nLab walking isomorphism

Redirected from "free-standing isomorphism".

Contents

Idea

The walking isomorphism (Hadzihasanovic et al 2024, Ozornova & Rovelli 2024), interval groupoid, interval object in groupoids or free-standing isomorphism, is the groupoid:

I βˆΌβ‰”{aβ‡†βˆΌb} I_\sim \;\coloneqq\; \big\{ a \overset{\;\; \sim \;\;}{\leftrightarrows} b \big\}

with precisely two objects and (besides their identity morphisms) one isomorphism and its inverse morphism connecting them.

This is such that for π’ž\mathcal{C} any category, a functor of the form

I βˆΌβŸΆπ’ž I_\sim \longrightarrow \mathcal{C}

is precisely a choice of isomorphism in π’ž\mathcal{C}.

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.

Definition

Definition

The walking isomorphism is the groupoid generated by two objects 00 and 11 and one morphism 0β‰…10 \cong 1, which is an isomorphism by definition of a groupoid.

Definition

Equivalently, the walking isomorphism is the category generated by two objects 00 and 11, one arrow 0β†’10 \rightarrow 1, one arrow 1β†’01 \rightarrow 0, and the equations:

0β†’1β†’0=id 0,1β†’0β†’1=id 1, 0 \to 1 \to 0 = \mathrm{id}_0, \quad 1 \to 0 \to 1 = \mathrm{id}_1,

which make 0β†’10 \to 1 an isomorphism.

The arrow 0β†’10 \rightarrow 1 is an isomorphism, whose inverse is the arrow 1β†’01 \rightarrow 0. Since these are the only arrows in the category, the interval groupoid is a groupoid.

Remark

The interval groupoid can also be described as the free groupoid on the interval category.

Remark

The interval groupid is an interval object for Cat and Grpd.

Remark

The interval groupoid can also be described as the complete graph with two vertices K 2K_2.

Representing of isomorphisms

Proposition

Let π’œ\mathcal{A} be a category. Let ℐ\mathcal{I} denote the walking isomorphism. Evaluation at the arrow 0β†’10\to 1 establishes a natural bijective correspondence between functors β„β†’π’œ\mathcal{I}\to\mathcal{A} and isomorphisms in π’œ\mathcal{A}. Thus, for any isomorphism ff of π’œ\mathcal{A} there is a unique functor F:β„β†’π’œF: \mathcal{I} \rightarrow \mathcal{A} such that the arrow 0β†’10 \rightarrow 1 of ℐ\mathcal{I} maps under FF to ff.

Proof

Immediate from the definitions.

Properties

Proposition

{1}βŠ†β„\{ 1 \} \subseteq \mathcal{I} is the full subcategory classifier of the 1-category CatCat. That is, for any small category π’œ\mathcal{A}, there is a bijective correspondence between full subcategories of π’œ\mathcal{A} and functors π’œβ†’β„\mathcal{A} \to \mathcal{I}, with the reverse direction given by taking the pullback of the inclusion {1}βŠ†β„\{ 1 \} \subseteq \mathcal{I}.

Proof

Functors into ℐ\mathcal{I} are uniquely determined by functions on the sets of objects. {1}\{ 1 \} is a full subcategory of ℐ\mathcal{I}, and any pullback of a full subcategory can be given as a full subcategory.

In fact, this generalizes. If XX is a simplicial set, say that a full subspace of XX is a subsimplicial set SβŠ†XS \subseteq X with the property there is some subset S 0βŠ†X 0S_0 \subseteq X_0 such that SS contains exactly the simplices whose vertices are all contained in S 0S_0.

Proposition

The nerve N(ℐ)N(\mathcal{I}) is the full subspace classifier for sSetsSet, and thus N(ℐ)N(\mathcal{I}) represents the subpresheaf FullSubβŠ†SubFullSub \subseteq Sub of full subspaces.

Proof

This can be determined from the explicit description of N n(ℐ)β‰…{0,1} n+1N_n(\mathcal{I}) \cong \{ 0, 1 \}^{n+1} given by listing the vertices of a path. However, it’s more informative to observe that N(ℐ)=indisc({0,1})N(\mathcal{I}) = indisc(\{ 0, 1 \}), where indiscindisc is the indiscrete space functor, which is the direct image part of the geometric embedding SetβŠ†sSetSet \subseteq sSet whose inverse image is X↦X 0X \mapsto X_0.

Remark

This also implies N(ℐ)N(\mathcal{I}) is the full subcategory classifier of qCatqCat, the 1-category of quasi-categories, since those are given by full subspaces of simplicial sets.

References

The phrase β€œwalking isomorphism” appears in:

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.