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You have the following two results in the book Simplicial homotopy theory of Goerss and Jardine: Corollary III.2.7. Suppose that A is a simplicial abelian group. Then there are isomorphisms $$\pi_n(A,0) \cong H_n(NA) \cong H_n(A),$$ where $H_n(A)$ is the $n$th homology group of the Moore complex associated to $A$. These isomorphisms are natural ...

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Allegedly, one can prove this directly, but I have never seen the details. (For instance, the same fact is asserted without proof as Theorem 1.9 in [Moore, Semi-simplicial complexes and Postnikov systems].) Here is a more high-level proof. Recall that the class of anodyne extensions of simplicial sets is defined by induction: Any horn inclusion is an ...

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The grammar is "a category is tensored over a monoidal category"; this is a generalization of a set being equipped with an action of a monoid, or an abelian group being equipped with an action of a ring. In full generality you should provide the tensoring, but sometimes if you require enough it exists uniquely. The general pattern of the uniqueness results ...

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Assume that your notion of "object" is internalizable into arbitrary topoi. This is almost always the case. Then a simplicial object of the given type is usually (always?) an object of the given type internal to the topos of simplicial sets. For example, a simplicial group is a group internal to the topos of simplicial sets. A simplicial module (over a ...

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Oups I see, Hovey means that we can always choose T such that it contains all degenerate simplices of $X$ and this choice is made possible thanks to lemma 3.5.8.

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