Types of Induced Maps

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Hi again, I am trying to get a better grasp of induced maps, and trying to see the
results that are used/assumed in defining these maps.

Until recently, I only knew of one type of induced map, described like this:

We have groups G, G' , with respective subgroups N,N' with ##N \triangleleft G , N'\triangleleft G'##and a homomorphism h:G→ G' . Then this somehow induces a map ##h_*##


##h_* G/N → G'/N' : h_*([a]_N ):=([h(a)]_N' ) ## , i.e., the coset class of a ( we can show the map is well-defined, i.e., it is independent of the choice of representative ) is sent to the coset class of the image ##h(a)## .

This is how induced maps in , e.g., homology, homotopy are defined, or where these maps come from.

But now I have run into some other induced maps that don't seem to have the same "source".

These are the maps:

1) We're given Abelian groups A,B , G, and a homomorphism ##f: A→ B##

Then this somehow induces a homomorphism ## f_*## with:

##f_*: = f\otimes 1 : A \times G → B \otimes G ## defined by:

## f_*(a,g):= f(a)\otimes g ##


2)Same setup, we have Abelian groups A,B,G, a homomorphism ##k: A→B## , then

we get the induced map ##k^*:=Hom(f,1): Hom(B,G)→ Hom(A,G) ## , defined by:

##k^*(\Phi)(a):= \Phi(k(a))## , for ##\Phi## in Hom(B,G), a in A.


I suspect the map 2 is just an extension to Abelian groups of the induced map on the duals of vector spaces:

Given a linear map ##L: V→ W ##, where V,W are finite-dimensional vector spaces over the
same field, we get the induced map ## L*: W*→ V* : L*(w*(w)):=w*(L(v) ##

Is that it? Where does the induced map 1) come from? I know both maps are functorial (1 is covariant and 2 is contravariant): are all induced maps functorial?

Thanks.
 
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Answers and Replies

  • #2
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Hi again, I am trying to get a better grasp of induced maps, and trying to see the
results that are used/assumed in defining these maps.

Until recently, I only knew of one type of induced map, described like this:

We have groups G, G' , with respective subgroups N,N' with ##N \triangleleft G , N'\triangleleft G'##and a homomorphism h:G→ G' . Then this somehow induces a map ##h_*##


##h_* G/N → G'/N' : h_*([a]_N ):=([h(a)]_N' ) ## , i.e., the coset class of a ( we can show the map is well-defined, i.e., it is independent of the choice of representative ) is sent to the coset class of the image ##h(a)## .

This is how induced maps in , e.g., homology, homotopy are defined, or where these maps come from.
I think you need ##h(N)\subseteq N'## as well.
But now I have run into some other induced maps that don't seem to have the same "source".

These are the maps:

1) We're given Abelian groups A,B , G, and a homomorphism ##f: A→ B##

Then this somehow induces a homomorphism ## f_*## with:

##f_*: = f\otimes 1 : A \times G → B \otimes G ## defined by:

## f_*(a,g):= f(a)\otimes g ##


2)Same setup, we have Abelian groups A,B,G, a homomorphism ##k: A→B## , then

we get the induced map ##k^*:=Hom(f,1): Hom(B,G)→ Hom(A,G) ## , defined by:

##k^*(\Phi)(a):= \Phi(k(a))## , for ##\Phi## in Hom(B,G), a in A.


I suspect the map 2 is just an extension to Abelian groups of the induced map on the duals of vector spaces:

Given a linear map ##L: V→ W ##, where V,W are finite-dimensional vector spaces over the
same field, we get the induced map ## L*: W*→ V* : L*(w*(w)):=w*(L(v) ##

Is that it?
Basically.
Where does the induced map 1) come from?
What is ##\otimes## in this context? Certainly no tensor product and equal to ##\times##. In this case, where is the problem. We always can extend a mapping by any direct terms on which we do nothing at all.
I know both maps are functorial (1 is covariant and 2 is contravariant): are all induced maps functorial?
No, Induced isn't a property in itself. It is more like "extends naturally" than something which can be uniquely defined. I searched "induced homomorphism" on nLab and many, many examples showed up, but not a definition.
 
  • #3
WWGD
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I think you need ##h(N)\subseteq N'## as well.
Basically.
What is ##\otimes## in this context? Certainly no tensor product and equal to ##\times##. In this case, where is the problem. We always can extend a mapping by any direct terms on which we do nothing at all.
No, Induced isn't a property in itself. It is more like "extends naturally" than something which can be uniquely defined. I searched "induced homomorphism" on nLab and many, many examples showed up, but not a definition.
Here ##\otimes ## is the tensor product of (Abelian, I believe) groups. And I assume the extensions have some special properties we care to extend or define.
 
  • #5
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As ##\mathbb{Z}-##modules?
I guess, but not sure. I don't know if there is a definiton of tensor product for general groups. EDIT: I was surprised to see that we can even define a tensor product for Chain Complexes in Singular, IIRC, Homology. Edit2: I believe thete isca tensor product for topological spaces without additional structure.
 
  • #6
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I guess, but not sure. I don't know if there is a definiton of tensor product for general groups. EDIT: I was surprised to see that we can even define a tensor product for Chain Complexes in Singular, IIRC, Homology. Edit2: I believe thete isca tensor product for topological spaces without additional structure.
Me neither. I have no idea what it should mean in such a case, esp. what would make it different from a direct product? My definition explicitly says modules. On the other hand, the tensor product is defined as the solution of a co-universal mapping problem, and this only requires a covariant and contravariant functor on the product category into Set to set it up. But maybe it cannot be solved in general.
 

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