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

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

as usual.

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

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

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

It sends

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

where

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

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

$$\array{ \mathbf{B} (\mathbb{Z}/{100}) &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} (\mathbb{Z}/{10}) &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) } \,.$$

That now carries a 2-cocycle

$$\mathbf{B} (\mathbb{Z}/{100}) \to \mathbf{B}^2 (\mathbb{Z}/{10}) \,,$$

and so on.

$$\array{ \vdots \\ \downarrow \\ \mathbf{B} (\mathbb{Z}/{1000}) &\stackrel{c}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) \\ \downarrow \\ \mathbf{B} (\mathbb{Z}/{100}) &\stackrel{c}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) \\ \downarrow \\ \mathbf{B} (\mathbb{Z}/{10}) &\stackrel{c}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) }$$

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 $\mathbb{Z}_{10}$. 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)