Unbiased monoidal categories
Many algebraic definitions are biased : they exist at every finite arity but are explicitly defined only at arities 2 and 0 (binary and nullary operators). In an “unbiased” definition, all finite arities are directly postulated.
Advantages of unbiasedness:
- Helpful in computational category theory: debiasing leads to normal forms
- Common in higher category theory (MO )
- Possibly generalizes to infinitary operations (MO )
Disadvantages:
- Uncommon in most areas of math
- Poorly documented in the literature (MO , Math.SE , nForum )
- Not finitely axiomatized, hence not straightforwardly algebraic theories
Examples
First examples of biased definitions: variations on the monoid structure
- Monoid (classical): binary product and unit
- Monoidal product of a monoidal category: binary product and monoidal unit
- Internal monoid in monoidal category (e.g. coproduct category): merging and creation
- Internal comonoid in monoidal category (e.g. cartesian category): duplication and deletion
Algorithmically, debiasing monoids leads to a list-based normal form, where the associativity and unit axioms are implicit in the representation.
More examples of biased definitions:
- Compositions in a category: binary composition and identity
- Action of braid category on braided monoidal category: braiding and identity
- Action of permutation category on symmetric monoidal category: braiding and identity
The last two formulations are sloppy: I mean the “permutation-like maps” formed by taking arbitrary compositions and products of braidings and identities, generalizing the transposition decomposition of a permutation.
Literature
- Deligne and Milne, 1982: Tannakian Categories, Proposition 1.5 (pdf)
- Leinster, 2004: Operads in higher-dimensional category theory, Sec 1.2: Unbiased bicategories (pdf, arxiv)
- Leinster, 2004: Higher Operads, Higher Categories, Sec 3.1: Unbiased monoidal categories and Appendix A: Symmetric structures (arxiv)
- Brandenburg, 2011 (unpublished notes): Unbiased symmetric monoidal categories (Math.SE , pdf)