Universal Properties IV: Cones and a first look at Limits

Sorry for the delay since my last post (to those who actually read this…)

So I stumbled across a really nice way of looking at universal properties that is equivalent to specifying them as a terminal (or initial) object in a suitable comma category, but it has a much nicer “intuitive feel.”

Cones (and co-Cones)

Let C be some category. Let $\{d_i\}_{i \in I}$ be a collection of objects of C indexed by some set $I$ , and let $\{g_{ij}: d_i \to d_j\}_{i,j \in I}$ be a collection of morphisms in C (we do not require that there is a morphism for any two $i,j$ , we only allow for the possibility of there being one). We call this collection of objects and morphisms a diagram in C.

Let $D$ be a diagram in C. A cone on $D$ is a C-object $c$ and collection of morphisms $f_i : c \to d_i$ , such that for all $i,j \in I$ , $f_j \circ g_{ij} = f_i$ . If we have two such “cones,” $c$ and $c'$ on this diagram, we say $h: c' \to c$ such that the appropriate diagram commutes (try to figure it out! It’s a good idea to get an intuitive feel for these things). It follows pretty quickly then that we have a category of cones over $D$ , call it $\textbf{C}_D$ . A limit of the diagram $D$ is then just a terminal object in $\textbf{C}_D$ .

Similarly, a co-cone is just a cone with all the arrows reversed (i.e. an object $c$ together with maps $f_i : d_i \to c$ for each $i$ ). A co-limit of such a co-cone is an initial object in the category of co-cones over the appropriate diagram.

Examples………………………

Say we’re working in the category R-Mod for some ring with unity $R$ .

Pullbacks: Let $A,B,C$ be three $R$ -modules, and consider the diagram $A \overset{f}{\to} C \overset{g}{\leftarrow} B$ . The limit of this diagram is then just the ordinary pullback (or fiber product), the module $A \times_C B = \{ (a,b) | f(a) = g(b)\}$
Products: Let $A,B$ be $R-$ modules. Then the limit of the “diagram” consisting of just $A$ and $B$ and no morphisms between them is the product $A \times B$ .
co-Products: consider the same diagram used for the product. The co-limit of this is then the co-product of $A$ and $B$ , $A \oplus B$ .
Terminal objects: are just the limit of the “empty diagram.”
Initial objects: are just the co-limit of the “empty diagram.”

and so on.

Having “Finite (co-) Limits”

Notice that all the above limits and co-limits were taken over a “finite” diagram. That is, there were only finitely many objects and morphisms in each diagram. Such (co-)limits are referred to as “finite” (co-)limits (I wonder why…). It turns out that it’s a highly desirable property for a category to “have all (finite) (co-)limits.” It took me a lonnnnnggg time to grok this.

Remember when we first started talking about universal properties? When you specify that an object satisfies a certain universal property, it is unique up to unique isomorphism if it actually exists. These objects don’t have to exist. The property of having, say, all finite limits or co-limits means that whenever you specify a universal property for an object with a finite diagram, that object actually exists. It’s a theorem (that I don’t currently know how to prove) that a category C with a terminal object and all pullbacks has all finite limits. Dually, if C has an initial object and all pushouts, it has all finite co-limits. Is this so unreasonable? Look at the list of examples again.

Back? Good. Suppose we’ve got all pullbacks and a terminal object, call it 1. Then the product is just the limit of the pullback diagram $A \to \textbf{1} \leftarrow B$ . The equalizer of two parallel maps $f,g : A \to B$ is the pullback of $A \overset{f}{\to} B \overset{g}{\leftarrow} A$ . The kernel of $f: A \to B$ (we’re still working with $R-$ modules) is the pullback of $A \overset{f}{\to} B \overset{0}{\leftarrow} A$ . Get the picture?