nLab n-congruence

relation, internal relation

Rel, bicategory of relations, allegory

Types of Binary relation

• reflexive, irreflexive

• symmetric, antisymmetric asymmetric

• transitive, comparison;

• left and right euclidean;

• total, connected

• extensional, well-founded relations.

• functional relations,

• entire relations,

• equivalence relations, congruence

• apartness relations

• simple graph

In higher category theory

• 2-congruence

• (n,r)-congruence

2-category theory

Definitions

• 2-category

• strict 2-category

• bicategory

• enriched bicategory

Transfors between 2-categories

• 2-functor

  • pseudofunctor

  • lax functor

  • equivalence of 2-categories

• 2-natural transformation

  • lax natural transformation

  • icon

• modification

• Yoneda lemma for bicategories

Morphisms in 2-categories

• fully faithful morphism

• faithful morphism

• conservative morphism

• pseudomonic morphism

• discrete morphism

• eso morphism

Structures in 2-categories

• adjunction

• mate

• monad

• cartesian object

• fibration in a 2-category

• codiscrete cofibration

Limits in 2-categories

• 2-limit

• 2-pullback

• comma object

• inserter

• inverter

• equifier

Structures on 2-categories

• 2-monad

  • lax-idempotent 2-monad

  • pseudomonad

  • pseudoalgebra for a 2-monad

• monoidal 2-category

  • cartesian bicategory

• Gray tensor product

• proarrow equipment

category object in an (∞,1)-category, groupoid object

In a 1-category

• congruence

• internal category, internal groupoid

  weak equivalence of internal categories

• internal functor, internal profunctor

• internal diagram, internal presheaf

• n-fold category, cat-n-group

In a (2,1)-category

• 2-congruence

In ∞Grpd

• semi-Segal space, Segal space, complete Segal space,

• reduced Segal space

In $Cat(Cat(\cdots Cat(\infty Grpd)))$

• higher Segal space

• n-fold complete Segal space

• Theta space

In (n,r)-categories

• (n,r)-congruence

Model category presentations

• model structure for complete Segal spaces

Internal $n$-category

• n-category object in an (∞,1)-category

• Theta_n space

Operadic case

• model structure for dendroidal complete Segal spaces