Initial objects, final objects, and zeros

An object is initial if there’s a unique function from it to anything. An object if final if there’s a unique function from anything to it.

To be more precise, an object I in a category C is initial if for every object A in C, there exists a unique morphism from I to A. In terms of Hom sets, Hom(IA) has exactly one element for every A.

The dual notion of initial is final. (Some authors say terminal.) An object F in C is terminal if for every object A there is a unique morphism from A to F. In other words, Hom(AF) has exactly one element for every A.

Although initial and final are simple ideas, they play out differently in different categories. We’ll give lots of examples. Each example illustrates something different.

Sets

In the category of sets, the empty set is initial. It takes a little effort to wrap your head around the idea of a unique function defined on the empty set. It’s the function that takes nothing to nothing! It’s unique, because a function can’t map nothing to something.

Any one-element set is final: between a set A and a one-element set {x}, there’s one and only one function, namely the one that takes every element of A to x.

The category of sets illustrates a couple features of initial and final sets. First, it shows that initial and final objects can be different (they aren’t always, as we’ll see soon). Second, it shows that initial and final elements are not necessarily unique, though they are isomorphic. All one-element sets are isomorphic.

Groups

In the category of groups and group homomorphisms, the trivial group is both initial and final. A group has to have an identity element, so the smallest possible group, the trivial group, is the group that only has an identity element.

A group homomorphism must take the identity of one group to the identity of the other, so there’s a unique homomorphism from the trivial group to any other group: the only element of the trivial group is mapped to the identity element of the other group. Also, there’s a unique homomorphism from any group to the trivial group: map everything to the one element of the trivial group.

When an object is both initial and final, it is called a zero object. This can be confusing because the categorical use of the word “zero” might not coincide with the conventional terminology in that category. In Abelian groups, the identity element is typically called 0, and so there’s no confusion. In general groups, the identity element is usually called 1.

So the category of groups has zero objects, but the category of sets does not.

Preorders

You can think of a preordered set as a category by saying there exists a morphism between objects A and B if and only if A ≤ B. A minimum element is an initial object and a maximum element is a final object.

This example shows that a category may or may not have an initial object, and it may or may not have a final object. To see this, pick two real numbers a and b with aand consider the intervals (ab), (ab], [ab), and [ab].

Rings

So far initial objects have been the smallest thing in their category: the empty set, the one-element group, the minimum element in a pre-order. Things are different in the category of rings and ring homomorphisms. (Here we assume rings must have a multiplicative identity element. Unfortunately there’s not universal agreement on this definition.)

The zero ring consists of one element, the additive identity 0. Because there’s only one element, the additive identity is also the multiplicative identity.

The zero ring is not intial in the category of rings! There is no homomorphism between the zero ring and any non-zero ring because a ring homomorphism must preserve both the additive and multiplicative identities. That is, the 0 (additive identity) of one ring must go to the 0 of the other, and the 1 (multiplicative identity) of one ring to the 1 of the other. Since 0 = 1 in the zero ring, but not in any non-zero ring, there cannot be a homomorphism from the zero ring to a non zero ring: the only element of the zero ring would have to go to two different elements, the 0 and the 1 of the other ring.

The integers are initial in the category of rings. To see that there exists a unique ring homomorphism from the integers Z to any other ring R, note that 0 must go to the 0 of R, and 1 must go to the 1 of R. The image of every other integer follows.

The zero ring is terminal even though it is not initial. Since the initial and terminal objects are different, there is no zero object.

Fields

There is no field homomorphism between two fields with different characteristics. Pick any field F and let p be its characteristic. F cannot be initial because there are only homomorphisms from F to other fields of characteristic p. Similarly, F cannot be final because there are only homomorphisms from fields of characteristic p into F. So fields are an example of a category with no intial or final objects.

However, if we restrict ourselves to the category of fields with a given characteristic, then there are initial objects. The rational numbers are initial in fields of characteristic 0, and the integers mod p are initial in any field of characteristic p.

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