# 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 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 $Ho([C^{op},A]) \simeq Ho([\Delta^{op},A]) \simeq Ho(Ch^{+}_{\bullet}(A)) = D(A)$?

• I definitely remember reading a paper that answered this positively (I recall this was the main point of the paper). I read it recently, but that doesn't mean it's a recent paper. I'll try to remember, but others might answer first. – David White Dec 11 '15 at 20:49
• Okay, I remembered, but it doesn't answer this question. I was thinking of Schwede-Shipley Equivalences of Monoidal Model Categories, which lifts Dold-Kan to monoids, but not general diagram categories. Sorry that doesn't help. – David White Dec 11 '15 at 21:07
• Thanks David, I was partly inspired to ask the question when I read this on another question here on MO: "A simplicial object in a category C is a functor from the Delta category into C, but for almost all purposes the Delta category could be replaced with any test category in the sense of Grothendieck" (link: mathoverflow.net/questions/691/simplicial-objects). I thought there'd be some sort of generalization. – user44644 Dec 11 '15 at 21:55
• A recent paper by Lack and Street (Combinatorial categorical equivalences of Dold-Kan type) does generalize Dold-Kan to settings other than simplicial ones. – Dmitri Pavlov Dec 12 '15 at 12:15
• That paper looks very interesting, will check it out, thanks Dmitri. – user44644 Dec 12 '15 at 18:34