10
$\begingroup$

There is a basic construction in category theory which I've only just recently become acquainted with, that is the comma category.

It seems to be a quite basic construction for which, however, I've seen really few "real life" examples.

I know the slice, coslice and arrow categories are particular cases of a comma category. This is in MacLane, or in the wikipedia article. In that article there are also the examples of pointed sets or graphs, which are of a more concrete nature.

I'm asking, then, for more examples of this construction in mathematics. Examples of (co)slice categories are also welcome.

Here's one example I've come up with. The completion of a metric space $M$ consists of a pair $(\overline{M},i)$ where $\overline{M}$ is a complete metric space and $i:M\to \overline{M}$ is a uniformly continuous function which satisfies the following universal property: if $N$ is another complete metric space and $g:M\to N$ is a uniformly continuous function, then there exists a unique uniformly continuous function $h:\overline{M}\to N$ such that the following diagram commutes:

Completion

I claim this completion is an initial object in a suitable comma category. Consider the functors enter image description here where $\mathbf{Met_u}$ is the category of metric spaces with uniformly continuous functions and $\mathbf{CompMet_u}$ is the category of complete metric spaces with uniformly continuous functions. The functor $F$ is such that $F(\star)=M$ where $\star$ is the sole object of $\mathbf{1}$, and $U$ is the inclusion functor.

Then an initial object of $(F\downarrow U)$ is exactly a completion of $M$.

Bonus question: is this approach to the completion not interesting/not useful? I ask this because it seems the categorial approach to completions has nothing to do with comma categories. (I can see, though, that the universal property of this completion can also be seen as an adjunction).

  • 0
    I'm sorry for the "bonus question" part, I know it is not good practice to ask more than one question per question (hmm, that's slightly redundant). If it is a more interesting question than it seemed to me and there is someone willing to give a comprehensive answer that is not suitable for a comment here, then I will happily split it into another question.2012-04-21
  • 0
    Have you read the proof of the general adjoint functor theorem? It constructs left adjoints via initial objects of certain comma categories...2012-04-21
  • 0
    @ZhenLin: I haven't (but plan on doing so relatively shortly). That's very interesting to me, and it can't be just a coincidence that both my example and the one given in the answer by Matt N. are both left adjoints... Thank you.2012-04-21
  • 0
    Did you want things that weren't slice/coslice categories? I assume you know that you can think of $R\text{-}\mathbf{Alg}$ as the slice category $(R\downarrow \mathbf{CRing}$.2012-04-21
  • 0
    @AlexYoucis: no, I didn't mean to say that, I'm also interested in "concrete" examples of (co)slice categories! I didn't know what you mention, perhaps you could post an answer?2012-04-21
  • 0
    @BrunoStonek Done.2012-04-21
  • 0
    An augmented $k$-algebra is a $k$-algebra with a morphism of algebras to $k$. These naturally form a comma category.2012-04-23

5 Answers 5