## Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories

Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed.

Here is the correct statement of (*):

For any cofibration $f:A\to B$ and any trivial fibration $g:X\to Y$ in $C$, the induced morphism:

$$\operatorname{Map}(B,X)\to \operatorname{Map}(B,Y)\times_{\operatorname{Map}(A,Y)} \operatorname{Map}(A,X)$$

is a trivial Kan fibration. (Where $\operatorname{Map}$ is the (sSet)-enriched $\operatorname{Hom}$).

Let $C$ be a simplicially-enriched category with a model structure, not necessarily compatible with the simplicial enrichment.

Suppose that all objects of $C$ are cofibrant, that $C$ is tensored and cotensored over $SSet$, and that the class of weak equivalences of $C$ is closed under filtered colimits.

(*)Suppose further that for any cofibration $f:A\to B$ and any fibration $g:X\to Y$ in $C$, the induced morphism:

$$\operatorname{Map}(B,X)\to \operatorname{Map}(B,Y)\times_{\operatorname{Map}(A,Y)} \operatorname{Map}(A,X)$$

is a Kan fibration. (Where $\operatorname{Map}$ is the (sSet)-enriched $\operatorname{Hom}$).

Lastly, assume that $A\otimes \Delta^n\tilde{\to}A\otimes \Delta^0=A$ is a weak equivalence for any object $A$ in $C$ and any $n\in \mathbf{N}$. (Here, the tensor $A\otimes K$, where $A$ is in $C$ and $K$ is in $sSet$, is the object of $C$ representing the functor $Map(A,-)^K$).

Let $L\subseteq K$ be an inclusion of simplicial sets. Suppose $\sigma:\Delta^n\hookrightarrow K$ is a nondegenerate simplex of $K$ with all of its faces living in $L$. That is, we can factor the map $\partial\sigma:=\sigma|_{\partial\Delta^n}$ through the inclusion $L\subseteq K$ (in fact, we will assume that the target of this map actually is $L$).

Then for any object $D$ in $C$, the pushout $$D\otimes \Delta^n\coprod_{D\otimes\partial\Delta^n} D\otimes L\cong D\otimes (\Delta^n\coprod_{\partial\Delta^n}L)$$ is a homotopy pushout. Now, the question here is, why is this the case?

The proof I'm reading says that it follows from the line marked (*) above (and the fact that $C$ is left-proper (which follows from the fact that all objects of $C$ are cofibrant)), but it's not clear to me how to apply that hypothesis.

That is, how does the line marked (*) imply anything relevant?

If you'd like to look up the original source, it is Higher Topos Theory Proposition A.3.1.7 (in the appendix).

Edit: It's probable that a few of the hypotheses are unrelated to the actual question. I included everything because I'm not sure what's important.

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Here's an easy step, which may or may not be in the right direction: You want to show that tensoring with D preserves cofibrations, or equivalently that its right adjoint preserves trivial fibrations. It's clear that tensoring with D preserves trivial cofibrations, or equivalently that its right adjoint preserves fibrations, because that's what (*) says when A is initial and B is D. – Tom Goodwillie Jul 10 2010 at 17:12
Well, here's something funny: Not only did my comment not use the assumption that the map $A\otimes\Delta^n\to A\otimes \Delta^0$ is a weak equivalence for any object $A$ -- it shows that that assumption is superfluous, because a one-sided inverse $A\otimes\Delta^0\to A\otimes \Delta^n$ is a trivial cofibration. – Tom Goodwillie Jul 10 2010 at 19:00
Rereading the question, I find that it looks like you did not write exactly what you meant. I am guessing that $K$ was meant to be the pushout of the simplex and $L$ along the boundary of the simplex -- that is, that $\sigma$ was meant to be the only nondegenerate simplex of $K$ not in $L$. $K$ does not appear in the conclusion. – Tom Goodwillie Jul 11 2010 at 1:37
I'd just like to comment that the entire question becomes trivial with the correction. – Harry Gindi Jul 11 2010 at 7:46

Assuming I understand the question, isn't the following a counterexample? Let $\cal C$ be sSet with the usual model structure, but with the trivial simplicial enrichment in which the simplicial set $Map(A,X)$ is the discrete (constant) set of sSet morphisms $A\to X$. So $A\otimes K$ is coproduct of $\pi_0(K)$ copies of $A$. Then the inclusion $\partial\Delta^1\to \Delta^1$ induces a map $A\otimes \partial\Delta^1\to A\otimes \Delta^1$ that is not in general a cofibration, and the pushout $A\otimes * \leftarrow A\otimes \partial\Delta^1\to A\otimes \Delta^1$ is not a homotopy pushout.

Or am I making some mistake about the meaning of "enriched, tensored, and cotensored", or about the meaning of "homotopy pushout" in the present context?

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well, the example you gave doesn't fit exactly, since both $\partial\Delta^1$ should embed in both $L$ and $\Delta^1$. That wouldn't stop you from coming up with another counterexample, of course. – Harry Gindi Jul 11 2010 at 2:17
I didn't interpret "living in $L$" as embedding in $L$. But let's make it $\Delta^1\leftarrow\partial\Delta^1\to\Delta^1$ instead. – Tom Goodwillie Jul 11 2010 at 2:23
Don't take my word for it. Do I have the definitions right? I think that if $C$ is enriched over $V$ then it's said to be tensored if for every $C$-object $A$ the functor $Map(A,-)$ from $C$ to $V$ has a left adjoint; and cotensored if for every $C$-object $B$ the functor $Map(-,B)$ from $C$ to $V^{op}$ has a right adjoint. – Tom Goodwillie Jul 11 2010 at 3:29
Alright, I've e-mailed Lurie, and he said that the statement of the proposition is incorrect (and hence your counterexample works)! – Harry Gindi Jul 11 2010 at 5:07
Oh, good. I was almost ready to sign myself "enriched, tensored, and bewildered". – Tom Goodwillie Jul 11 2010 at 11:32
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