nLab
carrying

Contents

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 10=/10\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

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

as usual.

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

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

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

It sends

a = b a+bmod10 id c(a,b) id id , \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)={1 a+b10 0 a+b<10. 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 100\mathbb{Z}_{100}

B 100 * B 10 c B 2 10. \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 100B 2 10, \mathbf{B} \mathbb{Z}_{100} \to \mathbf{B}^2 \mathbb{Z}_{10} \,,

and so on.

B 1000 c B 2 10 B 100 c B 2 10 B 10 c B 2 10 \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 1010-adic integers. This makes perfect sense in terms of carrying: the 1010-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)
Revised on June 18, 2012 08:49:03 by Urs Schreiber (89.204.138.106)