Abstract Nonsense

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

Point of Post: This post is mostly focused on motivating and defining limits and colimits, and defining representable functors only to make this process easier.

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Motivation

We have, up until this point, discussed various kinds of “constructions”, in particular we have discussed products and equalizers and their appropriate dual notion. Despite their apparent differences, all of these constructions were made via a very similar procedure. Namely, we had some “diagram” in mind, and we were looking for the object and arrows to (or from) that object to make the diagram a reality. For example, the product of $X_1$ and $X_2$  could be phrased as trying to find an object $X_1\times X_2$ and maps $X_1\overset{\pi_1}{\leftarrow}X_1\times X_2\overset{\pi_2}{\rightarrow}X_2$ such that given an object $Y$ and maps $X_1\overset{f_1}{\leftarrow}Y\overset{f_2}{\rightarrow}X_2$ we have (uniquely) the following commuting diagram

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Similarly, we could write down a certain diagram for equalizers involving known objects/arrow, desired objects/arrows, and variable objects/arrows which tells us (unsurprisingly) what we have, what we want, and what our wanted things should do. This is the general idea of a limit, or perhaps better, the “limit of a diagram”. Namely, we have some set of objects and maps between these objects and some diagram (such as the one above) in mind for which there is some imaginary (desired) object and a variable object that tells us what this imaginary object does (in relation to the known objects, and maps from the known objects to the variable objects) [was this the same sentence as two sentences ago?]. Of course, this is all intuition and we need a firm, rigorous grounding to set this heavy intuition upon. So, how do we do this? The key was the observation we made during our discussion of equalizers, that an equalizer could very well be thought of as a “representing element” that takes a $\mathbf{Set}$-valued functor and turns it into a covariant (or contravariant) Hom functor. Roughly the idea behind this, is that we can state in $\mathbf{Set}$ what diagram we want with very little problem (because $\mathbf{Set}$ is such a manageable category) and finding a Hom functor that “represents” this diagram (which, secretly, is itself a functor) is equivalent to finding an object that makes everything work out the way we want it to.

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The above was very wishy-washy, and perhaps (to some–or most) not at all helpful, but it’s what makes sense in my head. If the above doesn’t suffice as motivation perhaps some we’ve-secretly-already-done-this magic will help. Indeed, the limits and colimits we are about to consider can be thought of as generalizations to general categories (and more general diagrams) of the notion of direct and inverse limits of modules or rings.

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Limits and Colimits

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Ok, so now we can finally get down to brass tacks. Let us just go ahead and charge into the actual technical details of a limit and colimit of a diagram.

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For us a diagram in $\mathcal{C}$ over $\mathcal{I}$ shall be a functor $F:\mathcal{I}\to\mathcal{C}$ where $\mathcal{I}$ is a small category. Why should this be our definition of a diagram? A category is just a collection of dots and arrows (right?) and a functor can be thought of as placing the objects of $\mathcal{C}$ on top of the dots and morphisms in $\mathcal{C}$ on top of the arrows. For example, if we have the category $\mathcal{I}$ which consists of two objects $\bullet$ and $\bold\star$ and two non-identity arrows $\bullet\overset{\displaystyle \longrightarrow}{\longrightarrow}\bold\star$ a functor $F:\mathcal{I}\to\mathbf{Top}$ is nothing more than two continuous topological spaces $F(\bullet)=X$ and $F(\bold{\star})=Y$ and two continuous maps $f,g:X\to Y$, or graphically

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$\bullet\overset{\displaystyle \longrightarrow}{\longrightarrow}\bold{\star}\;\;\overset{F}{\mapsto}\;\; X\overset{\displaystyle \overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}Y\quad\mathbf{(1)}$

Hopefully this makes clear why this definition of diagram makes sense.

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Ok, so we are going to want to take the “limit” of diagrams. What exactly should this mean? As we said in the introduction a limit should be some kind of external object that “completes” the diagram in the most efficient way. In particular, let’s say that a cone for a diagram $F:\mathcal{I}\to\mathcal{C}$ shall be an object $C$ of $\mathcal{C}$ and a set of arrows $C\xrightarrow{\alpha_i}F(i)$ for each $i$ an object of $\mathcal{I}$ with the property that whenever one has a morphism $i\xrightarrow{f}j$ in $\mathcal{I}$ the diagram

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$\begin{array}{ccc}C & \xrightarrow{\alpha_i} & F(j)\\ _{\alpha_j}\big\downarrow & ^{F(f)}\nearrow & \\ F(i) & & \end{array}$

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commutes.  We shall usually denote a cone as $(C,\{\alpha_i\})$.

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Why is a cone called a cone? Well, think about the diagram $F$ pictorally (i.e. envision it as a set of $\mathcal{C}$ objects and arrows). Then, $C$ sits external to this diagram–for the sake illustrative purposes let’s imagine it sits above the diagram. Then the arrows from $C$ down to the vertices (objects) of the diagram creates a “cone” where the vertex is $C$, the base is the diagram, and the connecting part is this set of arrows.

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Anyways, let’s say that a morphism of cones , say from $(C,\{\alpha_i\})$ to $(D,\{\beta_i\})$, to be an arrow $C\xrightarrow{\nu}D$ with the property that the resulting triangles

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$\begin{array}{ccc}C & \xrightarrow{\nu} & D\\ _{\alpha_i}\big\downarrow & \swarrow_{\beta_i} & \\ F(i) & & \end{array}$

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Intuitively one can think about $C$ as lying farther away from the diagram $F$ and the morphism of cones as being an arrow down from $D$ such that it doesn’t matter if we go straight down from $C$ to the diagram or go from $C$ to $D$ and then down to the diagram.

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Ok, it’s easy to see that the composition of cone morphisms is a cone morphism and that the identity arrow $C\to C$ serves as a cone morphism $(C,\{\alpha_i\})\to(C,\{\alpha_i\})$. Thus, the set of all cones of $F$ and cone morphisms between such cones forms a category $\mathbf{Cone}(F)$. We then define a limit of $F$ to be a terminal object in $\mathbf{Cone}(F)$.

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While this definition is clean and quick it is slightly opaque, so let’s lay it out for the world to see. A limit of $F$ should be an object $L$ along with arrows $L\xrightarrow{\alpha_i}F(i)$ such that the resulting diagrams commute. Moreover, any time we have another object $C$ and a set of arrows $C\xrightarrow{\beta_i}F(i)$ which makes the resulting diagrams commutes, there exists a unique arrow $L\xrightarrow{\nu}C$ such that the resulting diagrams commute.

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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 13, 2012 -