### The Dold–Kan Theorem: Two Questions

#### Posted by Tom Leinster

The Dold–Kan Theorem states that the category of simplicial abelian groups is equivalent, in a particular way, to the category of chain complexes. I was never particularly captivated by it until André Joyal explained to me that it can be viewed as a categorification of the fundamental theorem of Newton’s finite difference calculus. He spoke about this again at Category Theory 2010 in Genova last month, but there are no notes available online and that’s not what this post is about—although his talk is what reignited my interest. So that’ll just have to be a teaser.

The purpose of this post is to ask two questions. First I’ll explain why I’m asking them. Chain complexes (of abelian groups) can be regarded as strict $\infty$-categories in $\mathbf{Ab}$, the category of abelian groups. So the Dold–Kan Theorem states that in $\mathbf{Ab}$, strict $\infty$-categories are the same as simplicial objects. I’m sure other people have contemplated this: hence the following questions.

Ross Street defined a functor $J: \Delta \to \infty\mathbf{-Cat}$ assigning to $[n]$ the $n$th oriental. (Here $\infty\mathbf{-Cat}$ is the category of strict $\infty$-categories.) Since $\Delta$ is small and $\infty\mathbf{-Cat}$ cocomplete, there is an induced adjunction $- \otimes J: \mathbf{Set}^{\Delta^{op}} \stackrel{\longrightarrow}{\leftarrow} \infty\mathbf{-Cat}: Hom(J, -).$ The right adjoint $Hom(J, -): \infty\mathbf{-Cat} \to \mathbf{Set}^{\Delta^{op}}$ assigns to an $\infty$-category what is sometimes called its simplicial nerve. The left adjoint $- \otimes J$ is a kind of ‘realization’ functor. Many similar ‘realization’ functors preserve finite products.

Q1Does this left adjoint preserve finite products?

If the answer is no, this tentative train of thought is over.

If the answer is yes, there is an induced adjunction on abelian group objects. Since abelian groups in $\infty\mathbf{-Cat}$ are chain complexes, this means that we have an adjunction $\mathbf{Ab}^{\Delta^{op}} \stackrel{\longrightarrow}{\leftarrow} \mathbf{Ch}(\mathbf{Ab})$ between simplicial abelian groups and chain complexes.

Q2Is this adjunction the equivalence of Dold and Kan?

I don’t feel that this is a very deep or original train of thought, so I’m optimistic that someone has been through it already and knows the answers—even if they’re ‘no’.

## Re: The Dold–Kan Theorem: Two Questions

I am not quite sure concerning your specific questions

Q1andQ2, but I can offer maybe a good reason to be captivated by the Dold-Kan correspondence in the sense of: a good way to understand its relation to strict $\infty$-categories:Strict $\infty$-groupoids are equivalently crossed complexes and this equivalence is transparent: over a basepoint $x$ the crossed complex is in degree $k \geq 2$ the group of $k$-morphisms in the $\infty$-groupoid whose source is the identity on the identity on the identity, etc. on $x$.

Chain complexes of abelian groups (in non-negative degree) are just special cases of crossed complexes. And the sequence of inclusions

$Ch_\bullet^+ \hookrightarrow Crs \simeq Str \infty Grpd \hookrightarrow \infty Grpd \simeq KanCplx$

of strict and strictly abelian into strict into general $\infty$-groupoids is the Dold-Kan correspondence.

(This is in Ronnie Brown’s collected works, of course, though I realize we are lacking a good $n$Lab summary at the moment. The best reference I have energy for digging out at this time of night is somewhat curious: take footnote 116 on page 316 of his book

Nonabelian algebraic topologyand work yourself backwards from there…