# Abstract Nonsense

## Crushing one theorem at a time

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

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The Limit and Colimit Functors

As a last example we are going to assume we have a category $\mathcal{C}$ and a small category $\mathcal{I}$ for which every diagram $F:\mathcal{I}\to\mathcal{C}$ has a limit and a colimit. We want to define functors $\lim:\mathcal{C}^\mathcal{I}\to\mathcal{C}$ and $\text{colim}:\mathcal{C}^\mathcal{I}\to\mathcal{C}$ where $\mathcal{C}^\mathcal{I}$ is the functor category. We already know how this should act on objects–it just takes a diagram $F:\mathcal{I}\to\mathcal{C}$ and assigns to it either $\lim F$ or $\text{colim }F$. The question then is exactly how to figure out what the direct limit of a morphism should be.

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To this end suppose that we have an arrow $\eta:F\implies G$ in $\mathcal{C}^\mathcal{I}$. We then have arrows $F(i)\xrightarrow{\eta_i}G(i)$ for each $i$ an object of $\mathcal{I}$. Note then though that we have arrows $\lim F\xrightarrow{\alpha_i}F(i)\xrightarrow{\eta_i}G(i)$ which by the definition of limits gives us a unique arrow $\lim F\xrightarrow{\lim \eta}\lim G$ with $\beta_i\circ\lim\eta=\eta_i\circ\alpha_i$ (where of course, the $\alpha_i$ and $\beta_i$ are the cone maps associated to $\lim F$ and $\lim G$ respectively). Of course, it’s easy to see that if we have the composition $\eta\circ\varepsilon$ of two natural transformations $F\overset{\varepsilon}{\implies}G\overset{\eta}{\implies}H$ then $\lim(\varepsilon\circ\eta)$ satisfies

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\begin{aligned}\gamma_i\circ\lim(\eta\circ\varepsilon) &=(\eta_i\circ\varepsilon_i)\circ\alpha_i\\ &=\eta_i\circ(\varepsilon_i\circ\alpha_i)\\ &=\eta_i\circ(\beta_i\circ\lim\varepsilon)\\ &= (\eta_i\circ\beta_i)\circ\lim\varepsilon\\ &= (\gamma_i\circ\lim\eta)\circ\lim\varepsilon\\ &=\gamma_i\circ(\lim\eta\circ\lim\varepsilon)\end{aligned}

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and since $\lim(\eta\circ\varepsilon)$ and $\lim\eta\circ\lim\varepsilon$ agree on $\gamma_i$ (obviously the cone maps $\lim H\to H(i)$) we may conclude that they are equal. Noticing finally that if $1_F:F\implies F$ then $\alpha_i\circ\lim 1_F=1_{F(i)}\circ \alpha_i=\alpha_i=\alpha_i\circ 1_{\lim F}$ and thus $\lim 1_F=1_{\lim F}$. Thus, we may conclude that, with these definitions, $\lim:\mathcal{C}^\mathcal{I}\to\mathcal{C}$ is a functor.

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Very similarly we may construct a functor $\text{colim}:\mathcal{C}^\mathcal{I}\to\mathcal{C}$ which on objects is obviously $\text{colim }F$ and on arrows sends $F\overset{\eta}{\implies}G$ to the unique arrow $\text{colim }F\xrightarrow{\text{colim }\eta}\text{colim }G$ with $\text{colim }\eta\circ\nu_i=\omega_i\circ\eta_i$ where $G(i)\xrightarrow{\omega_i}\text{colim }G$ and $F(i)\xrightarrow{\nu_i}\lim F$ are the cocone maps.

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Ok, so with all of this defined we’d like to discuss a certain series of adjunctions involving these two functors and another one we’ve seen before. Namely, define the diagonal functor $\Delta:\mathcal{C}\to\mathcal{C}^\mathcal{I}$ by taking each object $X$ in $\mathcal{C}$ to the functor $\Delta_X:\mathcal{I}\to\mathcal{C}$ with $\Delta_X(i)=X$ for all $i$ and $\Delta_X(s)=1_X$ for all arrows $i\xrightarrow{s}j$, and taking each arrow $X\xrightarrow{f}Y$ to the natural transformation $\Delta_X\overset{\Delta_f}{\implies}\Delta_Y$ with $X\xrightarrow{\eta_i=f}Y$ for all $i$ (it’s easy to verify this really is a natural transformation).

