0
$\begingroup$

I'm currently looking at the differences between the treatment of the notion of isomorphism in category theory and set theory. One interesting difference is how the two systems treat isomorphism. An important class of differences are where bijective homomorphisms are not always isomorphisms (e.g. Top), however, I'm interested in the case where we have a (category-theoretic) isomorphism that is not bijective. The two examples I've managed to come up with so far are:

(1.) The incredibly trivial: e.g. The category consisting of a single set $\{0,1\}$, whose only morphism is defined by $f(0)=0$ and $f(1) = 0$.

(2.) The not-really-involving-functions-at-all-trivial: e.g. The category of Proofs, where objects are sentences and arrows are equivalence classes of proofs. Isomorphisms here are equivalence classes of proofs between equivalent propositions, so aren't bijections, but only because they aren't really functions.

Are there interesting examples of non-bijective isomorphisms from elsewhere in category theory (where the example has interesting mathematical content, but this isn't just because the arrows do not have underlying set-theoretic functions)?

$\endgroup$
  • 1
    $\begingroup$ Functors preserve isos, so of constructs (concrete categories with underlying functor to category $\mathbf{Sets}$) nothing can be expected. All isos will be sent to isos in $\mathbf{Sets}$ which are the bijections. $\endgroup$ – drhab Sep 27 '17 at 8:11
  • 1
    $\begingroup$ I don't understand your first example. What category are you considering here ? $\endgroup$ – Arnaud D. Sep 27 '17 at 8:58
  • $\begingroup$ @drhab Thanks: What about the case where there's no composition preserving functor back to set, but nonetheless an underlying set? Any interesting mathematics lurking there? $\endgroup$ – Neil Barton Sep 27 '17 at 9:46
  • 1
    $\begingroup$ Might be, but I have never been on that territory (and also never encountered it) so sorry, but I can't help you. $\endgroup$ – drhab Sep 27 '17 at 9:51
  • 1
    $\begingroup$ Arguably, it makes no sense to speak of "underlying sets" when you are not refering to a functor into $\mathsf{Set}$. You could map objects 'randomly' to any sets you like; I don't see what the point of this is. Morphisms between things (categories e.g.) should respect all the structure, not just some of it (only objects e.g.). - Isomorphisms may only be non-bijective simply because they don't have underlying functions. I don't think you can say anything more meaningful than that. $\endgroup$ – Stefan Perko Sep 27 '17 at 10:25
3
$\begingroup$

Consider the homotopy category, where the objects are topological spaces, and the morphisms are homotopy classes of continuous functions. Then the inclusion of the unit circle into the punctured plane is an iso in this category, with inverse the radial projection map.

Freyd showed that this category is not concrete - there is no nice, faithful functor from the homotopy category to the category of sets, which is why drhab's point that functors preserve isos doesn't get in the way. Indeed, the natural underlying set mapping from the way I've described the category is not functorial.

$\endgroup$

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.