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If $\mathcal{C}$ has finite products, then the class of projective cofibrations is also closed under finite products. Indeed, since the cartesian product in the category of simplicial presheaves prese …

You can do this for any category of presheaves. Let $\mathcal{C}$ be a small category. For a presheaf $X$ on $\mathcal{C}$, we write $\mathbf{El} (X)$ for the category of elements of $X$, i.e. the com …

The class of morphisms having the right lifting property with respect to $\Lambda^1_k \times \Delta^n \hookrightarrow \Delta^1 \times \Delta^n$ (for all $k \in \{ 0, 1 \}$ and all $n \ge 0$) is strict …

Here is a proof for the case where $X$ is a Hausdorff space. Note that each $U_n$ is also Hausdorff in this case. A standard argument shows that the face operators of $U_\bullet$ are (surjective) loc …

Yes, $\wedge$ is invariant under weak homotopy equivalence in $\mathbf{sSet}_*$. It suffices to show that it sends anodyne extensions (= trivial cofibrations) to weak homtoopy equivalences: indeed, ev …

The nerve functor does not preserve homotopy colimits. Indeed, take any simplicial set $X$ with non-trivial $\pi_n$ ($n > 1$) and consider $X$ as a simplicial diagram of sets. In $\mathbf{sSet}$, its …

As pointed out, the answer is no – because we cannot hope to have fibrant replacements in general. On the other hand, by following the programme of van Osdol [1977, Simplicial homotopy in an exact cat …

I will write $[B, C]$ instead of $\underline{\mathrm{Hom}}(B, C)$. Recall the tensor–hom adjunction: $$\mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, [B, C])$$ Thus there is a canonical bijection …

It is well known that the underlying simplicial set of a simplicial group is a Kan complex. However, the underlying simplicial set of the canonical simplicial resolution (call it $F_\bullet G$) you de …

Since you have functorial factorisations you should exploit that to the hilt. If $\mathcal{M}$ is a model category with functorial factorisations then the category $\mathbf{c}\mathcal{M}$ of cosimplic …

Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones. We can make $\textbf{Top}$ into a simplicially enriched category as follows: Given topological spaces $X$ and $Y$ …