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Is there a name for those categories where objects posses a given structure and every bijective morphism determines an isomorphism between the corresponding objects?

Examples of categories of that type abound: Gr, Set, ...

An specific example of a category where the constraint doesn't hold is given by Top: a morphism there is a continuous function between topological spaces. Now, it is easy to give here a concrete example of a bijective morphism between [0,1) and $\mathbb{S}^{1}$ that fails to be an isomorphism of topological spaces. In fact, much more is known in this case, right?

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There's a subtlety in this question. A category doesn't have a notion of bijection unless it's concrete, i.e. has a distinguished forgetful functor, and the same category may have different concretizations. – Qiaochu Yuan Dec 8 '09 at 20:46
*forgetful functor to Set. – Qiaochu Yuan Dec 8 '09 at 20:48
... notion of bijection unless it's concrete. - Really? – J. H. S. Dec 8 '09 at 20:51
Yes, really. This is not an issue about sets versus classes, either. The point is that you can't define the property of a map being a bijection in terms purely of objects and morphisms. In some cases, "left and right cancellable" is a good generalization of "bijection". – David Speyer Dec 8 '09 at 20:56
For example, one definition of an abelian category is an additive category, with kernels and cokernels, such that any map which is left and right cancellable is invertible. – David Speyer Dec 8 '09 at 20:58

4 Answers 4

up vote 33 down vote accepted

The comments on the question point out that it's not really well-posed: the property "bijective" isn't defined for morphisms of an arbitrary category.

However, for maps between sets, "bijective" means "injective and surjective". A common way to interpret "injective" in an arbitrary category is "monic", and a common way to interpret "surjective" in an arbitrary category is "epic". So we might interpret "bijective" as "monic and epic".

Then JHS's question becomes: is there a name for categories in which every morphism that is both monic and epic is an isomorphism? And the answer is yes: balanced.

It's not a particularly inspired choice of name, nor does it seem to be a particularly important concept. But the terminology is quite old and well-established, in its own small way.

Incidentally, you don't have to interpret "injective" and "surjective" in the ways suggested. You could, for instance, interpret "surjective" as "regular epic", and indeed that's often a sensible thing to do. But then the question becomes trivial, since any morphism that's both monic and regular epic is automatically an isomorphism.

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This isn't quite the question you asked, but does address the notion of ''bijective'' morphisms in categories, so I hope you'll forgive this digression.

The examples you've mentioned - Set, Gp, Top - are all concrete, meaning they are equipped with a forgetful functor U to Set. We say a morphism f in a concrete category C is injective if its image Uf is injective, i.e., monic in the category Set. Dually, f is surjective if Uf is surjective. One usually thinks of concrete categories as "sets with structure", so these definitions coincide with the common use of such terminology: e.g., we call a map of spaces surjective when the underlying map of sets is.

So we have four adjectives to use for arrows in C: monic, epic, injective, surjective. It's an easy exercise to see that all injections are monic and all surjections are epic. The converse is not true in general, but finding examples of monos that aren't injective and epis that aren't surjective can be tricky, and here's why.

Often, particularly in ''algebraic'' examples, the functor U : CSet has a left adjoint F. When this is the case, it is an easy exercise to see that every mono must be injective. Dually, if U has a right adjoint, then every epi is surjective. So for example, the forgetful functor U : TopSet has both adjoints, and hence for spaces the notions injective/surjective and monic/epic coincide, at which point Tom's post answers your question.

Here are some examples of concrete categories where these concepts differ, all of which can be found in Francis Borceux's Handbook of Categorical Algebra (I think). In the category of divisible abelian groups, the quotient map $\mathbb{Q} \rightarrow \mathbb{Q}/\mathbb{Z}$ is monic, though it's clearly not injective. In the category of monoids, the inclusion $\mathbb{N} \rightarrow \mathbb{Z}$ is epic, though not surjective. In the category of Hausdorff spaces, the epis are continuous functions with dense image, so also need not be surjective.

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"every bijective morphism determines an isomorphism"

I think you mean that the forgetful functor reflects invertibility.

Let $\bf A$ be the "category of objects with structure" and their structure-preserving maps (homomorphisms), $\bf S$ the category of carriers (maybe sets and functions) and $U:{\bf A}\to{\bf S}$ the "forgetful" functor between them. In fact, just let $U:{\bf A}\to{\bf S}$ be any functor you like.

Now let $f:X\to Y$ be any morphism of $\bf A$. You are saying that, whenever $U f:U X\to U Y$ is an isomorphism (such as a bijection) then $f$ was already an isomorphism.

The forgetful functor from any category of algebras has this property, as more generally does the right adjoint of any monadic adjunction. However, the underlying set functor from the usual category of topological spaces does not, because there are many different topologies that can be put on a set.

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As other people commented, the language of categories is richer than the language of sets with structures (Bourbaki structures). There are many categories, where objects don't have an underlying set.

However, one can restate the property you formulate as follows: the category C admits a faithful conservative functor to Sets. Then we can interpret the fiber of this functor over a given set S as the set of structures on S and call the functor a forgetful functor. By faithfulness the homs in C will be subsets of those in Sets, and we can say that the homs in C preserve the structure.

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