nLab notions of walking structure

• walking structure

  • the boolean domain $2 \coloneqq \partial \Delta^1$; i.e. the walking pair of objects

  • the directed interval category $I \coloneqq \Delta^1$; i.e. the walking morphism

  • the (2,0)-horn category $\Lambda_0^2$; i.e. the walking cospan

  • the (2,1)-horn category $\Lambda_1^2$; i.e. the walking composable pair

  • the (2,2)-horn category $\Lambda_2^2$; i.e. the walking span

  • the 2-simplex category $\Delta^2$; i.e. the walking commutative triangle

  • the walking parallel pair

  • the walking isomorphism

  • Adj; i.e. the walking adjunction

  • the walking equivalence

  • the walking adjoint equivalence

  • the walking 2-isomorphism with trivial boundary

  • the syntactic category of a theory $T$; i.e. the walking $T$-model

• In dependent type theory, many higher inductive types can be considered to be walking structures:

  • the unit type is the walking element

  • the boolean type is the walking pair of elements

  • the interval type is the walking identification

  • one definition of the circle type is the walking self-identification

  • another definition of the circle type is the walking parallel pair of identifications.

  • the suspension type of a type $I$ is the walking parallel $I$-indexed family of identifications.

  • the natural numbers type is the walking sequence

  • the integers type is the walking bi-infinite sequence

  • the walking bridge

  • the directed interval type is the walking morphism in the sense of element of a hom type