Homotopy Type Theory category theory > history (Rev #28, changes)

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Here we collect articles about doing category theory in HoTT. This is based off of the HoTT Book.

Definitions

Morphisms

• isomorphism in a precategory
• epimorphism in a precategory?
• monomorphism in a precategory?

Categories

• precategory
• category
• equivalence of precategories
• weak equivalence of precategories
• isomorphism of precategories?
• opposite precategory
• product precategory

Functors

• functor
• natural transformation
• functor precategory
• left adjoint
• faithful functor
• full functor
• split essentially surjective
• essentially surjective
• hom functor
• representable functor

Limits and colimits

• product?
• coproduct?
• equaliser?
• directed colimit?

Dagger morphisms

• unitary isomorphism in a dagger precategory
• dagger epimorphism in a dagger precategory
• dagger monomorphism in a dagger precategory

Dagger categories

• dagger precategory
• dagger category
• monoidal dagger category
• braided monoidal dagger category
• symmetric monoidal dagger category
• cartesian monoidal dagger category
• cocartesian monoidal dagger category
• semiadditive dagger category
• finitely complete dagger category?
• finitely cocomplete dagger category?
• compact closed dagger category
• HilbR

Order theory

• preorder
• directed type
• net
• poset
• join-semilattice
• meet-semilattice
• omega-complete poset
• partial function classifier
• Sierpinski space
• lattice
• distributive lattice
• sigma-complete lattice
• sigma-frame
• frame
• Heyting algebra
• Boolean algebra?
• monotonic function
• monotonic function preorder?
• sigma-continuous function?
• opposite preorder
• antitonic function
• posite?
• antitonic predicate?
• monotonic predicate?
• ideal?
• filter
• order
• strict order
• dense strict order
• open interval
• closed interval
• Dedekind cut

Higher categories

• 2-preorder
• 2-poset
• (2,1)-precategory?
• (2,1)-category?
• (2,2)-precategory?
• (2,2)-category?
• 3-preorder?
• 3-poset?

Theorems

• Rezk completion
• Yoneda lemma

References

• HoTT Book
• Univalent categories and the Rezk completion