Functor preserving long exact sequences

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











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?







share|cite|improve this question




















  • Sorry if this question is dumb, I think maybe I am wrong at step 0
    – 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.
    – Berci
    Aug 22 at 8:11










  • @Berci Is it true that it preserves limits and colimits or something like that?
    – 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
    – Max
    Aug 22 at 8:40










  • Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
    – Jendrik Stelzner
    Aug 25 at 10:21














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?







share|cite|improve this question




















  • Sorry if this question is dumb, I think maybe I am wrong at step 0
    – 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.
    – Berci
    Aug 22 at 8:11










  • @Berci Is it true that it preserves limits and colimits or something like that?
    – 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
    – Max
    Aug 22 at 8:40










  • Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
    – Jendrik Stelzner
    Aug 25 at 10:21












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?







share|cite|improve this question












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?









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 22 at 7:41









user586231

1




1











  • Sorry if this question is dumb, I think maybe I am wrong at step 0
    – 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.
    – Berci
    Aug 22 at 8:11










  • @Berci Is it true that it preserves limits and colimits or something like that?
    – 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
    – Max
    Aug 22 at 8:40










  • Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
    – Jendrik Stelzner
    Aug 25 at 10:21
















  • Sorry if this question is dumb, I think maybe I am wrong at step 0
    – 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.
    – Berci
    Aug 22 at 8:11










  • @Berci Is it true that it preserves limits and colimits or something like that?
    – 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
    – Max
    Aug 22 at 8:40










  • Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
    – Jendrik Stelzner
    Aug 25 at 10:21















Sorry if this question is dumb, I think maybe I am wrong at step 0
– user586231
Aug 22 at 7:41




Sorry if this question is dumb, I think maybe I am wrong at step 0
– 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.
– 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.
– Berci
Aug 22 at 8:11












@Berci Is it true that it preserves limits and colimits or something like that?
– user586231
Aug 22 at 8:26




@Berci Is it true that it preserves limits and colimits or something like that?
– 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
– 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
– Max
Aug 22 at 8:40












Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
– Jendrik Stelzner
Aug 25 at 10:21




Possible duplicate of showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
– Jendrik Stelzner
Aug 25 at 10:21















active

oldest

votes











Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);








 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2890736%2ffunctor-preserving-long-exact-sequences%23new-answer', 'question_page');

);

Post as a guest



































active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes










 

draft saved


draft discarded


























 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2890736%2ffunctor-preserving-long-exact-sequences%23new-answer', 'question_page');

);

Post as a guest













































































這個網誌中的熱門文章

How to combine Bézier curves to a surface?

Mutual Information Always Non-negative

Why am i infinitely getting the same tweet with the Twitter Search API?