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\mathbb{Z}/10 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(/10)B 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 } \,,


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(/100)B 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 10\mathbb{Z}_{10}. 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”.


  • 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 January 13, 2017 00:48:50 by Toby Bartels (