Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?

Solution 1:

I was hoping to find something more definitive, but it doesn't look like it's going to happen. And I suppose it's good hygiene to have questions that have been more-or-less answered more visibly "marked" as answered. So, prodded by Martin's comment, here are my comments in answer form:

According to the nlab, the dual notion is a subterminal object (the idea being that if there is a terminal object, then the property is equivalent to being a subobject of the terminal object). How to dualize the terminology is not clear to me. "Subinitial object" has the unfortunate connotation that the object should be a subobject of the initial object. You could say "co-subterminal", I suppose. What you really need is a prefix that means "a quotient object of", and put it before "initial object".

The closest thing I can think of to a prefix meaning "a quotient object of" is the suffix "-ly indexed", in analogy with "finitely-indexed" objects. So you could try "initially-indexed object". A variant would be "initially-generated" object, in analogy with e.g. finitely generated groups. But maybe these possibilities are taking the notion of "that's what it would be if there were an initial object" too far. Maybe something simple like "pre-initial object" would do...

Also related: if a category $\mathcal{C}$ has a set of objects $S$ such that for every $X \in \mathcal{C}$ there is a unique object $I \in S$ admitting an arrow $I\to X$, and moreover that arrow $I \to X$ is unique, then the set S is called a multi-initial object (in particular, the objects in S are "co-weakly initial" / co-subterminal/ initially indexed). For example, you exhibit a multi-initial object in the category of fields.