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# Exercise 78 - Chapter 3

edited June 2018

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].