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Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to do homotopy theory, we know that $Ho(sSet) \simeq Ho(Top)$ for the standard model categoriesstructures on $sSet$ and $Top$, a test category $X$ is one for which there's a model structure on $[X^{op},Set]$ such that there's an equivalence $Ho([X^{op},Set]) \simeq Ho(Top)$.

Now let $A$ be an abelian category, let $[C^{op},A]$ be the category of simplicial objects in $A$, $Ch^{+}_{\bullet}(A)$ is the category of $\mathbb{N}$-graded chain complexes in $A$, these two categories are equivalent according to the Dold-Kan correspondence, they're also Quillen equivalent as model categories (at least for abelian categories with enough projectives?). Is there a similar "test-category"-ish result or construction for Dold-Kan correspondences? That is, can the category $\Delta$ be replaced in this context too? Are there categories $C$ for which a model structure on $[C^{op},A]$ can be defined such that $Ho([C^{op},A]) \simeq Ho([\Delta^{op},A]) \simeq Ho(Ch^{+}_{\bullet}(A)) = D(A)$?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to do homotopy theory, we know that $Ho(sSet) \simeq Ho(Top)$ for the standard model categories on $sSet$ and $Top$, a test category $X$ is one for which there's a model structure on $[X^{op},Set]$ such that there's an equivalence $Ho([X^{op},Set]) \simeq Ho(Top)$.

Now let $A$ be an abelian category, let $[C^{op},A]$ be the category of simplicial objects in $A$, $Ch^{+}_{\bullet}(A)$ is the category of $\mathbb{N}$-graded chain complexes in $A$, these two categories are equivalent according to the Dold-Kan correspondence, they're also Quillen equivalent as model categories (at least for abelian categories with enough projectives?). Is there a similar "test-category"-ish result or construction for Dold-Kan correspondences? That is, can the category $\Delta$ be replaced in this context too? Are there categories $C$ for which a model structure on $[C^{op},A]$ can be defined such that $Ho([C^{op},A]) \simeq Ho([\Delta^{op},A]) \simeq Ho(Ch^{+}_{\bullet}(A)) = D(A)$?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to do homotopy theory, we know that $Ho(sSet) \simeq Ho(Top)$ for the standard model structures on $sSet$ and $Top$, a test category $X$ is one for which there's a model structure on $[X^{op},Set]$ such that there's an equivalence $Ho([X^{op},Set]) \simeq Ho(Top)$.

Now let $A$ be an abelian category, let $[C^{op},A]$ be the category of simplicial objects in $A$, $Ch^{+}_{\bullet}(A)$ is the category of $\mathbb{N}$-graded chain complexes in $A$, these two categories are equivalent according to the Dold-Kan correspondence, they're also Quillen equivalent as model categories (at least for abelian categories with enough projectives?). Is there a similar "test-category"-ish result or construction for Dold-Kan correspondences? That is, can the category $\Delta$ be replaced in this context too? Are there categories $C$ for which a model structure on $[C^{op},A]$ can be defined such that $Ho([C^{op},A]) \simeq Ho([\Delta^{op},A]) \simeq Ho(Ch^{+}_{\bullet}(A)) = D(A)$?

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user44644
user44644
  • 211
  • 1
  • 2

Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to do homotopy theory, we know that $Ho(sSet) \simeq Ho(Top)$ for the standard model categories on $sSet$ and $Top$, a test category $X$ is one for which there's a model structure on $[X^{op},Set]$ such that there's an equivalence $Ho([X^{op},Set]) \simeq Ho(Top)$.

Now let $A$ be an abelian category, let $[C^{op},A]$ be the category of simplicial objects in $A$, $Ch^{+}_{\bullet}(A)$ is the category of $\mathbb{N}$-graded chain complexes in $A$, these two categories are equivalent according to the Dold-Kan correspondence, they're also Quillen equivalent as model categories (at least for abelian categories with enough projectives?). Is there a similar "test-category"-ish result or construction for Dold-Kan correspondences? That is, can the category $\Delta$ be replaced in this context too? Are there categories $C$ for which a model structure on $[C^{op},A]$ can be defined such that $Ho([C^{op},A]) \simeq Ho([\Delta^{op},A]) \simeq Ho(Ch^{+}_{\bullet}(A)) = D(A)$?