Multicategories
Multicategories are like categories, except that the morphisms can have multiple inputs (and a single output). A symmetric multicategory with one object is better known as an operad.
Multicategories are environment in which the coherence data of monoidal categories acquire a universal property. The representable multicategories are equivalent to monoidal categories (Hermida 2000). In the literature, multicategories have been sadly neglected compared to monoidal categories.
The combinatorics of free multicategories, involving planar trees, are considerably more complicated than the combinatorics of categories. See the page on trees for more.
Literature
Books and surveys
- Leinster, 2004: Higher operads, higher categories (doi, arxiv), Ch. 2: Classical operads and multicategories
- Shulman, 2016, lecture notes: Categorical logic from a categorical point of
view (GitHub , pdf)
- Although mainly about categorical logic, multicategories play a big role
- Chapter 2 has a good introduction to bare, symmetric, and cartesian multicategories
General theory
- Hermida, 2000: Representable multicategories (doi)
- Standard reference for the important fact that representable multicategories are equivalent to monoidal categories
- Manzyuk, 2009: Closed categories vs. closed multicategories (arxiv)
Symmetric multicategories admit a tensor product, the Boardman-Vogt tensor product , making them into a symmetric closed monoidal category. Note that the symmetry is crucial for this construction!
- Boardman & Vogt, 1973: Homotopy invariant algebraic structures on topological spaces (doi)
- Elmendorf & Mandell, 2009: Permutative categories, multicategories and
algebraic K-theory (doi)
- Definition 2.2: clear description of closed structure on symmetric multicategories, which is more elementary than the tensor product
- Tronin, 2011: Natural multitransformations of multifunctors (doi)
- Tronin, 2016: On algebras over multicategories (doi)
- Seems to rediscover the internal hom for symmetric multicategories
- Pisani, 2014: Sequential multicategories (pdf, arxiv)
- Section 3: The monoidal closed structure of the category of symmetric multicategories
Fibrations of multicategories
The literature seems to include several notions of a fibration of multicategories .
- Hermida, 2004: Fibrations for abstract multicategories (pdf)
- First paper on the topic, unfortunately closer to an extended abstract
- Title notwithstanding, only defines opfibrations of multicategories, which are called “covariant fibrations”
- Hörmann, 2017: Fibered multiderivators and (co)homological descent (pdf,
arxiv)
- Fibrations and opfibrations of multicategories in Appendix A
- See also (Hörmann 2018) below
- Licata, Shulman, Riley, 2017: A fibrational framework for substructural and
modal logics (doi, extended )
- Fibrations and opfibrations of cartesian (2-)multicategories in Section 5
- Blanco & Zeilberger, 2020: Bifibrations of polycategories and classical linear
logic (doi)
- The nice Introduction points out that the notion of “contravariant fibration (and of bifibration) of multicategories was not addressed in (Hermida 2004)” but was given later by (Hörmann 2017) and (Licata, Shulman, Riley 2017)
Higher-dimensional multicategories