# (Almost) co-free objects?

another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions.

According to my book (The Joy of Cats, [AdamekHerrlichSchrecker]), in a concrete category, an object $F$ is co-free over a set $X$, with canonical function $f: |F| \rightarrow X$, if for every other object $G$ and function $g : |G| \rightarrow X$ there is a unique morphism $u : G \rightarrow F$ such that $f \cdot |u| = g$ (with |.| denoting the forgetfull functor).

Now, in my own favourite concrete category, co-free objects are not always there, but for every set $X$ I am able to create a (for me meaningful) object $F$ with the property that for every injection $g : |G| \rightarrow X$ there is a unique morphism $u : G \rightarrow F$ such that $f \cdot |u| = g$.

What I would like to know is whether there is a name in literature for this property as well?

Concrete example: my favourite category is the category of 'prefix orders' or 'generalized trees', i.e. the category of partial orders in which every downward closed set $x^- = \{ y \mid y \leq x \}$ is totally ordered. Morphisms in this category are partial history preserving maps, i.e. partial functions $f : X \rightarrow Y$ such that $f(x^-) = f(x)^-$ for all $x$. In this category, for a given set X, the set $X^*$ of all total orders over subsets of X that have a maximum can be considered as a tree (using the natural prefix ordering on those total orders). This tree $X^*$ has the property I suggest, but is not co-free. I would like to call it the 'free tree' over $X$ first, but then found out it is actually more (but not exactly) like a 'co-free tree'. The canonical function in this case is max() of course.

• Given a set $X$ and a concrete category, I'm considering to define an $X$-induced object to be the smallest object that has the above property (i.e. it has the property and there is a unique map into any other object that has the property such that the obvious diagram commutes). Do you think this is an appropriate name? The intuition seems to be that the total structure may be large, but each of the elements that make up the structure are of at most the size of $X$... Does that make sense? – Pieter Cuijpers May 15 '14 at 12:37