# What does it mean that terminal object have morphism to other object?

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?

• are you familiar with the category of vector spaces over some field? in that case the terminal and initial objects are the same. Commented Jul 23, 2021 at 6:31
• Here you go: ncatlab.org/nlab/show/global+element Commented Jul 23, 2021 at 8:23
• "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. Commented Jul 23, 2021 at 14:30
• 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 $\mathscr{F}$. Commented Jul 23, 2021 at 15:38

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 $$f(*) \in X$$.