Understanding the definition of left derived functor Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)How does slice category help create functor?Covariant functor, and left exactWhat is the trivial module functor?Confusion regarding definition of adjoint functor - Hilton and StammbachEilenberg-Mac Lane functor being an embeddingWhat is an alternating functor?Problem with functor Sym and OrdExistence of adjugant with making given natural transformation be the counitDefining contravariant left/right-exact functor with opposite category?Derived functors and induced functors
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Understanding the definition of left derived functor
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)How does slice category help create functor?Covariant functor, and left exactWhat is the trivial module functor?Confusion regarding definition of adjoint functor - Hilton and StammbachEilenberg-Mac Lane functor being an embeddingWhat is an alternating functor?Problem with functor Sym and OrdExistence of adjugant with making given natural transformation be the counitDefining contravariant left/right-exact functor with opposite category?Derived functors and induced functors
$begingroup$
I'm studying homological algebra. But I stuck in a place in Rotman's Advanced Algebra II, 3rd ed. What does $Thatf$ mean in the paragraph? The point made me confused is that $T$ is a functor from $_RtextbfMod$ to $_StextbfMod$, how can $T$ apply on the chain map $hatf$? Is the symbol the author used problematic? And what does the $Thatf$ really mean in the context?
abstract-algebra category-theory homological-algebra
asked Apr 1 at 13:57
EricEric
1,846615
$endgroup$
add a comment |
$begingroup$
I'm studying homological algebra. But I stuck in a place in Rotman's Advanced Algebra II, 3rd ed. What does $Thatf$ mean in the paragraph? The point made me confused is that $T$ is a functor from $_RtextbfMod$ to $_StextbfMod$, how can $T$ apply on the chain map $hatf$? Is the symbol the author used problematic? And what does the $Thatf$ really mean in the context?
abstract-algebra category-theory homological-algebra
asked Apr 1 at 13:57
EricEric
1,846615
$endgroup$
1
$begingroup$
$T$ takes $R$-modules like each $P_A$ and $P'_A'$ to $S$-modules like $TP_A$ and $TP'_A'$. The chain map $checkf: P_A to P'_A'$. So how to denote the induced map from $TP_A to TP'_A'$? $Tcheckf$ seems a natural choice (with some natural abuse of notation)
$endgroup$
– bounceback
Apr 1 at 17:31
add a comment |
$begingroup$
I'm studying homological algebra. But I stuck in a place in Rotman's Advanced Algebra II, 3rd ed. What does $Thatf$ mean in the paragraph? The point made me confused is that $T$ is a functor from $_RtextbfMod$ to $_StextbfMod$, how can $T$ apply on the chain map $hatf$? Is the symbol the author used problematic? And what does the $Thatf$ really mean in the context?
abstract-algebra category-theory homological-algebra
asked Apr 1 at 13:57
EricEric
1,846615
$endgroup$
I'm studying homological algebra. But I stuck in a place in Rotman's Advanced Algebra II, 3rd ed. What does $Thatf$ mean in the paragraph? The point made me confused is that $T$ is a functor from $_RtextbfMod$ to $_StextbfMod$, how can $T$ apply on the chain map $hatf$? Is the symbol the author used problematic? And what does the $Thatf$ really mean in the context?
abstract-algebra category-theory homological-algebra
abstract-algebra category-theory homological-algebra
asked Apr 1 at 13:57
EricEric
1,846615
asked Apr 1 at 13:57
EricEric
1,846615
edited Apr 1 at 17:17
Eric
asked Apr 1 at 13:57
EricEric
1,846615
asked Apr 1 at 13:57
EricEric
1,846615
asked Apr 1 at 13:57
EricEric
1,846615
1,846615
1
$begingroup$
$T$ takes $R$-modules like each $P_A$ and $P'_A'$ to $S$-modules like $TP_A$ and $TP'_A'$. The chain map $checkf: P_A to P'_A'$. So how to denote the induced map from $TP_A to TP'_A'$? $Tcheckf$ seems a natural choice (with some natural abuse of notation)
$endgroup$
– bounceback
Apr 1 at 17:31
add a comment |
1
$begingroup$
$T$ takes $R$-modules like each $P_A$ and $P'_A'$ to $S$-modules like $TP_A$ and $TP'_A'$. The chain map $checkf: P_A to P'_A'$. So how to denote the induced map from $TP_A to TP'_A'$? $Tcheckf$ seems a natural choice (with some natural abuse of notation)
$endgroup$
– bounceback
Apr 1 at 17:31
1
1
$begingroup$
$T$ takes $R$-modules like each $P_A$ and $P'_A'$ to $S$-modules like $TP_A$ and $TP'_A'$. The chain map $checkf: P_A to P'_A'$. So how to denote the induced map from $TP_A to TP'_A'$? $Tcheckf$ seems a natural choice (with some natural abuse of notation)
$endgroup$
– bounceback
Apr 1 at 17:31
$begingroup$
$T$ takes $R$-modules like each $P_A$ and $P'_A'$ to $S$-modules like $TP_A$ and $TP'_A'$. The chain map $checkf: P_A to P'_A'$. So how to denote the induced map from $TP_A to TP'_A'$? $Tcheckf$ seems a natural choice (with some natural abuse of notation)
$endgroup$
– bounceback
Apr 1 at 17:31
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It is an abuse of notation : $T$ induces an obvious functor from chain complexes to chain complexes, and the author is just using the same notation for the induced functor.
answered Apr 1 at 17:30
MaxMax
16.3k11144
$endgroup$
add a comment |
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$begingroup$
It is an abuse of notation : $T$ induces an obvious functor from chain complexes to chain complexes, and the author is just using the same notation for the induced functor.
answered Apr 1 at 17:30
MaxMax
16.3k11144
$endgroup$
add a comment |
$begingroup$
It is an abuse of notation : $T$ induces an obvious functor from chain complexes to chain complexes, and the author is just using the same notation for the induced functor.
answered Apr 1 at 17:30
MaxMax
16.3k11144
$endgroup$
add a comment |
$begingroup$
It is an abuse of notation : $T$ induces an obvious functor from chain complexes to chain complexes, and the author is just using the same notation for the induced functor.
answered Apr 1 at 17:30
MaxMax
16.3k11144
$endgroup$
It is an abuse of notation : $T$ induces an obvious functor from chain complexes to chain complexes, and the author is just using the same notation for the induced functor.
answered Apr 1 at 17:30
MaxMax
16.3k11144
answered Apr 1 at 17:30
MaxMax
16.3k11144
answered Apr 1 at 17:30
MaxMax
16.3k11144
answered Apr 1 at 17:30
MaxMax
16.3k11144
16.3k11144
add a comment |
add a comment |
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$begingroup$
$T$ takes $R$-modules like each $P_A$ and $P'_A'$ to $S$-modules like $TP_A$ and $TP'_A'$. The chain map $checkf: P_A to P'_A'$. So how to denote the induced map from $TP_A to TP'_A'$? $Tcheckf$ seems a natural choice (with some natural abuse of notation)
$endgroup$
– bounceback
Apr 1 at 17:31