# 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?

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Hmm...what about $\emptyset$, i.e. the subcategory with object $\emptyset$ and the unique morphism $\emptyset \rightarrow \emptyset$? – uncookedfalcon Jan 12 at 20:30

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@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. – PatrickR Jan 12 at 20:41
Ahh...fantastic point! – uncookedfalcon Jan 12 at 20:48

## 1 Answer

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

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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? – PatrickR Jan 12 at 21:00
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. – PatrickR Jan 12 at 21:08
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