Abstract Nonsense

Crushing one theorem at a time

Adjoint Functors (Pt. II)


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

\text{ }

As a second example consider the forgetful functor U:R\text{-}\mathbf{Mod}\to\mathbf{Set}. We claim that this functor is right adjoint with a left adjoint being the free module functor F:\mathbf{Set}\to R\text{-}\mathbf{Mod} which takes a set X to the free (left) R-module R^{\oplus X} and to each set map f:X\to Y gives the R-module map F(f):R^{\oplus X}\to R^{\oplus Y} given by applying the universal property of free modules to the inclusion X\to Y\hookrightarrow R^{\oplus Y}. To see this we define \eta_{X,M}:\text{Hom}_\mathbf{Set}(X,U(M))\xrightarrow{\approx}\text{Hom}_R(R^{\oplus X},M) by merely taking the set map \varphi to the R-map \widetilde{\varphi}:R^{\oplus X}\to M 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 f:Y\to X and we check that \eta_{Y,M}\circ f^\ast=F(f)^\ast\circ \eta_{X,M}. To do this we let \varphi be any set map X\to U(M)=M and (r_y)\in R^{\oplus Y} be arbitrary. We then check that

\text{ }

\displaystyle \begin{aligned}(\eta_{Y,M}\circ f^\ast)(\varphi)((r_y)) &=\eta_{Y,M}(f^\ast(\varphi))((r_y))\\ &=\eta_{Y,M}(\varphi\circ f)((r_y))\\ &=\sum_{y\in Y}r_y(\varphi\circ f)(y)\\ &=\sum_{y\in Y}r_y \varphi(f(y))\end{aligned}

\text{ }

and

\text{ }

\displaystyle \begin{aligned}(F(f)^\ast\circ\eta_{X,M})(\varphi)((r_y)) &= F(f)^\ast(\widetilde{\varphi})((r_y))\\ &= \widetilde{\varphi}(F(f)((r_y)))\\ &= \widetilde{\varphi}\left(\sum_{y\in Y}r_y (\delta_{f(y),x})\right)\\ &=\sum_{y\in Y}r_y \varphi(f(y))\end{aligned}

\text{ }

Now, to show naturality in the other coordinate we let g:M\to N be any R-map, we then need to check that \eta_{X,N}\circ g_\ast=g_\ast\circ \eta_{X,M}. To do this we once again let \varphi be any set map X\to U(M)=M and (r_x)\in R^{\oplus X} be arbitrary. We then need check that

\text{ }

\displaystyle (\eta_{X,N}\circ g_\ast)(\varphi)((r_x))=\eta_{X,N}(g\circ\varphi)(r_x)=\sum_{x\in X}r_x g(\varphi(x))

\text{ }

and

\displaystyle \begin{aligned}(g_\ast\circ \eta_{X,M})(\varphi)((r_x)) &=(g\circ \widetilde{\varphi})((r_x))\\ &=g(\widetilde{\varphi}((r_x)))\\ &=g\left(\sum_{x\in X}r_x \varphi(x)\right)\\ &=\sum_{x\in X}r_x g(\varphi(x))\end{aligned}

\text{ }

Thus, we may conclude that F\dashv U as claimed.

\text{ }

\text{ }

As just a particular case of a more general theorem we have previously discussed the tensor functor \bullet\otimes_R N:R\text{-}\mathbf{Mod}\to R\text{-}\mathbf{Mod} and \text{Hom}_R(N,\bullet):\text{R}\text{-}\mathbf{Mod}\to R\text{-}\mathbf{Mod} where R is some commutative ring and N is some R-module (thought of in the usual way as an (R,R)-bimodule) then \bullet\otimes_R N\dashv \text{Hom}_R(N,\bullet).

\text{ }

\text{ }

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:

\text{ }

The forgetful functor U:R\text{-}\mathbf{CAlg}\to\mathbf{Set} (commuatative R-algebras) is right adjoint to the polynomial ring functor \mathbf{Set}\to R\text{-}\mathbf{CAlg} defined by sending S to the polynomial ring R[S] and f:S\to T to the R-algebra map R[S]\to R[T] which extends S\to T by

\text{ }

\displaystyle \sum_{s\in S}\sum_{i}r_{i,s}s^i\mapsto \sum_{s\in S}\sum_i r_{i,s}f(s).

\text{ }
The forgetful functor U:\mathbf{Grp}\to\mathbf{Set} is right adjoint to the free group functor \mathbf{Grp}\to\mathbf{Set} (which I assume everyone is familiar with).

\text{ }

The inclusion functor \mathbf{Ab}\to\mathbf{Grp} is right adjoint to the abelianization functor \mathbf{Grp}\to\mathbf{Ab} sending a group G to its abelianization and sending a group map G\to H to the map G^{\text{ab}}\to H^{\text{ab}} we get from the characterization of G^\text{ab} applied to the composition G\to H\xrightarrow{\pi} H^{\text{ab}}.    latex \text{ }$

The forgetful functor U:\mathbf{CompMet}\to\mathbf{Met} (complete metric spaces and metric spaces both as full subcategories of \mathbf{Top}) has the completion functor \mathbf{Met}\to\mathbf{CompMet} as a left adjoint.

\text{ }

\text{ }

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.


Advertisements

April 15, 2012 - Posted by | Algebra, Category Theory | , , , , , ,

2 Comments »

  1. […] Adjoint Functors (Pt. III) Point of Post: This is a continuation of this post. […]

    Pingback by Adjoint Functors (Pt. III) « Abstract Nonsense | April 15, 2012 | Reply

  2. […] limit functor was left exact. This now follows immediately since this is just a limit functor which we know is a right adjoint to the diagonal functor, and so continuous, and so finitely continuous and so […]

    Pingback by Left Exact, Right Exact, and Exact Functors « Abstract Nonsense | April 24, 2012 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: