one question about homotopy pushout

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This question arises when I'm reading Jacob Lurie's Higher Topos Theory, p814.




Suppose we are given a diagram $$A_0leftarrow Arightarrow A_1$$
in a model category $mathcal C$. In general, the pushout $A_0coprod_AA_1$ is poorly behaved in the sense that a map of diagrams $$beginarrayrcl
A_0&leftarrow& A&rightarrow &A_1\
downarrow&&downarrow&&downarrow\
B_0&leftarrow&B &rightarrow &B_1
endarray$$
need not induce a weak equivalence $A_0coprod _AA_1rightarrow B_0coprod_BB_1$, even if each of the vertical arrows in the diagram is individually a weak equivalence. To correct this difficulty, it is convenient to introduce the left derived functor of 'pushout'. The homotopy pushout of hthe diagram$$A_0leftarrow Arightarrow A_1$$
is defined to be the pushout $A_0'coprod_A'A_1'$, where we have chosen a commutative diagram
$$beginarrayrcl
A_0'&xleftarrowj& A'&xrightarrowi &A_1'\
downarrow&&downarrow&&downarrow\
A_0&leftarrow&A &rightarrow &A_1
endarray$$
where the vertical maps are weak equivalences, and the top row is cofibrant diagram in the sense that $A'$ is cofibrant and the maps $i$ and $j$ are both cofibrations. One can show that such a diagram exists and the pushout $A_0'coprod_A'A_1'$ depends on the choice of diagram only up to weak equivalence.




My question is, given two such diagrams, how to construct such a weak equivalence between the two pushouts?




homotopy-theory model-categories




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  • What precisely do you mean? Showing that $A'_0coprod _A' A'_1rightarrow B'_0coprod_B' B'_1$ is a weak equivalence if all vertical maps in the first diagram are weak equivalences?
    – Paul Frost
    2 days ago











  • @PaulFrost I mean if we the homotopy pushout defined as above is up to a weak equivalence.
    – user12580
    2 days ago














up vote
0
down vote

favorite












This question arises when I'm reading Jacob Lurie's Higher Topos Theory, p814.




Suppose we are given a diagram $$A_0leftarrow Arightarrow A_1$$
in a model category $mathcal C$. In general, the pushout $A_0coprod_AA_1$ is poorly behaved in the sense that a map of diagrams $$beginarrayrcl
A_0&leftarrow& A&rightarrow &A_1\
downarrow&&downarrow&&downarrow\
B_0&leftarrow&B &rightarrow &B_1
endarray$$
need not induce a weak equivalence $A_0coprod _AA_1rightarrow B_0coprod_BB_1$, even if each of the vertical arrows in the diagram is individually a weak equivalence. To correct this difficulty, it is convenient to introduce the left derived functor of 'pushout'. The homotopy pushout of hthe diagram$$A_0leftarrow Arightarrow A_1$$
is defined to be the pushout $A_0'coprod_A'A_1'$, where we have chosen a commutative diagram
$$beginarrayrcl
A_0'&xleftarrowj& A'&xrightarrowi &A_1'\
downarrow&&downarrow&&downarrow\
A_0&leftarrow&A &rightarrow &A_1
endarray$$
where the vertical maps are weak equivalences, and the top row is cofibrant diagram in the sense that $A'$ is cofibrant and the maps $i$ and $j$ are both cofibrations. One can show that such a diagram exists and the pushout $A_0'coprod_A'A_1'$ depends on the choice of diagram only up to weak equivalence.




My question is, given two such diagrams, how to construct such a weak equivalence between the two pushouts?




homotopy-theory model-categories




share|cite|improve this question





















  • What precisely do you mean? Showing that $A'_0coprod _A' A'_1rightarrow B'_0coprod_B' B'_1$ is a weak equivalence if all vertical maps in the first diagram are weak equivalences?
    – Paul Frost
    2 days ago











  • @PaulFrost I mean if we the homotopy pushout defined as above is up to a weak equivalence.
    – user12580
    2 days ago












up vote
0
down vote

favorite









up vote
0
down vote

favorite











This question arises when I'm reading Jacob Lurie's Higher Topos Theory, p814.




