# Morphism from object to initial object, meaning?

By the definition of the terminal object, $\{*\}$, in the category Set, $\{*\}$ is trivial when appearing on the codomain of a morphism, as then the morphism is unique. It gives non trivial information when appearing on the domain of morphism, for then $f : \{*\} \to B$ picks out a particular element of $B$. There are now as many morphisms as elements of $B$, and $f$ being monomorphism, picks unambiguously elements of the codomain.

I am interested in the case of an initial object. The initial object, call it $I$ is now trivial when appearing on the domain, as it is unique, but it is non-trivial when it appears on the codomain. My question is this:

How can we understand the meaning of $f:A \to I$?

Is there a general method to understand what "meaning" to assign to categorical definitions?

• Are you interested in arrows with codomain the initial object in general, or in the category of sets? Jan 16, 2018 at 16:54
• Both if possible...
– EEEB
Jan 16, 2018 at 16:56
• In the case of $\mathrm{Set}$, the initial object is the empty set $\emptyset$, so $\mathrm{Hom}(X,\emptyset)=\emptyset$. So this does not give a lot of information. Note that your observation about the terminal object does not really generalize to arbitrary categories. For instance, in the category of commutative rings with unit, the terminal object is $0$, but $\mathrm{Hom}(0,R)=\emptyset$.
– asdq
Jan 16, 2018 at 16:58
• @asdq: $\hom(X, \varnothing) = \varnothing$ only when $X \neq \varnothing$.
– user14972
Jan 16, 2018 at 16:59
• @Hurkyl You're right, the same should be added in the case of commutative rings and the zero ring.
– asdq
Jan 16, 2018 at 17:00

For example, if $k$ is a commutative ring then we can consider the category of commutative $k$-algebras. Here the initial object is $k$, and a morphism $f : R \to k$ of $k$-algebras corresponds to what is called a $k$-point or $k$-rational point of the affine scheme $\text{Spec } R$ over $k$. These can be very interesting; for example when $k = \mathbb{Z}$ it is possible to encode solutions of Diophantine equations this way.
In many cases, such as in toposes, the initial object is strict, so that any morphism $A \to 0$ is an isomorphism. What this means is "you can't quotient something to get nothing".