Abstract Nonsense

Limits, Colimits, and Representable Functors (Pt. II)

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

Similarly to cones we can define a cocone of $F$ to be an object $X$ and arrows $F(i)\xrightarrow{\rho_i}X$ which make the resulting diagrams

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$\begin{array}{ccc}X & \xleftarrow{\rho_j} & F(j)\\ _{\rho_i}\big\uparrow & \nearrow^{F(f)} & \\ F(i) & & \end{array}$

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I don’t think I need to tell you how to define a morphism of cocones and that we have a category $\mathbf{Cocone}(F)$. We define then that a colimit of $F$ shall be a terminal object of $\mathbf{Cocone}(F)$. It’s pretty easy to see that a cocone of $F$ is nothing more than a cone of the functor $F:\mathcal{I}^\text{op}$ considered as the composition $\mathcal{I}^{\text{op}}\xrightarrow{\text{op}}\mathcal{I}\xrightarrow{F}\mathcal{C}$.

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Since initial and terminal objects are unique up to isomorphism we shall often denote the objects associated to the limit and colimit (if they exist–and this isn’t a vaccuous statement) as $\varprojlim F$ and $\varinjlim F$ respectively–occasionally we may write $\lim F$ and $\text{colim} F$.

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So, let’s get our hands a little dirty and take a look at some real live limits and colimits.

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Ok, let $\mathbf{2}$ be the discrete category on two objects. What then is a limit for a functor $F:\mathbf{2}\to\mathcal{C}$? Well, we have two objects $x$ and $y$ and corresponding to the two objects in $\mathbf{2}$ and no non-identity arrows so that we have the (pretty unexciting) diagram

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$x\quad y$

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We see then that a limit of $F$ is an object $z$ of $\mathcal{C}$ with arrows $x\xleftarrow{\alpha_1}z\xrightarrow{\alpha_2}y$ with the property that given any pair of arrows $X\xrightarrow{\beta_1}x$ and $X\xrightarrow{\beta_2}y$ there exists a unique arrow $X\xrightarrow{\nu}z$ such that $\alpha_i\circ \nu=\beta_i$. Thus, we see that colimit of $F$ is nothing more than a product of $x$ and $y$. As is probably not hard to guess, a colimit of $F$ is a coproduct of $x$ and $y$.

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This simple example shows that limits/colimits need not exist in general for, as we have seen, products need not always exist in categories. This is something to keep in mind–not all diagrams have limits.

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Ok, let $\mathcal{I}$ be the category shown in $\mathbf{(1)}$. What is a limit of this diagram? Well, we see that it would need to be an object $E$ along with arrows $E\xrightarrow{\alpha}x$ and $E\xrightarrow{\beta}y$ such that $f\circ\alpha=g\circ\alpha=\beta$ and moreover any time we have an object $X$ along with arrows $X\xrightarrow{\alpha'}x$ and $X\xrightarrow{\beta'}y$ such that $f\circ\alpha'=g\circ\alpha'=\beta'$ there exists a unique arrow $X\xrightarrow{\nu}E$ such that $\alpha\circ\nu=\alpha'$ and $\beta\circ\nu=\beta'$. Take a minute to convince one’s self that the maps $\beta$ and $\beta'$ were incidental (there inclusion as information is redundant since we know what they have to be–$f\circ\alpha$) and that the only thing was that $f\circ\alpha=g\circ\alpha$. From this make the leap to conclude that a limit of this diagram is nothing more than an equalizer of $x\overset{\displaystyle \overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}y$. Shocking as this may come to you [sorry for those with a weak heart] but a colimit of this diagram is a coequalizer of $x\overset{\displaystyle \overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}y$.

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Representable Functors

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Ok, so what do limits and colimits have to do with making something work in $\mathbf{Set}$ and using this to define the object in $\mathcal{C}$?

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Well, let’s begin by showing that given an arbitrary functor $F:\mathcal{I}\to\mathbf{Set}$ that $\lim F$ exists. Indeed, define

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$\displaystyle \mathcal{L}(F)=\left\{(x_k)\in\prod_{k\in\text{obj}(\mathcal{I})}F(k):F(s)(x_i)=x_j\text{ for all }i\xrightarrow{s}j\right\}$

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We claim that $\mathcal{L}(F)$ along with the functions $\alpha_i:\mathcal{L}(F)\to F(i)$ given by $\alpha_i(x_k)=x_i$. What we claim is that $\left(\mathcal{L}(F),\{\alpha_i\}\right)$ is actually a limit of $F$.

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To see that $\left(\mathcal{L}(F),\{\alpha_i\}\right)$ is even a cone we need to check that $F(s)\circ\alpha_i=\alpha_j$ for all $i\xrightarrow{s}j$. But, this is simple. Let $(x_k)\in\mathcal{L}(F)$ be arbitrary, then $F(s)(\alpha_i((x_k)))=F(s)(x_i)=x_j=\alpha_j((x_k))$ from where the conclusion follows. Now, we need to prove that this cone really is a universal object in $\mathbf{Cone}(F)$. To do this, suppose that $(L,\{\beta_i\})$ is another cone of $F$ and define $\nu:L\to\mathcal{L}(F)$ by $x\mapsto (\beta_k(x))$. What we now need to check is that $\alpha_i\circ\nu=\beta_i$ for all $i$, but this is trivial since $\alpha_i(\nu(x))=\alpha_i((\beta_k(x))=\beta_i(x)$. Ok, so the last thing to check is that such a mapping is unique. But, this is clear by the definition of the product of sets–if two functions agree on the projections (for that’s all these are) for all possible coordinates the functions are equal. Thus, we see that $\mathcal{L}(F)=\lim F$.

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Ok, so now the idea is to use the fact that limits exist in $\mathbf{Set}$ to define them in arbitrary categories. The basic idea should be that we should be able to phrase the properties of a purported limit of a diagram $F:\mathcal{I}\to\mathbf{Set}$ merely by talking about sets–namely the Hom sets. How to do this?

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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.