Suppose we are given a diagram $$A_0leftarrow Arightarrow A_1$$
in a model category $mathcal C$. In general, the pushout $A_0coprod_AA_1$ is poorly behaved in the sense that a map of diagrams $$beginarrayrcl
A_0&leftarrow& A&rightarrow &A_1\
downarrow&&downarrow&&downarrow\
B_0&leftarrow&B &rightarrow &B_1
endarray$$
need not induce a weak equivalence $A_0coprod _AA_1rightarrow B_0coprod_BB_1$, even if each of the vertical arrows in the diagram is individually a weak equivalence. To correct this difficulty, it is convenient to introduce the left derived functor of 'pushout'. The homotopy pushout of hthe diagram$$A_0leftarrow Arightarrow A_1$$
is defined to be the pushout $A_0'coprod_A'A_1'$, where we have chosen a commutative diagram
$$beginarrayrcl
A_0'&xleftarrowj& A'&xrightarrowi &A_1'\
downarrow&&downarrow&&downarrow\
A_0&leftarrow&A &rightarrow &A_1
endarray$$
where the vertical maps are weak equivalences, and the top row is cofibrant diagram in the sense that $A'$ is cofibrant and the maps $i$ and $j$ are both cofibrations. One can show that such a diagram exists and the pushout $A_0'coprod_A'A_1'$ depends on the choice of diagram only up to weak equivalence.




My question is, given two such diagrams, how to construct such a weak equivalence between the two pushouts?




homotopy-theory model-categories




share|cite|improve this question













This question arises when I'm reading Jacob Lurie's Higher Topos Theory, p814.




Suppose we are given a diagram $$A_0leftarrow Arightarrow A_1$$
in a model category $mathcal C$. In general, the pushout $A_0coprod_AA_1$ is poorly behaved in the sense that a map of diagrams $$beginarrayrcl
A_0&leftarrow& A&rightarrow &A_1\
downarrow&&downarrow&&downarrow\
B_0&leftarrow&B &rightarrow &B_1
endarray$$
need not induce a weak equivalence $A_0coprod _AA_1rightarrow B_0coprod_BB_1$, even if each of the vertical arrows in the diagram is individually a weak equivalence. To correct this difficulty, it is convenient to introduce the left derived functor of 'pushout'. The homotopy pushout of hthe diagram$$A_0leftarrow Arightarrow A_1$$
is defined to be the pushout $A_0'coprod_A'A_1'$, where we have chosen a commutative diagram
$$beginarrayrcl
A_0'&xleftarrowj& A'&xrightarrowi &A_1'\
downarrow&&downarrow&&downarrow\
A_0&leftarrow&A &rightarrow &A_1
endarray$$
where the vertical maps are weak equivalences, and the top row is cofibrant diagram in the sense that $A'$ is cofibrant and the maps $i$ and $j$ are both cofibrations. One can show that such a diagram exists and the pushout $A_0'coprod_A'A_1'$ depends on the choice of diagram only up to weak equivalence.




My question is, given two such diagrams, how to construct such a weak equivalence between the two pushouts?





homotopy-theory model-categories





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited yesterday
























asked 2 days ago









user12580

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  • What precisely do you mean? Showing that $A'_0coprod _A' A'_1rightarrow B'_0coprod_B' B'_1$ is a weak equivalence if all vertical maps in the first diagram are weak equivalences?
    – Paul Frost
    2 days ago











  • @PaulFrost I mean if we the homotopy pushout defined as above is up to a weak equivalence.
    – user12580
    2 days ago
















  • What precisely do you mean? Showing that $A'_0coprod _A' A'_1rightarrow B'_0coprod_B' B'_1$ is a weak equivalence if all vertical maps in the first diagram are weak equivalences?
    – Paul Frost
    2 days ago











  • @PaulFrost I mean if we the homotopy pushout defined as above is up to a weak equivalence.
    – user12580
    2 days ago















What precisely do you mean? Showing that $A'_0coprod _A' A'_1rightarrow B'_0coprod_B' B'_1$ is a weak equivalence if all vertical maps in the first diagram are weak equivalences?
– Paul Frost
2 days ago





What precisely do you mean? Showing that $A'_0coprod _A' A'_1rightarrow B'_0coprod_B' B'_1$ is a weak equivalence if all vertical maps in the first diagram are weak equivalences?
– Paul Frost
2 days ago













@PaulFrost I mean if we the homotopy pushout defined as above is up to a weak equivalence.
– user12580
2 days ago




@PaulFrost I mean if we the homotopy pushout defined as above is up to a weak equivalence.
– user12580
2 days ago















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