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Exercise 78 - Chapter 3

edited June 2018 in Exercises

Not every category has a terminal object. Find one that doesn’t.

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  • 1.
    edited May 2018

    Maybe the simplest category without a terminal object is \( \textbf{2} := [ 1 \, 2 ] \) the discrete category of two objects.

    Does \( \textbf{0} \) qualify? It has no morphisms (to any object) but it has no objects from which to draw a terminal-object so it has no terminal-object.

    The category \( \textbf{1} \) has a terminal object but the category with one object and one morphism in addition to the identity-morphism does not have a terminal-object [uniqueness is violated].

    Comment Source:Maybe the simplest category without a terminal object is \\( \textbf{2} := [ 1 \, 2 ] \\) the discrete category of two objects. Does \\( \textbf{0} \\) qualify? It has no morphisms (to any object) but it has no objects from which to draw a terminal-object so it has no terminal-object. The category \\( \textbf{1} \\) has a terminal object but the category with one object and one morphism in addition to the identity-morphism does not have a terminal-object [uniqueness is violated].
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