smash product of pointed spaces preserve weak equivalences

Question

Consider the category of pointed simplicial sets with usual notion of weak equivalence. The question is does the functor $$Y \mapsto X\wedge Y$$ preserve weak equivalences? or at the very least does it preserve trivial cofibrations (monomorphisms+weak equivalences)?

Answer 1

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, every pointed simplicial set is cofibrant, so by Ken Brown's lemma, any endofunctor on $\mathbf{sSet}_*$ that sends anodyne extensions (= trivial cofibrations) to weak homotopy equivalences must preserve all weak homotopy equivalences.

Let $X \vee Y = X \times \{ * \} \cup \{ * \} \times Y \subseteq X \times Y$. It is easy to see that $X \vee (-)$ and $(-) \vee Y$ each preserve anodyne extensions, so $\vee$ is invariant under weak homotopy equivalence. Recall that $X \wedge Y$ is defined by the following pushout diagram in $\mathbf{sSet}_*$: $$\require{AMScd} \begin{CD} X \vee Y @>>> X \times Y \\ @VVV @VVV \\ \Delta^0 @>>> X \wedge Y \end{CD}$$ Suppose $X_0 \to X_1$ and $Y_0 \to Y_1$ are anodyne extensions. Consider the following diagram in $\mathbf{sSet}_*$, $$\begin{CD} X_0 \vee Y_0 @>>> X_1 \vee Y_1 @>>> X_1 \times Y_1 \\ @VVV @VVV @VVV \\ \Delta^0 @>>> Z_0 @>>> Z_1 \\ & @VVV @VVV \\ && \Delta^0 @>>> X_1 \wedge Y_1 \end{CD}$$ where every square is a pushout square. Since $X_0 \vee Y_0 \to X_1 \vee Y_1$ is an anodyne extension, the same is true of $\Delta^0 \to Z_0$, so $Z_0 \to \Delta^0$ is a weak homotopy equivalence. Moreover, $X_1 \vee Y_1 \to X_1 \times Y_1$ is a monomorphism (= cofibration), so $Z_0 \to Z_1$ is also a monomorphism. Since the model structure on $\mathbf{sSet}_*$ is left proper, it follows that $Z_1 \to X_1 \vee Y_1$ is a weak homotopy equivalence.

On the other hand, we also have the following diagram in $\mathbf{sSet}_*$, $$\begin{CD} X_0 \vee Y_0 @>>> X_0 \times Y_0 @>>> X_1 \times Y_1 \\ @VVV @VVV @VVV \\ \Delta^0 @>>> X_0 \wedge Y_0 @>>> Z_1 \end{CD}$$ where both squares are pushout squares, and $X_0 \times Y_0 \to X_1 \times Y_1$ is an anodyne extension (because the model structure on $\mathbf{sSet}$ is cartesian), so $X_0 \wedge Y_0 \to Z_1$ is also an anodyne extension. It is easy to check that the composite $X_0 \wedge Y_0 \to Z_1 \to X_1 \wedge Y_1$ is the morphism obtained by functoriality, so this completes the proof.

Answer 2

Pointed simplicial sets with the smash product form a symmetric monoidal model category. All objects are cofibrant in this category, so the functor Y→X∧Y is a left Quillen functor. Left Quillen functors preserve weak equivalences between cofibrant objects, and in our case all objects are cofibrant, hence all weak equivalences are preserved.

Answer 3

It also follows immediately from Proposition 4.2.9 in Hovey's "Model Categories".