Applications of the Dold-Kan correspondence

Question

The Dold-Kan correspondence says essentially that simplicial abelian groups and nonnegative chain complexes of abelian groups are equivalent objects. While this is a very natural statement, I am not aware of any serious applications of it. Does anyone know any?

Answer 1

In addition to the construction of (generalized) Eilenberg–MacLane spaces mentioned in the comments, there are many other examples.

For example, the Dold–Kan correspondence allows us an easy perspective on cohomology operations on simplicial cochains. If $X$ is a simplicial set and $A$ is an abelian group, then $$\def\C{{\sf C}}\def\N{{\sf N}}\def\Z{{\bf Z}}\def\Hom{\mathop{\sf Hom}}\C^*(X,A)=\Hom(\N\Z[X],A).$$ In this formulation, it may not be entirely obvious why we could have nontrivial cohomology operations $$\def\H{{\sf H}}\H^m(X,A)→\H^n(X,B).$$ Consider now rewriting $\H^m(X,A)$ using the Dold–Kan correspondence: $$\def\U{{\sf U}}\H^m(X,A)=[\N\Z[X],A[m]]=[\Z[X],Γ A[m]]=[X,\U Γ A[m]],$$ where $\U$ denotes the forgetful functor from simplicial abelian groups to simplicial sets. Of course, the term $\def\K{{\sf K}}\U Γ A[m]=\K(A,m)$ is the $m$th Eilenberg–MacLane space of $A$.

In addition to providing us with a very simple proof that singular cohomology is representable by Eilenberg–MacLane spaces (which can be seen as another application), we also see immediately that cohomology operations $$\H^m(X,A)→\H^n(X,B)$$ correspond to homotopy classes of maps $$\K(A,m)→\K(B,n),$$ which themselves can be computed as elements of $\H^n(\K(A,m),B)$. Thus, the cohomology groups of Eilenberg–MacLane spaces compute cohomology operations.

More generally, the Dold–Kan correspondence is the principal mediating tool between homological algebra and (nonabelian) homotopical algebra. Indeed, the former term is predominantly used when working with chain complexes, and the latter when working with simplicial objects.

For example, the monoidal Dold–Kan correspondence establishes a Quillen equivalence between the model categories of simplicial rings and differential graded algebras.

In the commutative case, things become more interesting. In characteristic 0, we can prove that simplicial commutative algebras are Quillen equivalent to commutative differential graded algebras.

This fails in characteristic $p>0$, where instead we have a Quillen equivalence between simplicial $\def\E{{\sf E}}\E_\infty$-algebras and differential graded $\E_\infty$-algebras, which again is established using the (appropriately refined) Dold–Kan correspondence.

Answer 2

The Dold Kan correspondence is also at the heart of the formalism of nonabelian derived functors (nowadays phrased in terms of "animation"). Loosely speaking, the idea is to replace projective resolutions by free simplicial resolutions. This works in larger generality, for example for non-additive functors between abelian categories, or even functors out of non-abelian categories, like $\Omega^1: \operatorname{CRing}\to\operatorname{Ab}$, but is only justified since it translates in the case of an additive functor between abelian categories, via Dold-Kan, to the usual story in terms of projective resolutions.

Answer 3

I would like to add some more details on Achim's answer. One of the most (and earliest) important applications of the Dold-Kan correspondence is to help us define the derived functors of non-additive functors. Let's come back to the classical construction:

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$ be an additive functor between abelian categories. Suppose that $\mathcal{A}$ has enough projectives then for every $A$ we can take a quasi-isomorphism $\mathbf{P} \overset{\sim}{\longrightarrow} A$ then define the left-derived functors by $$L_i F(A) = H_i(F\mathbf{P}) \ \forall \ i \geq 0.$$ What are the advantages of being additive in this definition?

