derived functor that preserves weak equivalences
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Suppose we have a functor $F:Arightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
2- Assume that F takes weak equivalences to weak equivalences, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
It will be great to see the construction of such derived functor in detail. Thank you.
reference-request model-categories
asked Nov 25 at 18:39
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up vote
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Suppose we have a functor $F:Arightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
2- Assume that F takes weak equivalences to weak equivalences, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
It will be great to see the construction of such derived functor in detail. Thank you.
reference-request model-categories
asked Nov 25 at 18:39
ABC
604
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Suppose we have a functor $F:Arightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
2- Assume that F takes weak equivalences to weak equivalences, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
It will be great to see the construction of such derived functor in detail. Thank you.
reference-request model-categories
asked Nov 25 at 18:39
ABC
604
Suppose we have a functor $F:Arightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
2- Assume that F takes weak equivalences to weak equivalences, can we define the derived functor: $$Ho(F): Ho(A)rightarrow Ho(B) $$
It will be great to see the construction of such derived functor in detail. Thank you.
reference-request model-categories
reference-request model-categories
asked Nov 25 at 18:39
ABC
604
asked Nov 25 at 18:39
ABC
604
asked Nov 25 at 18:39
ABC
604
asked Nov 25 at 18:39
ABC
604
asked Nov 25 at 18:39
ABC
604
604
add a comment |
add a comment |
1 Answer
1
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up vote
6
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The answer is yes.
Consider the functor $F:A to Ho(B)$ obtained by composing $F:A to B$ with the localization functor $B to Ho(B)$.
Then this functor takes weak equivalences in $A$ to isomorphisms in $Ho(B)$,
so by the universal property of localization, there is a functor $F':Ho(A) to Ho(B)$, such that $F cong F'circ Q$, where $Q:A to Ho(A)$ is the localization functor. This functor $F'$ is your derived functor.
answered Nov 25 at 19:01
the L
7501920
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
The answer is yes.
Consider the functor $F:A to Ho(B)$ obtained by composing $F:A to B$ with the localization functor $B to Ho(B)$.
Then this functor takes weak equivalences in $A$ to isomorphisms in $Ho(B)$,
so by the universal property of localization, there is a functor $F':Ho(A) to Ho(B)$, such that $F cong F'circ Q$, where $Q:A to Ho(A)$ is the localization functor. This functor $F'$ is your derived functor.
answered Nov 25 at 19:01
the L
7501920
add a comment |
up vote
6
down vote
accepted
The answer is yes.
Consider the functor $F:A to Ho(B)$ obtained by composing $F:A to B$ with the localization functor $B to Ho(B)$.
Then this functor takes weak equivalences in $A$ to isomorphisms in $Ho(B)$,
so by the universal property of localization, there is a functor $F':Ho(A) to Ho(B)$, such that $F cong F'circ Q$, where $Q:A to Ho(A)$ is the localization functor. This functor $F'$ is your derived functor.
answered Nov 25 at 19:01
the L
7501920
add a comment |
up vote
6
down vote
accepted
up vote
6
down vote
accepted
The answer is yes.
Consider the functor $F:A to Ho(B)$ obtained by composing $F:A to B$ with the localization functor $B to Ho(B)$.
Then this functor takes weak equivalences in $A$ to isomorphisms in $Ho(B)$,
so by the universal property of localization, there is a functor $F':Ho(A) to Ho(B)$, such that $F cong F'circ Q$, where $Q:A to Ho(A)$ is the localization functor. This functor $F'$ is your derived functor.
answered Nov 25 at 19:01
the L
7501920
The answer is yes.
Consider the functor $F:A to Ho(B)$ obtained by composing $F:A to B$ with the localization functor $B to Ho(B)$.
Then this functor takes weak equivalences in $A$ to isomorphisms in $Ho(B)$,
so by the universal property of localization, there is a functor $F':Ho(A) to Ho(B)$, such that $F cong F'circ Q$, where $Q:A to Ho(A)$ is the localization functor. This functor $F'$ is your derived functor.
answered Nov 25 at 19:01
the L
7501920
answered Nov 25 at 19:01
the L
7501920
answered Nov 25 at 19:01
the L
7501920
answered Nov 25 at 19:01
the L
7501920
7501920
add a comment |
add a comment |
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