-category $Multicat$ of multicategories in two ways:
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 $C_\blacktriangleright$ 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.
The discrete cocones construction was introduced in Example 2.2(2) of:
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.
An analogue of sequential multicategories for (infinity, 1)-categories is contained in §2.4.3 of Higher Algebra.
The sequential multicategory structure was rediscovered in the one-object setting as the “T construction” of:
A generalisation of the concept to generalised multicategories is given in:
Last revised on August 19, 2024 at 11:28:34. See the history of this page for a list of all contributions to it.