The walking isomorphism (Hadzihasanovic et al 2024, Ozornova & Rovelli 2024), interval groupoid, interval object in groupoids or free-standing isomorphism, is the groupoid:
with precisely two objects and (besides their identity morphisms) one isomorphism and its inverse morphism connecting them.
This is such that for any category, a functor of the form
is precisely a choice of isomorphism in .
The walking isomorphism can be categorified in a number of ways: to the walking equivalence, to the walking adjoint equivalence, or to the walking 2-isomorphism?. Related, though not quite a categorification in one of the usual senses, is the walking 2-isomorphism with trivial boundary.
The walking isomorphism is the groupoid generated by two objects and and one morphism , which is an isomorphism by definition of a groupoid.
Equivalently, the walking isomorphism is the category generated by two objects and , one arrow , one arrow , and the equations:
which make an isomorphism.
The arrow is an isomorphism, whose inverse is the arrow . Since these are the only arrows in the category, the interval groupoid is a groupoid.
The interval groupoid can also be described as the free groupoid on the interval category.
The interval groupid is an interval object for Cat and Grpd.
The interval groupoid can also be described as the complete graph with two vertices .
Let be a category. Let denote the walking isomorphism. Evaluation at the arrow establishes a natural bijective correspondence between functors and isomorphisms in . Thus, for any isomorphism of there is a unique functor such that the arrow of maps under to .
Immediate from the definitions.
is the full subcategory classifier of the 1-category . That is, for any small category , there is a bijective correspondence between full subcategories of and functors , with the reverse direction given by taking the pullback of the inclusion .
Functors into are uniquely determined by functions on the sets of objects. is a full subcategory of , and any pullback of a full subcategory can be given as a full subcategory.
In fact, this generalizes. If is a simplicial set, say that a full subspace of is a subsimplicial set with the property there is some subset such that contains exactly the simplices whose vertices are all contained in .
The nerve is the full subspace classifier for , and thus represents the subpresheaf of full subspaces.
This can be determined from the explicit description of given by listing the vertices of a path. However, itβs more informative to observe that , where is the indiscrete space functor, which is the direct image part of the geometric embedding whose inverse image is .
This also implies is the full subcategory classifier of , the 1-category of quasi-categories, since those are given by full subspaces of simplicial sets.
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 βwalking isomorphismβ appears in:
Amar Hadzihasanovic, FΓ©lix Loubaton?, Viktoriya Ozornova, Martina Rovelli A model for the coherent walking -equivalence (arXiv:2404.14509)
Viktoriya Ozornova, Martina Rovelli, What is an equivalence in a higher category, Bulletin of the London Mathematical Society, Volume 56, Issue 1, January 2024, Pages 1-58 (doi:10.1112/blms.12947)
A generalization of the notion of the interval groupoid to simplicial groupoids is considered in
and plays a key role in the discussion of the model structure on simplicial groupoids, see there.
Last revised on June 1, 2025 at 03:28:30. See the history of this page for a list of all contributions to it.