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Right now I am trying to understand initial and terminal objects in a category. The premier example of terminal and initial objects are used in the category of sets as being singleton sets and $\emptyset$, respectively. Why are these the unique terminal and initial objects?

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The set $\emptyset$ is an initial object because, for every set $A$, there is unique a map $\emptyset \to A$. Indeed, if such a map exists, then it is a subset of the set $\emptyset \times A=\emptyset$. Therefore, if this map exists, it must be $\emptyset$ and $\emptyset$ is actually a map of sets $\emptyset \to A$. Hence such a map exists and is unique.

If $X$ is a singleton, then, for every set $A$, there is unique a map $A \to X$: this is just the map sending every element of $A$ to the unique element of $X$. This means that $X$ is a terminal object.

Finally, since initial objects are pairwise isomorphic (that is, in this case, in bijection) and since the same holds for terminal objects, it is the case that every initial object is the empty set and that every terminal object is a singleton.

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