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In particular, we claim that $\text{colim}\dashv\Delta\dashv\lim$.

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To show that $\text{colim }\dashv \Delta$ we being by defining the isomorphisms $\Psi_{F,X}:\text{Hom}_{\mathcal{C}^\mathcal{I}}(F,\Delta_X)\to\text{Hom}_\mathcal{C}(\text{colim }F,X)$. How do we define such an isomorphism? Well, what exactly is a natural transformation $F\overset{\varepsilon}{\implies}\Delta_X$? Well, it’s a set of maps $F(i)\xrightarrow{\varepsilon_i}\Delta_X(i)=X$–and that’s it, since the commutativity is trivial since $\Delta_X(s)=1_X$ for all morphisms $s$. Ok, great! Clearly then we can make $\Psi_{F,X}$ into a bijection by merely defining $\Psi_{F,X}(\varepsilon)$ to be the unique arrow $\text{colim }F\to X$ with $\Psi_{F,X}(\varepsilon)\circ\nu_i=\varepsilon_i$. Let’s then verify that these isomorphisms are natural in each entry. To this end let $\eta:G\implies F$ be some natural transformation. We must then verify that $\Psi_{G,X}\circ\eta^\ast=(\text{colim }\eta)^\ast\circ\Psi_{F,X}$. To do this let $F\overset{\varepsilon}{\implies}\Delta_X$ be some natural transformation we see then that

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$(\Psi_{G,X}\circ\eta^\ast)(\varepsilon)\circ\omega_i=\Psi_{G,X}(\varepsilon\circ\eta)\circ\omega_i=\varepsilon_i\circ\eta_i$

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and

\begin{aligned}((\text{colim }\eta)^\ast\circ\Psi_{F,X})(\varepsilon)\circ\omega_i &=\Psi_{F,X}(\varepsilon)\circ\text{colim }\eta\circ\omega_i\\ &=\Psi_{F,X}(\varepsilon)\circ \nu_i\circ\eta_i\\ &=\varepsilon_i\circ\eta_i\end{aligned}

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and since our desired maps agree on $\omega_i$ they must be equal.

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Now, for naturality in the other variable let $X\xrightarrow{f}Y$ be an arrow in $\mathcal{C}$, we must verify then that $f^\ast\circ\Psi_{F,X}=\Psi_{F,Y}\circ\Delta(f)^\ast$. To this end, let $F\overset{\varepsilon}{\implies}\Delta_X$  be a natural transformation. We check then that

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$(f^\ast\circ\Psi_{F,X})(\varepsilon)\circ\nu_i=f\circ\Psi_{F,X}(\varepsilon)\circ\nu_i=f\circ\varepsilon_i$

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and

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$(\Psi_{F,Y}\circ\Delta(f)^\ast)(\varepsilon)\circ\nu_i=\Psi_{F,Y}(\Delta(f)\circ\varepsilon)\circ\nu_i=f\circ\varepsilon_i$

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Thus, we see that $\text{colim }\dashv\Delta$ as claimed.

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Of course, dualizing the proof one can show that $\Delta\dashv\lim$ and thus we may finally conclude that:

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Theorem: Let $\mathcal{C}$ be a category which admits all limits (resp. colimits) for diagrams $F:\mathcal{I}\to\mathcal{C}$. Then, the functor $\lim$ (resp. $\text{colim}$) is right adjoint (resp. left adjoint).

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As a last point we note that right or left adjoints to a given functor are necessarily unique. Indeed, if $F\dashv G$ and $F\dashv H$ then we see that

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$\text{Hom}(X,G(Y))\cong\text{Hom}(F(X),Y)\cong\text{Hom}(X,H(Y))$

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for all objects $X,Y$. We see then by Yoneda’s lemma that $G(Y)\cong H(Y)$ naturally for all objects $Y$. So, $G\cong H$. The other cases are similar.

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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.