What are the full isomorphism-closed reflective subcategories of $\textbf{Set}$?

In the book "The Joy of Cats" it is mentioned (p. 58) that there are precisely three such subcategories. I can see two of them are:

  1. the whole category $\textbf{Set}$
  2. the subcategory consisting of all singletons

What would be the third one?

  • $\begingroup$ Hmm...what about $\emptyset$, i.e. the subcategory with object $\emptyset$ and the unique morphism $\emptyset \rightarrow \emptyset$? $\endgroup$ – uncookedfalcon Jan 12 '13 at 20:30
  • 1
    $\begingroup$ @uncookedfalcon I don't think $\emptyset$ is a reflective subcategory of $\textbf{Set}$. Because given a nonempty set $A$, there is no morphism from $A$ to $\emptyset$, hence no reflection morphism. $\endgroup$ – PatrickR Jan 12 '13 at 20:41
  • $\begingroup$ Ahh...fantastic point! $\endgroup$ – uncookedfalcon Jan 12 '13 at 20:48

The subcategory containing the singletons and the empty set seems to work.

  • $\begingroup$ I think you are right. The category you mention works. But that makes me think, maybe I was mistaken: the subcategory consisting of only the singletons is not reflective? Because if there were a reflection arrow from $\emptyset$ to some singleton $A$, it would have to be the empty function. But the empty function $\emptyset \to \emptyset$ does not factor through that map $\emptyset \to A$. What do you think? $\endgroup$ – PatrickR Jan 12 '13 at 21:00
  • $\begingroup$ sorry about the confusion: the subcategory consisting of only the singletons is reflective after all. Only the functions to a object in the subcategory need to be considered in the definition of reflection arrow. $\endgroup$ – PatrickR Jan 12 '13 at 21:08

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