nLab
carrying

Idea

In the context of arithmetic, carrying is part of the operation of representing addition of natural numbers by digits with respect to a base.

In terms of cohomology

Write ${\mathbb{Z}}_{10}=\mathbb{Z}/10\mathbb{Z}$ for the abelian group of addition of integers modulo 10. In the following we identify the elements as

$${\mathbb{Z}}_{10}=\{0,1,2,\cdots ,9\},$$\mathbb{Z}_{10} = \{0,1,2, \cdots, 9\} \,,

as usual.

Being an abelian group, every delooping n-groupoid $B^n{\mathbb{Z}}_{10}$ exists.

Carrying is a 2-cocycle in the group cohomology, hence a morphism of infinity-groupoids

$$c:B{\mathbb{Z}}_{10}\to B^2{\mathbb{Z}}_{10}.$$c : \mathbf{B} \mathbb{Z}_{10} \to \mathbf{B}^2\mathbb{Z}_{10} \,.

It sends

$$\begin{array}{ccc}& & •\\ & {}^a\nearrow & {\Downarrow }^{=}& {\searrow }^b\\ •& & \stackrel{a+b\mathrm{mod}10}{\to }& & \end{array}\mapsto \begin{array}{ccc}& & •\\ & {}^{\mathrm{id}}\nearrow & {\Downarrow }^{c(a,b)}& {\searrow }^{\mathrm{id}}\\ •& & \stackrel{\mathrm{id}}{\to }& & •\end{array},$$\array{ && \bullet \\ & {}^{\mathllap{a}}\nearrow &\Downarrow^=& \searrow^{\mathrlap{b}} \\ \bullet &&\stackrel{a+b mod 10}{\to}&& } \;\;\; \mapsto \;\;\; \array{ && \bullet \\ & {}^{\mathllap{id}}\nearrow &\Downarrow^{c(a,b)}& \searrow^{\mathrlap{id}} \\ \bullet &&\stackrel{id}{\to}&& \bullet } \,,

where

$$c(a,b)=\{\begin{array}{cc}1& a+b\ge 10\\ 0& a+b<10.\end{array}$$c(a,b) = \left\{ \array{ 1 & a + b \geq 10 \\ 0 & a + b \lt 10 \,. } \right.

The central extension classified by this 2-cocycle, hence the homotopy fiber of this morphism is ${\mathbb{Z}}_{100}$

$$\begin{array}{ccc}B{\mathbb{Z}}_{100}& \to & *\\ \downarrow & & \downarrow \\ B{\mathbb{Z}}_{10}& \stackrel{c}{\to }& B^2{\mathbb{Z}}_{10}\end{array}.$$\array{ \mathbf{B}\mathbb{Z}_{100} &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} \mathbb{Z}_{10} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^2 \mathbb{Z}_{10} } \,.

That now carries a 2-cocycle

$$B{\mathbb{Z}}_{100}\to B^2{\mathbb{Z}}_{10},$$\mathbf{B} \mathbb{Z}_{100} \to \mathbf{B}^2 \mathbb{Z}_{10} \,,

and so on.

$$\begin{array}{c}⋮\\ \downarrow \\ B{\mathbb{Z}}_{1000}& \stackrel{c}{\to }& B^2{\mathbb{Z}}_{10}\\ \downarrow \\ B{\mathbb{Z}}_{100}& \stackrel{c}{\to }& B^2{\mathbb{Z}}_{10}\\ \downarrow \\ B{\mathbb{Z}}_{10}& \stackrel{c}{\to }& B^2{\mathbb{Z}}_{10}\end{array}$$\array{ \vdots \\ \downarrow \\ \mathbf{B}\mathbb{Z}_{1000} &\stackrel{c}{\to}& \mathbf{B}^2\mathbb{Z}_{10} \\ \downarrow \\ \mathbf{B}\mathbb{Z}_{100} &\stackrel{c}{\to}& \mathbf{B}^2\mathbb{Z}_{10} \\ \downarrow \\ \mathbf{B}\mathbb{Z}_{10} &\stackrel{c}{\to}& \mathbf{B}^2\mathbb{Z}_{10} }

This tower can be viewed as a sort of “Postnikov tower” of $\mathbb{Z}$ (although it is of course not a Postnikov tower in the usual sense). Note that it is not “convergent”: the limit of the tower is the ring of $10$-adic integers. This makes perfect sense in terms of carrying: the $10$-adic integers can be identified with “decimal numbers” that can be “infinite to the left”, with addition and multiplication defined using the usual carrying rules “on off to infinity”.

References

• Dan Isaksen, A cohomological viewpoint on elementary school arithmetic, The American Mathematical Monthly, Vol. 109, No. 9. (Nov., 2002), pp. 796-805. (jstor)