Multitensors

In category theory, a multitensor on a category V is a structure that generalizes the notion of a tensor product to operations of arbitrary finite arity. It consists of a family of functors E_n: V^n → V for each natural number n ≥ 0, together with a unit natural transformation i: Id_V → E_1 and a collection of substitution maps

σ: E_k(E_{n1}(X11,...,X1n1),...,E_{nk}(Xk1,...,Xknk)) → E_{n1+...+nk}(X11,...,Xknk),

natural in all variables. These data satisfy coherence axioms that extend the associativity and unit laws familiar from monoidal structures.

A multitensor is a generalization of a monoidal structure: a monoidal category (V, ⊗, I) induces a multitensor by setting E_n(X1,...,Xn) = X1 ⊗ ... ⊗ Xn, with the unit and associativity constraints provided by the monoidal coherence. Conversely, a multitensor need not come from a single binary product; it encodes operations of varying arity and their compositions, abstractly capturing “how to combine” any finite family of objects to produce another object.

Examples and variants include:

- Finite products in a category with products, where E_n(X1,...,Xn) = X1 × ... × Xn, with the terminal object serving as E_0 and canonical projections giving substitution maps.

- Symmetric multitensors, where each E_n carries an action of the symmetric group on n and the substitution maps respect these actions.

Algebras for a multitensor (E-algebras) generalize monoid-like structures and recover ordinary monoids in a monoidal category as a special case. Multitensors appear in the study of operads, multicategories, and enriched higher-category theory, providing a flexible framework for multi-input operations and their compositions.