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 third-year math postdoc at the University of Wisconsin-Madison, where I work as a member of the geometry and topology research group. Generally speaking, I think math is pretty neat; and, if you give me the chance, I'll talk your ear off. 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 to show you a world of pure imagination.

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