The concept of a subobject classifier is of course standard and ubiquitous. But is there any nontrivial example of an unrestricted slice classifier?


Specifically, what I mean by this is, is there any non-preorder category with pullbacks with a morphism m into an object X such that ALL other morphisms can be taken as a pullback of m along some morphism into X? And, if so, is it even possible to have furthermore that parallel morphisms from any object Y into X are equal just in case the pullbacks of m along them are isomorphic as objects of the slice category over Y?


Naturally, if we demand further structure on the category (e.g., local cartesian closure), this becomes impossible by Cantor type arguments in its internal logic, but if we only demand pullbacks, can it be done?



Is it possible for a nontrivial category to have a slice classifier?
It looks like such categories may be rather easy to construct. The following example should give the general idea: take the category of sets $V\_\alpha$ of cardinality less than or equal to $\alpha$, for some infinite cardinal $\alpha$. The morphism classifier will be the set $C$ of cardinals up to and including $\alpha$, and the universal morphism should be $S \to C$ where the fiber over a cardinal $\beta$ is a set of cardinality $\beta$. 


Given any function $f: Y \to X$, the classifying morphism $\chi\_f: X \to C$ takes $x$ to the cardinal number of $f^{-1}(x)$. 


Hopefully I haven't made any dumb mistakes... 


