nLab monoidal categories - contents

monoidal categories

• enriched monoidal category, tensor category

• string diagram, tensor network

With braiding

• braided monoidal category

• balanced monoidal category

  • twist

• symmetric monoidal category

With duals for objects

• category with duals (list of them)

• dualizable object (what they have)

• rigid monoidal category, a.k.a. autonomous category

• pivotal category

• spherical category

• ribbon category, a.k.a. tortile category

• compact closed category

With duals for morphisms

• monoidal dagger-category?

• symmetric monoidal dagger-category

• dagger compact category

With traces

• trace

• traced monoidal category

Closed structure

• closed monoidal category

• cartesian closed category

• closed category

• star-autonomous category

Special sorts of products

• cartesian monoidal category

• semicartesian monoidal category

• monoidal category with diagonals

• multicategory

Semisimplicity

• semisimple category

• fusion category

• modular tensor category

Morphisms

• monoidal functor

  (lax, oplax, strong bilax, Frobenius)

• braided monoidal functor

• symmetric monoidal functor

Internal monoids

• monoid in a monoidal category

• commutative monoid in a symmetric monoidal category

• module over a monoid

Examples

• tensor product

• closed monoidal structure on presheaves

• Day convolution

Theorems

• coherence theorem for monoidal categories

• monoidal Dold-Kan correspondence

In higher category theory

• monoidal 2-category

  • braided monoidal 2-category

• monoidal bicategory

  • cartesian bicategory

• k-tuply monoidal n-category

  • little cubes operad

• monoidal (∞,1)-category

  • symmetric monoidal (∞,1)-category

• compact double category