nLab walking commutative triangle

Redirected from "2-simplex category".

Context

Category theory

(,1)(\infty,1)-Category theory

Internal (,1)(\infty,1)-Categories

Directed homotopy type theory

Contents

Definition

Definition

The walking commutative triangle or triangle category or 2-simplex category is the category Δ 2\Delta^2 which consists of three objects 0,1,2Δ 20, 1, 2 \in \Delta^2 and three morphisms h 01:hom Δ 2(0,1)h_{01}:\mathrm{hom}_{\Delta^2}(0, 1), h 02:hom Δ 2(0,2)h_{02}:\mathrm{hom}_{\Delta^2}(0, 2), h 12:hom Δ 2(1,2)h_{12}:\mathrm{hom}_{\Delta^2}(1, 2), such that h 02=h 12h 01h_{02} = h_{12} \circ h_{01}.

Δ 2\Delta^2 is also called the 2-simplex poset or triangle poset because Δ 2\Delta^2 is a thin category and thus a poset.

Every commutative triangle in a category CC is represented by a functor F:Δ 2CF:\Delta^2 \to C from the walking commutative triangle to CC. By definition of a category, we have an equivalence of categories between the functor category C 1 Λ2C^\Lambda_1^2 of composable pairs in CC and the functor category C Δ2C^\Delta^2 of commutative triangles in CC.

In simplicial type theory

In simplicial type theory, given a directed interval 𝕀\mathbb{I}, the 2-simplex type representing the walking commutative triangle is defined as the type of pairs of elements of 𝕀\mathbb{I} such that iji \leq j

Δ 2 i:𝕀 j:𝕀ij\Delta^2 \coloneqq \sum_{i:\mathbb{I}} \sum_{j:\mathbb{I}} i \leq j

Properties

The walking commutative triangle is equivalent to the endofunctor category I II^I of the walking morphism II to itself.

References

The phrase “walking commutative triangle” appears in:

The phrase “2-simplex” appears in

Last revised on June 1, 2025 at 03:28:10. See the history of this page for a list of all contributions to it.