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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 [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.