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I would like to add some more details on Achim's answer. One of the most 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 one $L_i T(-,n)$. When $T$ is additive, one has $$L_i T(-,n) \simeq L_{i-n}T(-).$$ Two of the most important class 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.

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.

I would like to add some more details on Achim's answer. One of the most 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 (at least 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). 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 one $L_i T(-,n)$. When $T$ is additive, one has $$L_i T(-,n) \simeq L_{i-n}(T).$$ Two of the most important class 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.

I would like to add some more details on Achim's answer. One of the most 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 one $L_i T(-,n)$. When $T$ is additive, one has $$L_i T(-,n) \simeq L_{i-n}T(-).$$ Two of the most important class 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.

I would like to add some more details on Achim's answer. One of the most 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 (at least 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). 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 one $L_i T(-,n)$. When $T$ is additive, one has $$L_i T(-,n) \simeq L_{i-n}(T).$$ Two of the most important class 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 comments, you can consults works of Dold and Puppe.

I would like to add some more details on Achim's answer. One of the most 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 (at least 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). 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 one $L_i T(-,n)$. When $T$ is additive, one has $$L_i T(-,n) \simeq L_{i-n}(T).$$ Two of the most important class 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.