nLab walking 2-isomorphism with trivial boundary

Idea

The walking 2-isomorphism with trivial boundary is, roughly speaking, the minimal 2-category which contains a 2-isomorphism between identity 1-arrows. It is in fact a 2-groupoid, and a model of the homotopy type of the 2-truncation of 2-sphere.

It is an example of a walking structure, and can be compared for example with the walking 2-isomorphism?.

Definition and elementary observations

Definition

Let $F$ be the free strict 2-category on the 2-truncated reflexive globular set with exactly one object $\bullet$, no non-identity 1-arrows, a 2-arrow $\iota: id(\bullet) \rightarrow id(\bullet)$, and a 2-arrow $\iota^{-1}: id(\bullet) \rightarrow id(\bullet)$. The walking 2-isomorphism with trivial boundary is the strict 2-category obtained as the quotient of $F$ by the equivalence relation on 2-arrows generated by forcing the equations $\iota \circ \iota^{-1} = id$ and $\iota^{-1} \circ \iota = id$ to hold.

Remark

There are exactly $\mathbb{Z}$ 2-arrows $\id\left( \bullet \right) \rightarrow id\left( \bullet \right)$, namely one for each possible string of compositions of $\iota$ and $\iota^{-1}$, taking into account (strict) associativity. Here $\mathbb{Z}$ is of course the integers. This amounts to a computation of $\pi_{2}\left(S^{2}\right)$, the second homotopy group of the 2-sphere.

Remark

Let $\cdot$ denote horizontal composition. By the interchange law, we have that

$$\begin{aligned} \left( \iota^{-1} \cdot \iota \right) \circ \iota^{-1} &= \left( \iota^{-1} \cdot \iota \right) \circ \left( id \cdot \iota^{-1} \right) \\ &= \left(\iota^{-1} \circ id \right) \cdot \left( \iota \circ \iota^{-1} \right) \\ &= \iota^{-1} \circ id \\ &= \iota^{-1}. \end{aligned}$$

The only possibility, given Remark , is then that $\iota^{-1} \cdot \iota = id$.

An entirely analogous argument demonstrates that $\iota \circ \iota^{-1} = id$. Thus horizontal composition in the walking 2-isomorphism with trivial boundary is trivial.

Representing of 2-isomorphisms with trivial boundary

Proposition

Let $\mathcal{I}$ denote the walking 2-isomorphism with trivial boundary. Let $\mathcal{A}$ be a 2-category, and let $\phi$ be a 2-isomorphism $id(a) \rightarrow id(a)$ in $\mathcal{A}$, for some object $a$ of $\mathcal{A}$. Then there is a unique functor $\mathcal{I} \rightarrow \mathcal{A}$ such that $\iota$ maps to $\phi$.

Proof

Immediate from the definitions.