5
$\begingroup$

I got very lost in checking that simplicial sets with Quillen model structure are indeed a simplicial model category.

Recall that a model category $\mathcal{M}$ is simplicial if it is enriched in $\mathbf{SSet}$, powered by $\mathbf{SSet}$ and tensored by $\mathbf{SSet}$ so that the natural adjunctions exist and and also the enrichment is compatible with the model structure -- namely, for $i: A \to B$ a cofibration in $\mathcal{M}$ and $p: X \to Y$ a fibration in $M$, the map of simplicial sets $i^*\times p_*: \operatorname{Map}(B,X) \to \operatorname{Map}(A,X)\underset{\operatorname{Map}(A,Y)}{\times}\operatorname{Map}(B,Y)$ is required to be a fibration in $\mathbf{SSet}_{Quillen}$, and it is required to be trivial if either $i$ or $p$ is.

Recall that for simplicial sets, the enrichment is given by $\operatorname{Map}(K,M)_n = \mathbf{SSet}(\Delta^n \times K, M).$

$\mathbf{SSet}_{Quillen}$ is a cofibrantly generated model category, so to check the compatibility one must fill the horns: $\require{AMScd}$ \begin{CD} \Lambda^n_i @>h>> \operatorname{Map}(B,X) \\ @V \iota V V @VV i^* \times p_* V\\ \Delta^n @>(\alpha,\beta)>> \operatorname{Map}(A,X)\underset{\operatorname{Map}(A,Y)}{\times}\operatorname{Map}(B,Y) \end{CD}

The filling would be the lift in the diagram

$\require{AMScd}$ \begin{CD} \Delta^n \times A @>\alpha>> X \\ @V 1\times i V V @VV p V\\ \Delta^n \times B@>\beta>> Y \end{CD}

The natural idea is to consider the diagram

$\require{AMScd}$ \begin{CD} \Lambda^n_i \times B @>h>> X \\ @V \iota\times 1 V V @VV p V\\ \Delta^n \times B@>\beta>> Y \end{CD}

where the left map is a trivial cofibration, the right map is a fibration and thus from the lifting properties I obtain $\sigma: \Delta^n \times B \to X$ satisfying that $p\sigma = \beta$. However, if I want this $\sigma$ to be the answer, I must check $\sigma(1 \times i) = \alpha$... Would that be so?

I see that I have not made use of $i$ being an inclusion, but I don't see how it helps.

$\endgroup$
6
$\begingroup$

The trick is to check that the corner map $$\lambda^n_k\bar{\times}\delta^m:\Lambda^n_k \times \Delta^m \coprod_{\Lambda^n_k\times \partial \Delta^m} \Delta^n \times \partial \Delta^m \hookrightarrow \Delta^n\times \Delta^m$$ is anodyne for all $k, m, n$ appropriate. This is proven in Higher Topos Theory chapter 2, for your reference.

The fact that $i$ was an inclusion allows you to build it up as a relative cell complex of boundary inclusions, so it is enough to prove the case where $i$ is a boundary inclusion.

$\endgroup$
  • $\begingroup$ The proof that the corner product of a horn inclusion with a boundary inclusion is anodyne a classic combinatorial proof. It doesn't just arise for free from abstract nonsense. $\endgroup$ – Harry Gindi May 26 '18 at 16:42
  • $\begingroup$ There's also a clearer proof of the corner product statement in one of Joyal's books on quasicategories, but I don't remember the reference. $\endgroup$ – Harry Gindi May 26 '18 at 16:59
  • $\begingroup$ There is another, different proof using some machinery of Cisinski that proves this by different means (by proving that the class of simplicial weak equivalences is a cartesian localizer, but this requires much more machinery). $\endgroup$ – Harry Gindi May 26 '18 at 18:32
  • 1
    $\begingroup$ Appendix H ("Open Boxes and Prisms") of Joyal's The Theory of Quasi-Categories and its Applications $\endgroup$ – Daniel Gerigk Jul 19 '18 at 4:19

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.