Functor preserving long exact sequences
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Let $mathcalC$ and $mathcalD$ be abelian categories.
An exact functor $F:mathcalCtomathcalD$ preserves exactness of short exact sequences:
$$0to Ato Bto Cto 0$$
goes to
$$0to F(A)to F(B)to F(C)to 0$$
I don't believe that this implies long exact sequences are sent to long exact sequences.
0) Am I wrong? If not:
1) What tools do we have to measure the failure of $F$ in taking a long exact to a long exact?
2) Is there a name for a functor that takes long exact sequences to long exact sequences?
homology-cohomology homological-algebra exact-sequence
add a comment |Â
up vote
0
down vote
favorite
Let $mathcalC$ and $mathcalD$ be abelian categories.
An exact functor $F:mathcalCtomathcalD$ preserves exactness of short exact sequences:
$$0to Ato Bto Cto 0$$
goes to
$$0to F(A)to F(B)to F(C)to 0$$
I don't believe that this implies long exact sequences are sent to long exact sequences.
0) Am I wrong? If not:
1) What tools do we have to measure the failure of $F$ in taking a long exact to a long exact?
2) Is there a name for a functor that takes long exact sequences to long exact sequences?
homology-cohomology homological-algebra exact-sequence
Sorry if this question is dumb, I think maybe I am wrong at step 0
â user586231
Aug 22 at 7:41
0) It's wrong: if $F$ preserves short exact sequences, then it preserves kernels and images so it preserves exactness at any node of a long exact sequence.
â Berci
Aug 22 at 8:11
@Berci Is it true that it preserves limits and colimits or something like that?
â user586231
Aug 22 at 8:26
1
It preserves kernels and binary products hence finite limits; and it preserves cokernels and binary coproducts hence finite colimits
â Max
Aug 22 at 8:40
Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
â Jendrik Stelzner
Aug 25 at 10:21
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $mathcalC$ and $mathcalD$ be abelian categories.
An exact functor $F:mathcalCtomathcalD$ preserves exactness of short exact sequences:
$$0to Ato Bto Cto 0$$
goes to
$$0to F(A)to F(B)to F(C)to 0$$
I don't believe that this implies long exact sequences are sent to long exact sequences.
0) Am I wrong? If not:
1) What tools do we have to measure the failure of $F$ in taking a long exact to a long exact?
2) Is there a name for a functor that takes long exact sequences to long exact sequences?
homology-cohomology homological-algebra exact-sequence
Let $mathcalC$ and $mathcalD$ be abelian categories.
An exact functor $F:mathcalCtomathcalD$ preserves exactness of short exact sequences:
$$0to Ato Bto Cto 0$$
goes to
$$0to F(A)to F(B)to F(C)to 0$$
I don't believe that this implies long exact sequences are sent to long exact sequences.
0) Am I wrong? If not:
1) What tools do we have to measure the failure of $F$ in taking a long exact to a long exact?
2) Is there a name for a functor that takes long exact sequences to long exact sequences?
homology-cohomology homological-algebra exact-sequence
asked Aug 22 at 7:41
user586231
1
1
Sorry if this question is dumb, I think maybe I am wrong at step 0
â user586231
Aug 22 at 7:41
0) It's wrong: if $F$ preserves short exact sequences, then it preserves kernels and images so it preserves exactness at any node of a long exact sequence.
â Berci
Aug 22 at 8:11
@Berci Is it true that it preserves limits and colimits or something like that?
â user586231
Aug 22 at 8:26
1
It preserves kernels and binary products hence finite limits; and it preserves cokernels and binary coproducts hence finite colimits
â Max
Aug 22 at 8:40
Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
â Jendrik Stelzner
Aug 25 at 10:21
add a comment |Â
Sorry if this question is dumb, I think maybe I am wrong at step 0
â user586231
Aug 22 at 7:41
0) It's wrong: if $F$ preserves short exact sequences, then it preserves kernels and images so it preserves exactness at any node of a long exact sequence.
â Berci
Aug 22 at 8:11
@Berci Is it true that it preserves limits and colimits or something like that?
â user586231
Aug 22 at 8:26
1
It preserves kernels and binary products hence finite limits; and it preserves cokernels and binary coproducts hence finite colimits
â Max
Aug 22 at 8:40
Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
â Jendrik Stelzner
Aug 25 at 10:21
Sorry if this question is dumb, I think maybe I am wrong at step 0
â user586231
Aug 22 at 7:41
Sorry if this question is dumb, I think maybe I am wrong at step 0
â user586231
Aug 22 at 7:41
0) It's wrong: if $F$ preserves short exact sequences, then it preserves kernels and images so it preserves exactness at any node of a long exact sequence.
â Berci
Aug 22 at 8:11
0) It's wrong: if $F$ preserves short exact sequences, then it preserves kernels and images so it preserves exactness at any node of a long exact sequence.
â Berci
Aug 22 at 8:11
@Berci Is it true that it preserves limits and colimits or something like that?
â user586231
Aug 22 at 8:26
@Berci Is it true that it preserves limits and colimits or something like that?
â user586231
Aug 22 at 8:26
1
1
It preserves kernels and binary products hence finite limits; and it preserves cokernels and binary coproducts hence finite colimits
â Max
Aug 22 at 8:40
It preserves kernels and binary products hence finite limits; and it preserves cokernels and binary coproducts hence finite colimits
â Max
Aug 22 at 8:40
Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
â Jendrik Stelzner
Aug 25 at 10:21
Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
â Jendrik Stelzner
Aug 25 at 10:21
add a comment |Â
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Sorry if this question is dumb, I think maybe I am wrong at step 0
â user586231
Aug 22 at 7:41
0) It's wrong: if $F$ preserves short exact sequences, then it preserves kernels and images so it preserves exactness at any node of a long exact sequence.
â Berci
Aug 22 at 8:11
@Berci Is it true that it preserves limits and colimits or something like that?
â user586231
Aug 22 at 8:26
1
It preserves kernels and binary products hence finite limits; and it preserves cokernels and binary coproducts hence finite colimits
â Max
Aug 22 at 8:40
Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
â Jendrik Stelzner
Aug 25 at 10:21