monoidal category

Idea

• : a category equipped with some notion of "tensor product" of its objects
• Example: category Vect of vector spaces
• The same category can often be made into a monoidal category in more than one way
  • informal: Set + cartesian product as tensor product
  • informal: Set + disjoint union as tensor product
  • informal: Vect + traditional tensor product as tensor product
  • informal: Vect + direct sum as tensor product
• For any monoidal category \(M\), the operation of tensor product is actually a functor
  • \(M \times M \otimes M \to M\).
  • it makes \(M\) a vertically categorified version of a monoid, which explains the term "monoidal category".

Definition

• A monoidal category is a cateogry \(C\), equipped with
  • a functor: \(\otimes: C \times C \to C\)
    • from the product category \(C \times C\) to \(C\)
    • called the tensor product
  • an object: \(1\in C\)
    • called the unit object or tensor unit
  • a natural isomorphism
    • : \(a:((-)\otimes(-))\otimes(-)\xrightarrow{\simeq}(-)\otimes((-)\otimes(-))\)
    • with component \(a_{x,y,z}{:}(x\otimes y)\otimes z\to x\otimes(y\otimes z)\)
    • called an associator
  • a natural isomorphism
    • \(\lambda:(1\otimes(-))\xrightarrow{\simeq}(-)\)
    • with component \(\lambda_x:1\otimes x\to x\)
    • called a left unitor
  • a natural isomorphism
    • \(\rho:(-)\otimes 1\xrightarrow{\simeq}(-)\)
    • with component \(\rho_x:x\otimes 1\to x\)
    • called a right unitor
• such that the following two kinds of diagrams commute, for all objects involved:
  • 1. triangle identity
  • 2. pentagon identity

Strict monoidal category

• A monoidal category is said to be strict if the associator, left unitor and right unitors are all identity morphisms
• In this case the pentagon identity and the triangle identities hold automatically.

Properties

• Coherence
• Closure