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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 exactly one morphism from $$I$$ to $$A$$. That is:

$\forall A \in \catobj{C}, \; \cathom{C}(I, A) \text{ is a singleton.}$

### 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 exactly one morphism from $$A$$ to $$F$$. That is:

$\forall A \in \catobj{C}, \; \cathom{C}(A, F) \text{ is a singleton.}$

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.

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

First, a lemma: terminal objects have a single endomorphism which is the identity morphism.

Proof. Let $$T$$ be a terminal object in category $$C$$. Then there is exactly one morphism in $$\cathom{C}(T, T)$$, as $$T$$ being initial or final asserts this. The definition of a category asserts that the identity morphism $$1_T$$ is a morphism of $$\cathom{C}(T, T)$$, so the single endomorphism is the identity morphism.

Now the main proof. Consider two initial objects $$I_1$$ and $$I_2$$. Consider possible morphisms between $$I_1$$ and $$I_2$$: being initial objects, there is a unique morphism $$f : I_1 \to I_2$$ and a unique morphism $$g: I_2 \to I_1$$. We can compose these morphisms to obtain endomorphisms, for example, $$g \, f : I_1 \to I_1$$, and $$f \, g : I_2 \to I_2$$. But it was just established in the above lemma that the identities $$1_{I_1}$$ and $$1_{I_2}$$ are the only endomorphisms in $$\cathom{C}(I_1, I_1) )$$ and $$\cathom{C}(I_2, I_2)$$, so $$g \, f = 1_{I_1}$$ and $$f \, g = 1_{I_2}$$, so $$f$$ and $$g$$ are isomorphic inverses.

### $$\mathbb{Z}$$ with relation $$\le$$, no terminal objects.

The category formulated by considering the relation $$\le$$ on $$\mathbb{Z}$$ (morphisms being the pairs of the relation's graph), has no initial or final object: an initial object one be an integer less that all other integers, and a final object would be an integer which all other integers are less than it.

In comparison, the slice category formulated from the relation $$\le$$ on $$\mathbb{R}$$ where objects are the morphisms of the category just mentioned–this category has a final object, namely $$(3, 3)$$.

This example is mentioned by Aluffi in "Example 5.2".

### Terminal objects of $$\mathrm{Set}$$

The category $$\mathrm{Set}$$ has just one initial object, the empty set $$\emptyset$$; the empty graph defines the unique morphism from $$\emptyset$$ to all other sets.

$$\mathrm{Set}$$ has many final objects: all singleton sets are a final object; for any set $$A$$ and any singleton set $$\{p\}$$, there is a unique function, the constant function, from $$A$$ to $$\{p\}$$.

It is a useful exercise to confirm that all these singleton sets are isomorphic.

p32