# Abstract Nonsense

## Crushing one theorem at a time

Point of Post: In this post we discuss the notion of adjoint functors, giving the two definitions via both Hom set adjunction and counit-unit adjunction.

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Motivation

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As is standard when talking about adjoint functors we begin with a quote by the late Saunders Mac Lane: “The slogan is ‘Adjoint functors arise everywhere'”. Mr. Mac Lane was surely not lying because (as I hope is clear by the end of this post) some of the functors we are most well-acquainted with are left or right adjoints. What exactly are adjoint functors? While there are tons-and-tons of motivations for what these ubiquitous little demons are, there is one though that stands forefront in my mind. The idea is that adjoint functors are kind of like generalized inverses–in the sense that while they are not actually invertible, they share many of the same functional properties of invertible functors. Namely, let’s assume that we have a functor $F:\mathcal{C}\to\mathcal{D}$. It is entirely unreasonable to assume that this is a literal isomorphism (it has a two-sided functor inverse). It is slightly less unreasonable to hope that $F$ is going to be an equivalence, which means that $FG\simeq 1_\mathcal{D}$ and $GF\simeq 1_\mathcal{C}$ for some functor $G:\mathcal{D}\to\mathcal{C}$. That said, having an equivalence of categories is a BIG deal thus we shouldn’t expect the average functor on the street to be an equivalence. Moreover, in both of the cases of a functor being an isomorphism (having a literal inverse) and being an equivalence the focus is really more on the categories. If I said to you that $F:\mathcal{C}\to\mathcal{D}$ is an isomorphism or $F:\mathcal{C}\to\mathcal{D}$ probably the most important thing that jumps to mind is “$\mathcal{C}$ and $\mathcal{D}$ are isomorphic” or $\mathcal{C}$ and $\mathcal{D}$ are equivalent”. Thus, if we are looking for generalizations of the functional properties of invertible or near invertible functors perhaps it behooves us to shy away from looking directly at the categories and instead look at how we want our functors to act on “elements” of the categories. Namely, let’s sit here for a second and try to think of what is a very desirable property that an invertible functor has.

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Well, recalling our mantra that we should only care about the morphisms in a category it seems then that a step in the discovery of this desirable is to figure out what invertible or near invertible functors do to morphisms. Well, let’s just mess around with the idea and see what comes up. Well, what we know is that there is a natural isomorphism $\eta_X:X\xrightarrow{\approx}G(F(X))$ for each $X$ an object $\mathcal{C}$. Ok, so, morphisms, morphisms. Hmm, well what does this tell us about looking at morphisms between $X$ and other objects of $Z$ $\mathcal{C}$. Ok, so we want to see what we can say about $\text{Hom}_\mathcal{C}(X,Z)$. Hmm, well nothing immediately jumps out. But, there is something we can do which is pretty nice. Since our $G$ is an equivalence $\mathcal{D}\to\mathcal{C}$ it is easy to prove that $Z$ must be isomorphic to some object $G(Y)$ in the image of $G$. Thus, we want to figure out what our equivalence $F:\mathcal{C}\to\mathcal{D}$ enable us to say about $\text{Hom}_\mathcal{C}(X,Z)\cong\text{Hom}_\mathcal{C}(X,G(Y))$. Well, since $\eta_X:X\xrightarrow{\approx}G(F(X))$ there is no harm in replacing $X$ with $G(F(X))$ so that we are really trying to figure out what we can say about $\text{Hom}_\mathcal{C}(G(F(X)),G(Y))$. But the cool thing is that since $G$ is an equivalence we have that $G$ induces bijections on Hom sets, and thus $\text{Hom}_\mathcal{C}(X,G(Y))\cong\text{Hom}_\mathcal{C}(G(F(X)),G(Y))\cong\text{Hom}_\mathcal{D}(F(X),Y)$. Moreover, if one follows the details of the construction above one can prove that this isomorphism is natural in both $X$ and $Y$.

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While this is clearly a very nice property for two functors to have it is not at all clear that it’s the correct generlization of invertible functors. Hopefully though, the ubiquitousness of functors satisfying this property (as illustrated below) will convince the reader.

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Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{C}$ to functors. We say that $(F,G)$ is an adjoint pair, denoted $F\dashv G$, if for each object $X$ in $\mathcal{C}$ and $Y$ in $\mathcal{D}$ there exists a bijection

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$\eta_{X,Y}:\text{Hom}_\mathcal{C}(X,G(Y))\xrightarrow{\approx}\text{Hom}_\mathcal{D}(F(X),Y)$

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which is natural in each variable–diagrammatically this means the following two diagrams commute

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$\begin{array}{ccc}\text{Hom}_\mathcal{C}(X,G(Y)) & \overset{\eta_{X,Y}}{\longrightarrow} & \text{Hom}_\mathcal{D}(F(X),Y)\\ ^{f^\ast}\big\downarrow & & \big\downarrow^{F(f)^\ast}\\ \text{Hom}_\mathcal{C}(X',G(Y)) & \underset{\eta_{X',Y}}{\longrightarrow} & \text{Hom}_\mathcal{D}(F(X'),Y)\end{array}$

