# Abstract Nonsense

## Preordered Sets as Categories, and their Functor Categories (Pt. II)

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

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Functors and Functor Categories out of Preordered Sets and

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Perhaps one of the most useful things thinking about preordered sets as categories gives us is how we can interpret functors from them into other, general categories. Indeed, suppose that we have a preordered set $\mathcal{I}=(I,\leqslant)$ and some other category $\mathcal{C}$. What does a functor $F:\mathcal{I}\to\mathcal{C}$ look like? Well, we can think about the object mapping $F:I\to\text{Obj}(\mathcal{C})$ as just being an indexed set $\{x_\alpha\}_{\alpha\in I}$ of $\mathcal{C}$-objects. What about the arrow mapping induced by $F$? Well, we see that given any $\alpha\leqslant\beta$ in $I$ we have an arrow $\alpha\xrightarrow{f}\beta$ in $\mathcal{I}$ which then produces an arrow $x_\alpha\xrightarrow{F(f)}x_\beta$ in $\mathcal{C}$. Now, since there is a UNIQUE arrow $\alpha\to\beta$ there is no harm in denoting $F(f)$ as $f_{\alpha,\beta}$ since there is no other arrow $F(\alpha)\to F(\beta)$ that is the image of an arrow under $F$. Thus, we see that as an arrow mapping is just an indexed set of maps $\{f_{\alpha,\beta}:x_\alpha\to x_\beta\}_{\alpha,\beta\in I,\;\alpha\leqslant\beta}$. But, not any old indexed set of maps will do, this indexed set of maps has to satisfy the functor properties. Indeed, the fact that $F(1_\alpha)=1_{x_\alpha}$ is equivalent to $f_{\alpha,\alpha}$ and the fact that if $\alpha\leqslant\beta\leqslant\gamma$ that $k_{\beta,\gamma}\circ k_{\alpha,\beta}=k_{\alpha,\gamma}$ is equivalent to $f_{\beta,\gamma}\circ f_{\alpha,\beta}=f_{\alpha,\gamma}$. So, all in all, we may provocatively identity functors $F:\mathcal{I}\to\mathcal{C}$ with orderered pairs $\left(\{x_\alpha\}_{\alpha\in I},\{x_\alpha\xrightarrow{f_{\alpha,\beta}}x_\beta\}_{\alpha,\beta\in I,\; \alpha\leqslant\beta}\right)$ which satisfy the following two axioms:

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\begin{aligned}&\mathbf{(1)}\quad f_{\alpha,\alpha}=1_{x_\alpha}\text{ for all }\alpha\in I\\&\mathbf{(2)}\quad f_{\beta,\gamma}\circ f_{\alpha,\beta}=f_{\alpha,\gamma}\text{ whenenver }\alpha,\beta,\gamma\in I\text{ and }\alpha\leqslant\beta\leqslant\gamma\end{aligned}

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But, this should look astonishingly familiar! Indeed, if $\mathcal{C}$ is $R\text{-}\mathbf{Mod}$ this is just the definition of a directed system of modules over $(I,\leqslant)$! So, we can basically identify directed systems of modules over some preordered set $\mathcal{I}=(I,\leqslant)$ as being functors $F:\mathcal{I}\to R\text{-}\mathbf{Mod}$. Of course, this allows us to readily generalized and define a directed system in $\mathcal{C}$ over $\mathcal{I}=(I,\leqslant)$ to be a functor $F:\mathcal{I}\to\mathcal{C}$ where we obviously make the identifications between the actual functor and the ordered pair with conditions on the morphisms as listed above.

