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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 morphismsmorphism:

$$Map(B,X)\to Map(B,Y)\times_{Map(A,Y)} Map(A,X)$$$\operatorname{Map}(B,X)\to \operatorname{Map}(B,Y)\times_{\operatorname{Map}(A,Y)} \operatorname{Map}(A,X)$$is a Kan fibration. (Where Map \operatorname{Map} is the (sSet)-enriched Hom).\operatorname{Hom}). Lastly, assume that A\otimes \Delta^n\tilde{\to}A\otimes \Delta^0=A is a weak equivalence for any object A in C and any n\in \mathbf{N}. (Here, the tensor A\otimes K, where A is in C and K is in sSet, is the object of C representing the functor Map(A,-)^K). Let L\subseteq K be an inclusion of simplicial sets. Suppose \sigma:\Delta^n\hookrightarrow K is a nondegenerate simplex of K with all of its faces living in L. That is, we can factor the map \partial\sigma:=\sigma|_{\partial\Delta^n} through the inclusion L\subseteq K (in fact, we will assume that the target of this map actually is L). Then for any object D in C, the pushout$$D\otimes \Delta^n\coprod_{D\otimes\partial\Delta^n} D\otimes L\cong D\otimes (\Delta^n\coprod_{\partial\Delta^n}L)$$is a homotopy pushout. Now, the question here is, why is this the case? The proof I'm reading says that it follows from the line marked (*) above (and the fact that C is left-proper (which follows from the fact that all objects of C are cofibrant)), but it's not clear to me how to apply that hypothesis. That is, how does the line marked (*) imply anything relevant? If you'd like to look up the original source, it is Higher Topos Theory Proposition A.3.1.7 (in the appendix). Edit: It's probable that a few of the hypotheses are unrelated to the actual question. I included everything because I'm not sure what's important. 5 added 179 characters in body; edited title; deleted 70 characters in body; deleted 7 characters in body HomotopypushoutsquaresgivenbytensoringVerifyingatechnicallemmaregardinghomotopypushoutsinthetheoryofsimplicialmodelcategories Let C be a simplicially-enriched category with a model structure, not necessarily compatible with the simplicial enrichment. Suppose that all objects of C are cofibrant, that C is tensored and cotensored over SSet, and that the class of weak equivalence equivalences of C is closed under filtered colimits. (*)Suppose further that for any cofibration f:A\to B and any fibration g:X\to Y in C, the induced morphisms:$$Map(B,X)\to Map(B,Y)\times_{Map(A,Y)} Map(A,X)$$is a Kan fibration. (Where Map is the (sSet)-enriched Hom). Lastly, assume that A\otimes \Delta^n\tilde{\to}A\otimes \Delta^0=A is a weak equivalence for any object A in C and any n\in \mathbf{N}. (Here, the tensor A\otimes K, where A is in C and K is in sSet, is the object of C representing the functor Map(A,-)^K). Let L\subseteq K be an inclusion of simplicial sets. Suppose \sigma:\Delta^n\hookrightarrow K is a nondegenerate simplex of K with all of its faces living in L. That is, we can factor the map \partial\sigma:=\sigma|_{\partial\Delta^n} through the inclusion L\subseteq K (in fact, we will assume that the target of this map actually is L). Then for any object D in C, the pushout$$D\otimes \Delta^n\coprod_{D\otimes\partial\Delta^n} D\otimes L\cong D\otimes (\Delta^n\coprod_{\partial\Delta^n}L)$$is a homotopy pushout. Since Now, the model structure question here isleft-proper, it's enough to show that one of the induced maps D\otimes\partial\Delta^n\to D\otimes\Delta^n or D\otimes\partial\Delta^n\to D\otimes L in the cocartesian square why is a cofibration.this the case? The proof I'm reading says that it follows from the line marked (*) above (and the fact that C is left-proper (which follows from the fact that all objects of C are cofibrant)), but it's not clear to me how to apply that hypothesis. That is, how does the line marked (*) imply anything relevant? If you'd like to look up the original source, it is Higher Topos Theory Proposition A.3.1.7 (in the appendix). Edit: It's probable that the majority a few of the hypotheses are unrelated to the actual question. I included everything because I'm not sure what's important. 4 added 26 characters in body; added 158 characters in body Let C be a simplicially-enriched category with a model structure, not necessarily compatible with the simplicial enrichment. Suppose that all objects of C are cofibrant, that C is tensored and cotensored over SSet, and that the class of weak equivalence of C is closed under filtered colimits. (*)Suppose further that for any cofibration f:A\to B and any fibration g:X\to Y in C, the induced morphisms:$$Map(B,X)\to Map(B,Y)\times_{Map(A,Y)} Map(A,X)$$is a Kan fibration. Lastly, assume that A\otimes \Delta^n\tilde{\to}A\otimes \Delta^0=A is a weak equivalence for any object A in C. C and any n\in \mathbf{N}. Let L\subseteq K be an inclusion of simplicial sets. Suppose \sigma:\Delta^n\hookrightarrow K is a nondegenerate simplex of K with all of its faces living in L. That is, we can factor the map \partial\sigma:=\sigma|_{\partial\Delta^n} through the inclusion L\subseteq K (in fact, we will assume that the target of this map actually is L). Then for any object D in C, the pushout$$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. Since the model structure is left-proper, it's enough to show that one of the induced maps$D\otimes\partial\Delta^n\to D\otimes\Delta^n$or$D\otimes\partial\Delta^n\to D\otimes L\$ in the cocartesian square is a cofibration.

The proof I'm reading says that it follows from the line marked (*) above, 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 the majority of the hypotheses are unrelated to the actual question. I included everything because I'm not sure what's important.