walking 2-isomorphism with trivial boundary



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


Let FF 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 ι:id()id()\iota: id(\bullet) \rightarrow id(\bullet), and a 2-arrow ι 1:id()id()\iota^{-1}: id(\bullet) \rightarrow id(\bullet). The walking 2-isomorphism with trivial boundary is the strict 2-category obtained as the quotient of FF by the equivalence relation on 2-arrows generated by forcing the equations ιι 1=id\iota \circ \iota^{-1} = id and ι 1ι=id\iota^{-1} \circ \iota = id to hold.


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


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

(ι 1ι)ι 1 =(ι 1ι)(idι 1) =(ι 1id)(ιι 1) =ι 1id =ι 1. \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 ι 1ι=id\iota^{-1} \cdot \iota = id.

An entirely analogous argument demonstrates that ιι 1=id\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


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


Immediate from the definitions.

Created on July 6, 2020 at 05:46:59. See the history of this page for a list of all contributions to it.