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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 C is said to be initial in C iff for every object A in C there is [what?]. That is:

[AObj(C), what?]

Final objects

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

[AObj(C), 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 C be a category.

  1. If I1 and I2 are both initial objects in C, then I1I2.
  2. If F1 and F2 are both final objects in C, then F1F2.

In addition, these isomorphisms are unique.