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Now, since we have identified functors $F:\mathcal{I}\to R\text{-}\mathbf{Mod}$ as being just directed systems, it’s seems natural to ask if whether the functor category $R\text{-}\mathbf{Mod}^\mathcal{I}$ of all functors $\mathcal{I}\to R\text{-}\mathbf{Mod}$ is the same as the category $\mathbf{Dir}_\mathcal{I}(R\text{-}\mathbf{Mod})$ of directed systems in $R\text{-}\mathbf{Mod}$ over $\mathcal{I}$. Basically this entails showing that the morphisms in $\mathbf{Dir}_\mathcal{I}(R\text{-}\mathbf{Mod})$ are really just natural transformations between the associated functors in the sense just described. Indeed, by definition a morphism between $(\{M_\alpha\},\{f_{\alpha,\beta}\})$ and $(\{N_\alpha\},\{g_{\alpha,\beta}\})$ is a set $\{w_\alpha\}$ of $R$-maps $w_\alpha:M_\alpha\to N_\alpha$ such that

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$\begin{matrix}M_\alpha & \overset{w_\alpha}{\longrightarrow} & N_\alpha\\ ^{f_{\alpha,\beta}}\big\downarrow & & \big\downarrow ^{g_{\alpha,\beta}}\\ M_\beta & \underset{w_\beta}{\longrightarrow} & N_\beta\end{matrix}$

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whenever $\alpha\leqslant\beta$. But, since there are only arrows $\alpha\xrightarrow{k_{\alpha,\beta}}\beta$ if and only if $\alpha\leqslant\beta$, and the corresponding arrows are $f_{\alpha,\beta}$. From this the fact that $\{w_\alpha\}$ is a natural transformation between $F:\alpha\mapsto M_\alpha, k_{\alpha,\beta}\mapsto f_{\alpha,\beta}$ and $G:\alpha\mapsto N_\alpha,k_{\alpha,\beta}\mapsto g_{\alpha,\beta}$ follows. Thus, the identification of $\mathbf{Dir}_\mathcal{I}(R\text{-}\mathbf{Mod})$ and $R\text{-}\mathbf{Mod}^\mathcal{I}$ is justified. We then extend this to define the category of directed systems in $\mathcal{C}$ over $\mathcal{I}$ to be the functor category $\mathcal{C}^\mathcal{I}$ with the functors identified with the ordered pairs as before.

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Ok, fine so an interesting (and I think, if one looks enough, fairly motivated question) is what do functors $F:\mathcal{I}^\text{op}\to\mathcal{C}$ (i.e. contravariant functors $\mathcal{I}\to\mathcal{C}$ look like? Well, just as the case of covariant functors we see that contravariant functors can be described as ordered pairs of indexed sets. Only, this time it’s easy to see that the ordered pairs $\left(\{M_\alpha\},\{f_{\alpha,\beta}:x_\beta\to x_\alpha\}_{\alpha,\beta\in I,\; \alpha\leqslant\beta}\right)$. Moreover, just checking the axioms of the contravariant functor it’s easy to see that this indexed set of arrows satisfies:

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\begin{aligned}&\mathbf{(1)}\quad f_{\alpha,\alpha}=1_{x_\alpha}\text{ for all }\alpha\in I\\&\mathbf{(2)}\quad f_{\alpha,\beta}\circ f_{\beta,\gamma}=f_{\alpha,\gamma}\text{ whenenver }\alpha,\beta,\gamma\in I\text{ and }\alpha\leqslant\beta\leqslant\gamma\end{aligned}

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If $\mathcal{C}=R\text{-}\mathbf{Mod}$ we’d call this an inverse system of modules, and so we define an inverse system in $\mathcal{C}$ over $\mathcal{I}$ to be a functor $F:\mathcal{I}^\text{op}\to\mathcal{C}$ where we naturally identify the functor with the associated ordered pair of indexed sets of objects/arrows. It’s not then hard to check that the category $\mathbf{IS}_\mathcal{I}(R\text{-}\mathbf{Mod})$ of inverse systems of modules over $\mathcal{I}$ is the same thing as $R\text{-}\mathbf{Mod}^{\mathcal{I}^\text{op}}$. We can thus define, in general, the category of inverse systems in $\mathcal{C}$ over $\mathcal{I}$ to be the category $\mathcal{C}^{\mathcal{I}^\text{op}}$ where the contravariant functors therein are identified with the ordered pair of indexed sets as mentioned above.

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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] Rotman, Joseph J. Introduction to Homological Algebra. Springer-Verlag. Print.

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