# Abstract Nonsense

## Crushing one theorem at a time

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

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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

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\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}

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and

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\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}

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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

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

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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}

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

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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 $\bullet\otimes_R N\dashv \text{Hom}_R(N,\bullet)$.

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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:

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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

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

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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).

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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.

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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.