# Full isomorphism-closed reflective subcategory of Set

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?

• Hmm...what about $\emptyset$, i.e. the subcategory with object $\emptyset$ and the unique morphism $\emptyset \rightarrow \emptyset$? Jan 12, 2013 at 20:30
• @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. Jan 12, 2013 at 20:41
• Ahh...fantastic point! Jan 12, 2013 at 20:48

• 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? Jan 12, 2013 at 21:00