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 be a collection of objects of **C** indexed by some set , and let be a collection of morphisms in **C** (we do not require that there is a morphism for any two , we only allow for the possibility of there being one). We call this collection of objects and morphisms a *diagram * in **C. **

Let be a diagram in **C. **A *cone* on is a **C**-object and collection of morphisms , such that for all , . If we have two such “cones,” and on this diagram, we say 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 , call it . A *limit *of the diagram is then just a terminal object in .

Similarly, a *co-cone* is just a cone with all the arrows reversed (i.e. an object together with maps for each ). 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 .

**Pullbacks:**Let be three -modules, and consider the diagram . The limit of this diagram is then just the ordinary pullback (or fiber product), the module**Products:**Let be modules. Then the limit of the “diagram” consisting of just and and no morphisms between them is the product .**co****-Products:**consider the same diagram used for the product. The co-limit of this is then the co-product of and , .**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 . The equalizer of two parallel maps is the pullback of . The kernel of (we’re still working with modules) is the pullback of . Get the picture?

A pretty good thing to try here would be to find out how these are equivalent to universal properties. So go try that. 🙂