## Category Theory: Mono, Epi, but not Iso?

### January 2, 2013

This post will require some very basic knowledge of category theory (like, what a category is, and how to make a poset into a category). For everything below, I will be a bit informal, but I will essentially mean that are objects in a category, and is some morphism between them which is also in the category.

The "natural" extension of the notion of a surjective map (in, say, the category of sets) is

**Definition**. A map is an *epimorphism* if, for each object and map we have that if then .

You should prove for yourself that this is, in fact, what a surjective map "does" in the category of sets. Pretty neat. Similarly, for injective maps (in, say, the category of sets) we have the more general notion:

**Definition**. A map is a *monomorphism* if, for each object and map we have that if then .

Again, you should prove for yourself that this is the property that injective mappings have in the category of sets. Double neat. There is also a relatively nice way to define an isomorphism categorically — which is somewhat obvious if you’ve seen some algebraic topology before.

**Definition**. A map is an *isomorphism* if there is some mapping such that and , where denote the identity morphism from the subscripted object to itself.

Now, naively, one might think, "Okay, if I have some certain kind of morphism in my category (set-maps, homomorphisms, homeomorphisms, poset relations, …) then if it is an epimorphism and a monomorphism, it should automatically be an isomorphism." Unfortunately, **this is not the case.** Here’s two simple examples.

**Example (Mono, Epi, but not Iso)**. The most simple category for which this works is the category **2**, which I’ve drawn below:

There are two objects, and three morphisms, the identites and the morphism . First, prove to yourself that this is actually a category. Second, we note that is an epimorphism: the only map from is the identity, and there is no mapping from , so the property trivially holds. Third, we note that is a monomorphism for the exact same reason as before. Last, we note that is *not *an isomorphism: we would need some which satisfied the properties in the definition above…but, there *is no map* from . Upsetting! From this, we must conclude that cannot be an isomorphism despite being a mono- and epimorphism.

**Similar Example (Mono, Epi, but not Iso). **Take the category , the natural numbers with morphisms as the relation . Which morphisms are the monomorphisms? Which morphisms are the epimorphisms? Prove that the *only *isomorphisms are the identity morphisms. Conclude that there are a whole bunch of morphisms which are mono- and epimorphisms but which are not isomorphisms.