Math and science::Algebra::Aluffi

# Initial and final objects

The description of universal properties is typically done by stating that an object of a category is terminal: it is either initial or final. These concepts are defined here.

### Initial objects

An object $$I$$ in category $$\cat{C}$$ is said to be initial in $$\cat{C}$$ iff for every object $$A$$ in $$\cat{C}$$ there is [what?]. That is:

[$\forall A \in \catobj{C}, \text{ what?}$]

### Final objects

An object $$F$$ in category $$\cat{C}$$ is said to be final in $$\cat{C}$$ iff for every object $$A$$ in $$\cat{C}$$ there is [what?]. That is:

[$\forall A \in \catobj{C}, \text{ what?}$]

An object is said to be a terminal object iff if is either an initial object or a final object.

### Unique up to a unique isomorphism. Proposition.

For any two initial objects in a category there is a single isomorphism between them. This statement is often phrased as: "initial objects are unique up to a unique isomorphism". The same is true for final objects.

Proof of this proposition is on the reverse side; I'd recommend trying to think of the proof before looking at it.

Here is different way of presenting the proposition, from Aluffi:

Let $$\cat{C}$$ be a category.

1. If $$I_1$$ and $$I_2$$ are both initial objects in $$\cat{C}$$, then $$I_1 \cong I_2$$.
2. If $$F_1$$ and $$F_2$$ are both final objects in $$\cat{C}$$, then $$F_1 \cong F_2$$.

In addition, these isomorphisms are unique.