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Let $\text{sPsh}(\mathcal{C})$ be the category of simplicial presheaves, which I want to see as $$\text{sPsh}(\mathcal{C})=[\mathcal{C}^{\text{op}}\times\Delta^{\text{op}},\text{Set}]=\text{Psh}( \mathcal{C}\times \Delta).$$

Let $y:\mathcal{C}\to \text{Psh}(\mathcal{C})$ be the Yoneda embedding, and let $d:\text{Psh}(\mathcal{C})\to \text{sPsh}(\mathcal{C})$ be the functor taking a presheaf $P$ to the constant simplicial presheaf having $P$ in every dimension $dP=(n \mapsto P[n]=P)$. Composing these two, we get an embedding $$r:\mathcal{C}\to \text{Psh}(\mathcal{C}) \to \text{sPsh}(\mathcal{C})$$ which we can also see as the composition $$r:\mathcal{C}\to \mathcal{C}\times{\Delta}\to \text{Psh}(\mathcal{C}\times{\Delta})$$ $$c\mapsto(c,0)\mapsto( \ (a,n)\mapsto\text{Hom}_{\mathcal{C}\times{\Delta}}((a,n),(c,0))\cong \text{Hom}_{\mathcal{C}}(a,c) \ ).$$ In other words, we take $c$ to $(c,0)$ and then to the representable functor $y(c,0)=h_{(c,0)},$ which, since $0$ is terminal in $\Delta,$ corresponds just the costant simplicial presheaf $n\mapsto h_c.$

So we have a full subcategory $$\{h_{(c,0)}: c\in \mathcal{C}\} \subset \text{sPsh}(\mathcal{C}).$$ Now a generic representable presheaf in $\text{sPsh}(\mathcal{C})$ will be of the form $$h_{(c,n)}:(a,m)\mapsto \text{Hom}((a,m),(c,n)).$$

I would like to prove (I don't know for sure it's true) that for every $(c,n)\in \mathcal{C}\times{\Delta},$ we have a weak equivalence in Bousfield-Kan model structure $$h_{(c,n)}\xrightarrow{\sim}h_{(c,0)}.$$

I was thinking about proving that the natural transformation $\eta:h_{(c,n)} \Rightarrow h_{(c,0)}$ given in each $(a,m)\in \mathcal{C}^{\text{op}}\times{\Delta^{\text{op}}}$ by the projection $$\text{Hom}_{\mathcal{C}\times{\Delta}}((a,m),(c,n))=\text{Hom}_{\Delta}(m,n) \times{\text{Hom}_{\mathcal{C}}}(a,c)\to \text{Hom}_{\mathcal{C}}(a,c)$$ is a weak equivalence.

This, in the B-K model structure would mean that for every $a \in \mathcal{C}$ the projection is a weak equivalence from the simplicial set $m\mapsto \text{Hom}_{\Delta}(m,n) \times{\text{Hom}_{\mathcal{C}}}(a,c)$ to the constant simplicial set $m\mapsto \text{Hom}_{\mathcal{C}}(a,c).$

This in turn would mean that the geometric realization of these is a weak homotopy equivalence of compactly generated weakly Hausdorff spaces.

I have no idea how to prove this though. I know geometric realization preserves products, but it does not get me very far.

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  • $\begingroup$ Geometric realization and honotopy groups preserve products, hence products of weak equivalences of simplicial sets are weak equivalences. Isn't that enough ? $\endgroup$ Commented Aug 22, 2020 at 21:51

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Since $\def\Hom{\operatorname{Hom}}\Hom_{\mathcal C}(a,c)$ is just a set, the product is also a disjoint union $$\Hom_\Delta(-,[n])\times\Hom_{\mathcal C}(a,c) = \coprod_{\Hom_{\mathcal C}(a,c)}\Hom_\Delta(-,[n])$$ and in this way, the projection onto $\Hom_{\mathcal C}(a,c)$ is the coproduct of many copies of the simplicial map $\Hom_\Delta(-,[n])\to*$; that is, the projection is a coproduct of several copies of $\Delta[n]\to*$.

The maps $\Delta[n]\to*$ are weak equivalences since the standard simplex $\Delta[n]$ is contractible, and all objects in $\mathbf{sSet}$ are cofibrant, so the coproduct of weak equivalences is again a weak equivalence by Ken Brown's lemma (coproducts preserve trivial cofibrations of cofibrant objects and thus preserve weak equivalences of cofibrant objects).

Therefore, we get that the map $$ \Hom_\Delta(-,[n])\times\Hom_{\mathcal C}(a,c)=\coprod_{\Hom_{\mathcal C}(a,c)}\Delta[n]\to\coprod_{\Hom_{\mathcal C}(a,c)}*=\Hom_{\mathcal C}(a,c) $$ is a weak equivalence for every $a\in\mathcal C$, allowing us to conclude that $h_{(c,n)}\simeq h_{(c,0)}$ in $\operatorname{sPSh}\mathcal C$.

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