6
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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 $\endgroup$ – Max Mar 11 at 13:45
  • $\begingroup$ @Max Sounds like isofibration to me, though Joy of Cats seems to use "transportable". $\endgroup$ – Arnaud D. Mar 11 at 13:55
  • $\begingroup$ @ArnaudD. Thanks for the link ! $\endgroup$ – Max Mar 11 at 14:45
  • $\begingroup$ @Max I posted a follow-up question based on your comment: math.stackexchange.com/q/3144177/71829 $\endgroup$ – Keshav Srinivasan Mar 12 at 3:39
2
+50
$\begingroup$

The answer, just as for so many questions which begin "in an arbitrary category, can I choose...", is that you need a very strong form of the axiom of choice to make things work.

First, note that not every category is well-powered. That is, there is no reason that we should be able to index the collection of subobjects of objects in a given category with any set, so if you're pedantic about wanting a function which does that indexing, you could have problems.

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.

$\endgroup$
0
$\begingroup$

In a Cartesian closed category $\mathcal C$ with a subobject classifier $\Omega$, you have, for any object $A$ of $\mathcal C$ has an exponential $\Omega^A,$ which is the object of $\mathcal C$ representing the subobjects of $A.$

Then any map from $1\to \Omega^A$

  • corresponds to a distinct map $A\times 1\to\Omega$ by the definition of Cartesian closed,
  • which correspponds to a distinct map $A\to\Omega,$
  • and this corresponds to a distinct class of monomorphism to $A,$ by the property of $\Omega$ being a subobject classifier.

These conditions are held by all topoi. This includes set theories without choice and set theories with intuitionist logic (i.e. without excluded middle.)

$\endgroup$

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.