Monoidal categories

Builds on

Required prerequisites, all the way down: 20

You will learn to

• Verify that (Vect_k, ⊗, k), (Set, ×, 1) and (Set, ⊔, ∅) are symmetric monoidal categories.
• State Mac Lane's coherence theorem and compute with string diagrams.
• Define monoid objects and identify them in examples: monoids, k-algebras, and monads as monoids in a category of endofunctors.

Definition

• A monoidal category is a category with a functor ⊗ : C × C → C, a unit object I, and natural isomorphisms expressing associativity and unit laws that satisfy the pentagon and triangle axioms.
• A braiding is a natural isomorphism A ⊗ B ≅ B ⊗ A satisfying the hexagon axioms; it is symmetric if braiding twice gives the identity.
• A monoid object is an object M with morphisms μ : M ⊗ M → M and η : I → M satisfying associativity and unit laws.

Theorem

• Mac Lane's coherence theorem: every monoidal category is monoidally equivalent to a strict one, and all formal diagrams built from associators and unitors commute.
• Monoid objects in (Ab, ⊗_ℤ, ℤ) are rings, and in (Vect_k, ⊗, k) they are k-algebras.
• Monads on C are monoid objects in the monoidal category of endofunctors of C under composition.

Common misconceptions

• Assuming every braided monoidal category is symmetric; categories of representations of quantum groups are braided but not symmetric.
• Thinking the monoidal product must be a cartesian product.

Leads to

Optional discoveries

• Mathematical foundations of quantum mechanics Symmetric monoidal categories with duals, such as finite-dimensional Hilbert spaces, are the setting of categorical quantum mechanics (Abramsky and Coecke).
• ← Advanced linear algebra

Further reading

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories (2015)
• Saunders Mac Lane, Categories for the Working Mathematician (1971)