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$\begin{array}{ccc}\text{Hom}_\mathcal{C}(X,G(Y)) & \overset{\eta_{X,Y}}{\longrightarrow} & \text{Hom}_\mathcal{D}(F(X),Y)\\ ^{g_\ast}\big\downarrow & & \big\downarrow^{G(g)_\ast}\\ \text{Hom}_\mathcal{C}(X,G(Y')) & \underset{\eta_{X,Y'}}{\longrightarrow} & \text{Hom}_\mathcal{D}(F(X),Y')\end{array}$

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for each arrow $X'\xrightarrow{f}X$ and each arrow $Y\xrightarrow{g}Y'$. In this case we say that $G$ is right adjoint to $F$ and $F$ is left adjoint to $G$ If $F:\mathcal{C}\to\mathcal{D}$ is any functor, we say $F$ is left adjoint if there exists $G:\mathcal{D}\to\mathcal{C}$ with $F\dashv G$. We similarly define a right adjoint functor.

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Let’s give some examples of adjoint functors:

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Let $\mathbf{Dom}$ be the category of integral domains with injective ring maps as morphisms. Since every ring map between fields is necessarily injective we see that there is a forgetful functor $U:\mathbf{Field}\to\mathbf{Dom}$. We claim that $U$ is right adjoint. What this means, again, is that we need to be able to find a functor $\mathbf{Dom}\to\mathbf{Field}$ which is left adjoint to $U$. Indeed, the functor we are seeking is the field of fraction functor $\text{Frac}:\mathbf{Dom}\to\mathbf{Field}$ which sends every integral domain $R$ to its field of fractions $\text{Frac}(R)$ and sends every injective ring map $R\to S$ to the induced ring map $\text{Frac}(R)\to\text{Frac}(S)$ guaranteed composing the map $\text{Frac}(R)\to S$ guaranteed by the universal property of fraction fields with the localization map $S\hookrightarrow \text{Frac}(S)$. Now to show that $\text{Frac}\dashv U$ we need to show that there is an isomorphism

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$\text{Hom}(R,U(F))\cong \text{Hom}(\text{Frac}(R),F)$

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which is natural in both $R$ and $F$. To see this merely take $\varphi:R\to U(F)=F$ to $\widetilde{\varphi}:\text{Frac}(R)\to F$ which is just the guaranteed map $\widetilde{\varphi}:\text{Frac}(R)\to\text{Frac}(F)$ after we make the identification $\text{Frac}(F)\cong F$. To prove that this is natural in both entries we first check that if we have an injective ring map $f:S\to R$ that $\eta_{S,F}\circ f^\ast=\text{Frac}(f)^\ast\circ\eta_{R,F}$. To do this take a $\varphi\in\text{Hom}(R,U(F))$ and some $\displaystyle \frac{x}{y}\in\text{Frac}(S)$, we then check that

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$\displaystyle (\eta_{S,F}\circ f^\ast)(\varphi)\left(\frac{x}{y}\right)=\eta_{S,F}(f^\ast(\varphi))\left(\frac{x}{y}\right)=\frac{f^\ast(\varphi)(x)}{f^\ast(\varphi)(y)}=\frac{\varphi(f(x))}{\varphi(f(y))}$

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and

\displaystyle \begin{aligned}\left(\text{Frac}(f)^\ast\circ\eta_{R,F}\right)(\varphi)\left(\frac{x}{y}\right)&=\text{Frac}(f)^\ast(\widetilde{\varphi})\left(\frac{x}{y}\right)\\ &=\widetilde{\varphi}\left(\text{Frac}(f)\left(\frac{x}{y}\right)\right)\\ &=\widetilde{\varphi}\left(\frac{f(x)}{f(y)}\right)\\ &=\frac{\varphi(f(x))}{\varphi(f(y))}\end{aligned}

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To check the other naturality we assume we have some field map $g:F\to F'$, we then need to check that $\eta_{R,F'}\circ g_\ast=g^\ast\circ\eta_{R,F}$ (since $U(g)=g$). To do this we again let $\varphi$ be arbitrary in $\text{Hom}(R,U(F))$ and check that

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$\displaystyle (\eta_{R,F'}\circ g_\ast)(\varphi)\left(\frac{x}{y}\right)=\eta_{R,F'}\left(g_\ast(\varphi)\right)\left(\frac{x}{y}\right)=\frac{g_\ast(\varphi)(x)}{g_\ast(\varphi)(y)}=\frac{g(\varphi(x))}{g(\varphi(y))}$

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and

$\displaystyle (g_\ast\circ\eta_{R,F})(\varphi)\left(\frac{x}{y}\right)=g_\ast(\widetilde{\varphi})\left(\frac{x}{y}\right)=\left(g\circ\widetilde{\varphi}\right)\left(\frac{x}{y}\right)=g\left(\frac{\varphi(x)}{\varphi(y)}\right)=\frac{g(\varphi(x))}{g(\varphi(y))}$

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Thus, we see that $\text{Frac}\dashv U$ as desired.

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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.

April 15, 2012 -