monoidal categories

With symmetry

• 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

• multicategory

Semisimplicity

• semisimple category

• fusion category

• modular tensor category

Morphisms

• monoidal functor

  (lax, oplax, strong bilax, Frobenius)

• braided monoidal functor

• symmetric monoidal functor

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

category theory

Concepts

• category

• functor

• natural transformation

• Cat

Universal constructions

• universal construction

  • representable functor

  • adjoint functor

  • limit/colimit

  • weighted limit

  • end/coend

  • Kan extension

Theorems

• Yoneda lemma

• Isbell duality

• Grothendieck construction

• adjoint functor theorem

• monadicity theorem

• adjoint lifting theorem

• Tannaka duality

• Gabriel-Ulmer duality

• small object argument

• Freyd-Mitchell embedding theorem

• relation between type theory and category theory

Extensions

• sheaf and topos theory

• enriched category theory

• higher category theory

Applications

• applications of (higher) category theory