nLab walking isomorphism

Idea

The interval object in groupoids or free-standing isomorphism or, if you insist, the “walking isomorphism”, is the groupoid:

$$I_\sim \;\coloneqq\; \big\{ a \overset{\;\; \sim \;\;}{\leftrightarrows} b \big\}$$

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

This is such that for $\mathcal{C}$ any category, a functor of the form

$$I_\sim \longrightarrow \mathcal{C}$$

is precisely a choice of isomorphism in $\mathcal{C}$.

The interval groupoid 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 free-standing isomorphism is the category generated by two objects $0$ and $1$, one arrow $0 \rightarrow 1$, one arrow $1 \rightarrow 0$, and the equations:

$$0 \to 1 \to 0 = \mathrm{id}_0, \quad 1 \to 0 \to 1 = \mathrm{id}_1,$$

which make $0 \to 1$ an isomorphism.

Remark

The arrow $0 \rightarrow 1$ is an isomorphism, whose inverse is the arrow $1 \rightarrow 0$.

Remark

The free-standing isomorphism is a groupoid.

Remark

The free-standing isomorphism can also be described as the free groupoid on the interval category, that is to say, the walking arrow. Because it is an interval object for Cat and Grpd, it is also known as the interval groupoid.

Representing of isomorphisms

Proposition

Let $\mathcal{A}$ be a category. Let $\mathcal{I}$ denote the free-standing isomorphism. Evaluation at the arrow $0\to 1$ establishes a natural bijective correspondence between functors $\mathcal{I}\to\mathcal{A}$ and isomorphisms in $\mathcal{A}$. Thus, for any isomorphism $f$ of $\mathcal{A}$ there is a unique functor $F: \mathcal{I} \rightarrow \mathcal{A}$ such that the arrow $0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$.

Proof

Immediate from the definitions.

Properties

Proposition

$\{ 1 \} \subseteq \mathcal{I}$ is the full subcategory classifier of the 1-category $Cat$. That is, for any small category $\mathcal{A}$, there is a bijective correspondence between full subcategories of $\mathcal{A}$ and functors $\mathcal{A} \to \mathcal{I}$, with the reverse direction given by taking the pullback of the inclusion $\{ 1 \} \subseteq \mathcal{I}$.

Proof

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

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

Proposition

The nerve $N(\mathcal{I})$ is the full subspace classifier for $sSet$, and thus $N(\mathcal{I})$ represents the subpresheaf $FullSub \subseteq Sub$ of full subspaces.

Proof

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

Remark

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

References

A generalization of the notion of the interval groupoid to simplicial groupoids is considered in

• William Dwyer, Daniel Kan, §2.8 of: Homotopy theory and simplicial groupoids, Indagationes Mathematicae (Proceedings) 87 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

and plays a key role in the discussion of the model structure on simplicial groupoids, see there.