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Wiki defines terminal object as the object towards which there is a single morphism from every other object of the category https://en.wikipedia.org/wiki/Initial_and_terminal_objects This definition is saying nothing about the possibility to have morphism from terminal object towards other object and how to interpret such morphism.

I have always thought that there are no morphisms from the terminal object, but, indeed, definition does not prohibit such morphisms.

There are use of such outgoing morphisms from the terminal object in the definition of fibrated categories and fibrations, see nice explanation in https://bartoszmilewski.com/2019/10/09/fibrations-cleavages-and-lenses/ where it is said that morphism from the terminal object into other object does the "selection of one element" (!) from this target object if it can be considered as a set of elements.

Is such interpretation true? How it is possible that terminal object have outgoing morphisms, what are their meaning and what are their application. Do such morphisms play the "selector" role indeed for the target object? Just trying to understand the notion of terminal object and why and where it is used?

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    $\begingroup$ are you familiar with the category of vector spaces over some field? in that case the terminal and initial objects are the same. $\endgroup$ Jul 23 at 6:31
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    $\begingroup$ Here you go: ncatlab.org/nlab/show/global+element $\endgroup$ Jul 23 at 8:23
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    $\begingroup$ "How it is possible that terminal object have outgoing morphisms" – Why would it be impossible for a terminal object to have outgoing morphisms? Everything is possible unless there's a reason why it's impossible. $\endgroup$ Jul 23 at 14:30
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    $\begingroup$ I don't know if this would be useful to you - but if you're looking at a category of sheaves, then the morphisms from the terminal object to a sheaf $\mathscr{F}$ are equivalent to the global sections of $\mathscr{F}$. $\endgroup$ Jul 23 at 15:38
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The best way to understand this is through concrete examples.

Let's first consider the category $\mathbf{Set}$ of sets and functions. Any singleton $\{*\}$ is a terminal object there (check this). So for any set $X$ an arrow $f: \{*\} \to X$ is essentially the same as picking one element $f(*) \in X$.

Another example to consider would be the category of groups and group homomorphisms, or the category of vector spaces over some field (say real vector spaces) as Jackozee suggested in the comments. In both these categories a terminal object is the same as an initial object: they are both given by the trivial group (respectively trivial vector space). Again, you should check this for yourself.

These are just some examples that should give some insight. You may want to work out more examples in categories you are interested in.

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