Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
category object in an (∞,1)-category, groupoid object
(directed enhancement of homotopy type theory with types behaving like -categories)
The walking commutative triangle or triangle category or 2-simplex category is the category which consists of three objects and three morphisms , , , such that .
is also called the 2-simplex poset or triangle poset because is a thin category and thus a poset.
Every commutative triangle in a category is represented by a functor from the walking commutative triangle to . By definition of a category, we have an equivalence of categories between the functor category of composable pairs in and the functor category of commutative triangles in .
In simplicial type theory, given a directed interval , the 2-simplex type representing the walking commutative triangle is defined as the type of pairs of elements of such that
The walking commutative triangle is equivalent to the endofunctor category of the walking morphism to itself.
the boolean domain ; i.e. the walking pair of objects
the directed interval category ; i.e. the walking morphism
the (2,0)-horn category ; i.e. the walking cospan
the (2,1)-horn category ; i.e. the walking composable pair
the (2,2)-horn category ; i.e. the walking span
the 2-simplex category ; i.e. the walking commutative triangle
Adj; i.e. the walking adjunction
the syntactic category of a theory ; i.e. the walking -model
The phrase “walking commutative triangle” appears in:
The phrase “2-simplex” appears in
Emily Riehl, Mike Shulman, A type theory for synthetic ∞-categories, Higher Structures 1 1 (2017) [arxiv:1705.07442, published article]
Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, Directed univalence in simplicial homotopy type theory (arXiv:2407.09146)
Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, The Yoneda embedding in simplicial type theory (arXiv:2501.13229)
Last revised on June 1, 2025 at 03:28:10. See the history of this page for a list of all contributions to it.