Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:20:42Z http://mathoverflow.net/feeds/question/31268 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31268/verifying-a-technical-lemma-regarding-homotopy-pushouts-in-the-theory-of-simplici Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories Harry Gindi 2010-07-10T04:45:28Z 2010-07-11T05:34:31Z <blockquote> <p><strong>Important Edit</strong>: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed.</p> <p>Here is the correct statement of (*):</p> <p>For any cofibration $f:A\to B$ and any <em>trivial</em> fibration $g:X\to Y$ in $C$, the induced morphism:</p> <p>$$\operatorname{Map}(B,X)\to \operatorname{Map}(B,Y)\times_{\operatorname{Map}(A,Y)} \operatorname{Map}(A,X)$$</p> <p>is a <em>trivial</em> Kan fibration. (Where $\operatorname{Map}$ is the (sSet)-enriched $\operatorname{Hom}$).</p> </blockquote> <p>Let $C$ be a simplicially-enriched category with a model structure, not necessarily compatible with the simplicial enrichment. </p> <p>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.</p> <p>(*)Suppose further that for any cofibration $f:A\to B$ and any fibration $g:X\to Y$ in $C$, the induced morphism:</p> <p>$$\operatorname{Map}(B,X)\to \operatorname{Map}(B,Y)\times_{\operatorname{Map}(A,Y)} \operatorname{Map}(A,X)$$</p> <p>is a Kan fibration. (Where $\operatorname{Map}$ is the (sSet)-enriched $\operatorname{Hom}$).</p> <p>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 $Map(A,-)^K$).</p> <p>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 $L$). </p> <p>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?</p> <p>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. </p> <p>That is, how does the line marked (*) imply anything relevant?</p> <p>If you'd like to look up the original source, it is Higher Topos Theory Proposition A.3.1.7 (in the appendix).</p> <p>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.</p> http://mathoverflow.net/questions/31268/verifying-a-technical-lemma-regarding-homotopy-pushouts-in-the-theory-of-simplici/31346#31346 Answer by Tom Goodwillie for Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories Tom Goodwillie 2010-07-11T01:48:44Z 2010-07-11T01:48:44Z <p>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.</p> <p>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?</p>