nLab
globe

Definition

The $n$-globe $G_n$ is the globular set represented by the object $[n]$ in the globe category $G$:

$$G_n:={\mathrm{Hom}}_G(-,[n]).$$G_n := Hom_G(-,[n]) \,.

There is a unique structure of a strict omega-category, an $n$-category in fact, on the $n$-globe. This makes the collection of $n$-globes arrange themselves into a co-globular $\omega$-category, i.e. a functor

$$G\to \omega \mathrm{Cat}$$G \to \omega Cat

$$[n]\mapsto G_n.$$[n] \mapsto G_n \,.

In this sense $n$-globes are the globular analog of the co-simplicial orientals.

• Figure of the 3-globe:

Could you say that, given a globular set that the $n$-globes are pointed $n$-spheres generated by repeated application of the identity assigning map, i.e.

$$g_v\in G_n\Rightarrow g_v=i^n(v),v\in G_0$$g_v\in G_n \implies g_v = i^n(v), v\in G_0

?

Or is an $n$-globe more like an $n$-loop?

Urs says: I seem to have convinced myself that the following is true, but hopefully somebody can check this:

I believe the $n$-globe is a “pointed $n$-sphere” as Eric is asking for in the following precise sense:

The $n$-globe is the double cone over the $(n-1)$-globe.

More in detail, I am thinking of the $n$-globes $G_n$ here in terms of strict $\omega$-categories. Let $\otimes$ be the Crans-Gray tensor product and write $\mathrm{pt}=G_0$ for the 0-globe (the point) and $I:=G_1$ for the 1-globe, the interval and let $s_0,s_1:\mathrm{pt}\to I$ be the two injections of the point to the endpoins of the interval. Then the above means that

$$\begin{array}{ccc}{G}_n\bigsqcup G_n& \stackrel{s_0\bigsqcup s_1}{\to }& G_n\otimes I\\ \downarrow & & \downarrow \\ \mathrm{pt}\bigsqcup \mathrm{pt}& \stackrel{s_0\bigsqcup s_1}{\to }& G_{n+1}\end{array}$$\array{ G_{n} \sqcup G_{n} &\stackrel{s_0 \sqcup s_1}{\to}& G_{n}\otimes I \\ \downarrow && \downarrow \\ pt \sqcup pt &\stackrel{s_0 \sqcup s_1}{\to}& G_{n+1} }

is a pushout diagram, which realizes the $(n+1)$-globes as the cylinder over the $n$-globe with top and bottom contracted to a point.

As an example, take the 2-globe

$$G_2=\left\{\begin{array}{cc}& \nearrow \searrow \\ 0& \Downarrow & 1\\ & \searrow \nearrow \end{array}\right\}$$G_2 = \left\lbrace \array{ & \nearrow \searrow \\ 0 &\Downarrow& 1 \\ & \searrow \nearrow } \right\rbrace

which is obtained from the cylinder over the 1-globe

$$G_1\otimes I=I\otimes I=\left\{\begin{array}{ccc}(\mathrm{0,0})& \to & (\mathrm{0,1})\\ \downarrow & \Downarrow & \downarrow \\ (\mathrm{1,0})& \to & (\mathrm{1,1})\end{array}\right\}$$G_1 \otimes I = I \otimes I = \left\lbrace \array{ (0,0) &\to& (0,1) \\ \downarrow &\Downarrow& \downarrow \\ (1,0) &\to& (1,1) } \right\rbrace

by contracting the two 1-globes on the left and the right to points.

Toby: Not answering your question, but I've seen this operation called ‘suspension’ in topology. A mathematical pun: $S^n$ (the $n$-sphere) is $S^n(2)$ (the $n$-fold suspension of the $2$-point space, which is of course the $0$-sphere). Also note that $2$ is the suspension of $0$, proving that $S(X)$ need not be a quotient of $X\times I$ (although it is such a quotient whenever $X$ is occupied).

Todd: Getting back to Eric’s question: usually people think of the $n$-globe as a combinatorial $n$-disk $D^n$. There are two standard inclusions of $D^{n-1}$ into $D^n$, which could be called “northern hemisphere” and southern hemisphere”, and which intersect at an “equator”. Going back the other way, there is a map $D^n\to D^{n-1}$ which smashes the globe flat onto an equatorial cross-section, so to speak.

Some Australians often write ”$n$-glob”, which used to annoy me (still does, a little bit). Anyway, I think of these things as solid globes, insofar as they have equators.

Eric: Thanks Todd and Urs. I see that globes are not pointed spheres the way I was thinking about them, i.e. images of the identity assigning map. Todd, your description helped a lot. Still it is kind of fun to visualize repeated applications of the identity assigning map generating spheres as they wrap around themselves even though it might not have much to do with globes.

Reference:

See the discussion in

Ronnie Brown, A new higher homotopy groupoid: the fundamental globular $\omega$-groupoid of a filtered space, Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.

To obtain the required structure of well defined compositions in this globular $\omega$-groupoid it has been found necessary to use the known structure in the cubical cases, and their relation with the globular case.