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.  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?

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

Author: brianhepler

I'm a second-year math postdoc at the University of Wisconsin-Madison, and I think math is pretty neat. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s