• First we have used $F(d) \circ F(d) = F(d^2) = F(0) = 0$ (being a functor and $F(0)=0$) to show that $F\mathbf{P}$ is indeed a complex.
• Second we want out derived functors defined uniquely up to isomorphisms. What we usually do in homological algebra is the comparison lemma which basically says that projective resolutions is unique in some sense (up to a chain homotopy). The proof of this lemma combines these following (just look at any book on homological algebra) $$f - g = ds + sd \ \ \text{and} \ \ \text{additivity of} \ F \Rightarrow F(f) - F(g) = F(d)F(s) + F(s)F(d).$$

In a word, additive functors preserve homotopies of chain complexes, and what we have done so far is taking the composition of the following (where $\mathbf{Ch}_{\geq 0}(F)$ is applying $F$ degree by degree, and only well-defined with $F$ additive) $$A \in \mathcal{A} \overset{\mathbf{P}}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{A}) \overset{\mathbf{Ch}_{\geq 0}(F)}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{B}) \overset{\text{homology}}{\longrightarrow} L_i F(A) \in \mathcal{B}.$$ Now if we lose the additivity of $F$, we must go other ways around, perhaps the most natural way is via the Dold-Kan correspondence (the category of simplicial sets - or abelian ones if you like - is not different that much from the category of chain complexes at least from the point of view of model categories theory, what we actually do is a homotopy theory). Let $N: \mathbf{Ch}_{\geq 0}(\mathcal{A}) \longrightarrow s\mathcal{A}$ (the right side is the category of simplicial objects on $\mathcal{A}$) be one-side equivalence in the Dold-Kan correspondence. Then we can go as in the following diagram $$A \in \mathcal{A} \overset{\mathbf{P}}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{A}) \overset{N}{\longrightarrow} s\mathcal{A} \overset{sF}{\longrightarrow} s\mathcal{B} \overset{N^{-1}}{\longrightarrow} \mathbf{Ch}_{\geq 0}(\mathcal{B}) \overset{\text{homology}}{\longrightarrow} L_i F(A) \in \mathcal{B}.$$ In this way, we can always talk about derived functor, say of an "arbitrary" $T$, but one should be careful since this new definition is a bi-degree theory $L_i T(-,n)$. When $T$ is additive, one has $$L_i T(-,n) \simeq L_{i-n}T(-).$$ Two of the most important classes of non-additive functors are the class of symmetric functors $\mathrm{Sym}$ and the class of free group-algebra functors $\mathbb{Z}$. Derived functors of these two classes coincide and give us the homology groups of Eilenberg-MacLane spaces. So at least theoretically, one knows why the (co)homology of Eilenbeg-MacLane spaces are legendarily difficult because they are derived functors of non-additive functors. Luckily, a full-fledged computation of these groups was done by H. Cartan in his famous Séminaire Henri Cartan (1954-1955). For more details on my comment, you can consult the work of Dold and Puppe.

Answer 4

I should preface with saying I am definitely not an expert on Dold-Kan. I am much more of an algebraist but have recently found the Dold-Kan correspondence to be a very helpful method of extending polynomial functors at the level of $R$-modules to functors on chain complexes of $R$-modules in a ``canonical" way.

The main point here (and I don't think I'm saying anything nontrivial) is that any endofunctor of $R$-modules naturally extends to an endofunctor of simplicial $R$-modules. This means, via the Dold-Kan correspondence, that any endofunctor on the category of $R$-modules naturally induces an endofunctor on the category of (nonnegatively graded) chain complex of $R$-modules. This endofunctor is ``natural" in the sense that it preserves homotopy equivalence -- this was already alluded to in Achim Krause's answer, since this means one can talk about deriving non-additive functors.

For my purposes, this allows me to work in a characteristic-free setting. For example, there is already a notion of ``symmetric powers" of chain complexes in commutative algebra, but this definition is actually rather unnatural since it does not preserve homotopy equivalence (in characteristic $2$, for instance, the second symmetric power of an exact complex of projective $R$-modules might have homology). On the other hand, using the Dold-Kan correspondence to define symmetric powers of complexes totally bypasses this issue (it does have the disadvantage of being much less amenable to explicit computation).

Actually, in the work of Gillet-Soule (``Intersection Theory using Adams Operations") one of the main tools the authors use to construct the $\lambda$-ring structure on the ring $\bigoplus K_0^Y (X)$ (K-theory of $X$ with support in $Y$) is to use the Dold-Kan version of exterior powers of complexes.