# Is there a notion of a transversal of subobjects?

If $$a$$ is an object in a category $$C$$, then a subobject of $$a$$ is an isomorphism class of monomorphisms with codomain $$a$$. The subobjects of all the objects in $$C$$ partitions the class of monomorphisms in $$C$$ into a bunch of subclasses. My question is, is there a notion of a transversal, i.e. a function which picks out a representative from each of the subclasses?

In the case of a concrete category, where you have a forgetful functor $$U:C\rightarrow Set$$, there is an obvious choice of representative for a given subobject of an object $$a$$: the unique monomorphism $$f$$ in the subobject such that $$U(f)$$ is an inclusion map from some subset of $$U(a)$$ to $$U(a)$$. But is it possible to define a transversal even for non-concrete categories, in a such a way that the representatives are “compatible” with one another?

Is there already a name for this structure?

• If your $U$ is not very nice, it may happen that no such $U(f)$ satisfies this. Isn't there a name for concrete categories where you can always lift structures along isomorphisms in the base category ? I seem to recall seeing that name in Joy of Cats. This isn't very important though Mar 11, 2019 at 13:45
• @Max Sounds like isofibration to me, though Joy of Cats seems to use "transportable". Mar 11, 2019 at 13:55
• @ArnaudD. Thanks for the link ! Mar 11, 2019 at 14:45
• @Max I posted a follow-up question based on your comment: math.stackexchange.com/q/3144177/71829 Mar 12, 2019 at 3:39

Even if you insist that your category be well-powered (or even small), it's easy to show that we need at least the axiom of choice. Given a cardinality $$\kappa$$, consider the category of sets of cardinality less than $$\kappa$$, with exactly one morphism between any pair of isomorphic sets, and morphisms from every set to $$1$$. Then all of the morphisms to the terminal object are monos since there are no non-trivial parallel pairs, and so an indexing of the subobjects is a choice of set of each cardinality less than $$\kappa$$.
Continuing in this vein, we can show that "every category has a transversal" is equivalent to the statement "every category has a skeleton". Given a category $$C$$, we can consider the category $$D$$ whose objects are objects of $$C$$, with exactly one morphism between any pair of isomorphic objects. Freely adjoin to this setup a terminal object. Then a transversal for $$D$$ provides a skeleton for $$C$$. Conversely, given $$D$$, we can consider the category $$\mathrm{mono}(D)$$ whose objects are monos in $$D$$ and whose morphisms are those completing triangles of monos with the same codomain; the isomorphism classes in this category are of course the subobjects of $$D$$, so a skeleton of $$\mathrm{mono}(D)$$ gives a transversal of $$D$$. You may be aware that "every category has a skeleton" is equivalent to the axiom of choice for proper classes; again, if we restrict ourselves to well-powered categories then the above construction doesn't hold water, but it should hopefully shed light on your question.