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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?

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  • $\begingroup$ Are you interested in arrows with codomain the initial object in general, or in the category of sets? $\endgroup$ – Arnaud D. Jan 16 '18 at 16:54
  • $\begingroup$ Both if possible... $\endgroup$ – EEEB Jan 16 '18 at 16:56
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    $\begingroup$ 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 $\mathrm{Hom}(0,R)=\emptyset$. $\endgroup$ – asdq Jan 16 '18 at 16:58
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    $\begingroup$ @asdq: $\hom(X, \varnothing) = \varnothing$ only when $X \neq \varnothing$. $\endgroup$ – user14972 Jan 16 '18 at 16:59
  • $\begingroup$ @Hurkyl You're right, the same should be added in the case of commutative rings and the zero ring. $\endgroup$ – asdq Jan 16 '18 at 17:00
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Often an object is defined by a universal property that specifies what maps into or out of it look like, but not the other way around. Then maps the other way around can do all sorts of things which aren't obviously determined by the universal property.

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 general it's more common to think about maps out of a terminal object rather than maps into an initial object (which are categorically dual); maps out of a terminal object are sometimes called global elements or global points because of various special cases where they generalize constructions like the global sections of a sheaf. Again, these can be very interesting in general and have to be understood on a case-by-case basis.

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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".

In other categories like abelian categories the initial object (zero object) is no longer interpreted as an empty set, so the interpretation does not hold.

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