0
$\begingroup$

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)?

$\endgroup$

3 Answers 3

3
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.