## Adjoint Functors (Pt. II)

**Point of Post: **This is a continuation of this post.

As a second example consider the forgetful functor . We claim that this functor is right adjoint with a left adjoint being the free module functor which takes a set to the free (left) -module and to each set map gives the -module map given by applying the universal property of free modules to the inclusion . To see this we define by merely taking the set map to the -map which results by applying the universal characterization of free modules. Ok, let’s go ahead and check that this is really natural in each entry. To do this we begin with a set map and we check that . To do this we let be any set map and be arbitrary. We then check that

and

Now, to show naturality in the other coordinate we let be any -map, we then need to check that . To do this we once again let be any set map and be arbitrary. We then need check that

and

Thus, we may conclude that as claimed.

As just a particular case of a more general theorem we have previously discussed the tensor functor and where is some commutative ring and is some -module (thought of in the usual way as an -bimodule) then .

I’m going to explicitly do one more example of an adjunction (well, truthfully two) but before I do I’d like to state some more without proof:

The forgetful functor (commuatative -algebras) is right adjoint to the polynomial ring functor defined by sending to the polynomial ring and to the -algebra map which extends by

.

The forgetful functor is right adjoint to the free group functor (which I assume everyone is familiar with).

The inclusion functor is right adjoint to the abelianization functor sending a group to its abelianization and sending a group map to the map we get from the characterization of applied to the composition latex \text{ }$

The forgetful functor (complete metric spaces and metric spaces both as full subcategories of ) has the completion functor as a left adjoint.

**References:
**

[1] Mac, Lane Saunders. *Categories for the Working Mathematician*. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. *Abstract and Concrete Categories: the Joy of Cats*. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. *Categories and Modules with K-theory in View*. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. *Abelian Categories.* New York: Harper & Row, 1964. Print.

[5] Mitchell, Barry. *Theory of Categories.* New York: Academic, 1965. Print.

[6] Herrlich, Horst, and George E. Strecker. *Category Theory: An Introduction*. Lemgo: Heldermann, 2007. Print.

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