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Monoidal category
Definition
A monoidal category is a category equipped with:
- a bifunctor (Product category and its projection functors, Covariant functor, identity functor, composite functor, and contravariant functor);
- an object of ;
- natural isomorphisms (Natural isomorphism)
such that the following two equations hold for all objects .
The pentagon axiom is
The triangle axiom is
This page imposes no further axiom. In particular, the equality is not built into the definition; it is derived later on this page.
Depends on
Used by
- Lax, strong, and strict monoidal functors Definition
- Monoid objects and comonoid objects in a monoidal category Definition
- Parenthesised tensor words and their evaluation functors Definition
- Strict monoidal category Definition
- The reverse and the opposite of a monoidal category Definition
- How Mac Lane's original coherence conditions reduce to this page's two axioms Remark
- Isbell's warning that isomorphic objects cannot simply be identified Remark
- Mac Lane writes the associator in the opposite direction Remark
- A category with finite products is monoidal Theorem
- Abelian groups are monoidal under the tensor product Theorem
- The endomorphisms of the tensor unit form a commutative monoid Theorem
- The left unitor of a tensor product is determined by the associator Theorem
- The pentagon axiom and the triangle axiom are independent Theorem
- The right unitor of a tensor product is determined by the associator Theorem
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 2.1.1 (standard reference, not scraped)
- S. Mac Lane, Categories for the Working Mathematician, Chapter VII.1 (standard reference, not scraped)