Understanding Induced Maps: Types and Applications

[WWGD]

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.

 

[fresh_42]

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.

 

[WWGD]

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.

 

[fresh_42]

As $\mathbb{Z}-$modules?

 

[WWGD]

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.

 

[fresh_42]

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.