When bijective morphism become ismorphism? Epi and mono morphism（morphism means here the morphism in the category）is not always isomorphism.
For example, in a category of topological space, bijection continuous map is not always homeomorphism.
But like in a category of group or formal group, we can declare that bijection hom is always isomorphism.
This can be proved easily by definition.
My question is : Are there any list or characterization of such （bijective morphism is always ismorphism）category? Are there known results?
Thank you in advance.
 A: Here is a result that I think is interesting.  In order to talk about some morphism $f : x \to y$ in an arbitrary category $\mathcal{C}$ as being a bijection, we need some way to interpret $f$ as a set function.  This is because the definition of "bijection" refers to how $f$ acts on elements.  One way we could interpret $f$ as a set function is if we had a functor $U : \mathcal{C} \to \mathrm{Set}$, for then any morphism $f : x \to y$ in $\mathcal{C}$ would give us a set function $Uf : Ux \to Uy$, and we could ask if $Uf$ is a bijection.  Many categories of interest "come equipped" with such an underlying set functor $U : \mathcal{C} \to \mathrm{Set}$.  Your example of topological spaces is one, posets are another, groups are another.  Algebra is full of examples.
If we have a functor $U : \mathcal{C} \to \mathrm{Set}$, then we can think of any morphism $f : x \to y$ in $\mathcal{C}$ as a set function, but it's usually only helpful to think this way if $U$ preserves and reflects things we care about.  For example, a group homomorphism $f : G \to H$ is monic/epic/iso (in $\mathrm{Grp}$) if and only if it's injective/surjective/bijective (in $\mathrm{Set}$).  So in this case we usually just think of a group homomorphism as a special kind of set function between the underlying sets of the groups.  Any functor $U : \mathcal{C} \to \mathrm{Set}$ sends isomorphisms to bijections, so there's always some connection.  But for an arbitrary functor $U : \mathcal{C} \to \mathrm{Set}$ there's no guarantee that monic/epic morphisms in $\mathcal{C}$ have anything to do with injective/surjective functions in $\mathrm{Set}$, and bijections in $\mathrm{Set}$ don't always give isos in $\mathcal{C}$.
So the natural question is: what conditions on $U$ ensure that monic/epic/iso morphisms in $\mathcal{C}$ have some relation to injective/surjective/bijective functions in $\mathrm{Set}$?  One direction of each case is often satisfied because $U$ is faithful: that is, if $f, g : x \to y$ are two morphisms such that $Uf = Ug$, then $f = g$.  This happens for topological spaces, groups, and lots of other examples.  This means, for example, that we can think of a continuous function as a set function that satisfies a certain property—in this case, continuity.  If $U$ is faithful, then we have one direction of each case: for any $f : x \to y$ in $\mathcal{C}$,

*

*If $Uf : Ux \to Uy$ is injective, then $f : x \to y$ is monic

*If $Uf : Ux \to Uy$ is surjective, then $f : x \to y$ is epic

*If $f : x \to y$ is an isomorphism, then $Uf : Ux \to Uy$ is a bijection

It is not too hard to look up examples where $U$ is faithful but the converses of the first two don't hold (the third, recall, holds for any functor $U$).  The converse holds in the third case if $\mathcal{C}$ is balanced, in the sense introduced in the comments.  For whatever reason, the converse fails for the second property much more often than for the first, in nature.
One natural question now is: what conditions can we put on a faithful functor $U : \mathcal{C} \to \mathrm{Set}$ such that $f : x \to y$ is iso when $Uf : Ux \to Uy$ is bijective?  Everything I've said so far has been pretty elementary, but for this question there's an interesting theorem.  A functor $U : \mathcal{C} \to \mathrm{Set}$ is monadic if there's an appropriate kind of "free" functor $F : \mathrm{Set} \to \mathcal{C}$.  Examples of monadic functors include the "underlying set" functor for groups, rings, modules, and the category of compact Hausdorff topological spaces.  Examples of nonmonadic functors include the "underlying set" functors for posets, fields, and the category of all topological spaces.  Monadic functors are interesting for our question because:

*

*There are lots of examples.  The category-theoretic approach to universal algebra is essentially the study of monads.

*There are various effective recognition results for monadic functors, usually called "monadicity" or "tripleability" theorems.  So it's easy to spot a monad when you meet it on the street.

*Suppose $U : \mathcal{C} \to \mathrm{Set}$ is monadic and $f : x \to y$ is a morphism in $\mathcal{C}$.  If $Uf : Ux \to Uy$ is a bijection, then $f$ is an isomorphism.  This is because any monadic functor is equivalent to the underlying set functor for the category of algebras for its monad, and this functor clearly reflects isomorphisms.

So this is one result of the kind you ask for.  If we're in a situation where we can talk about a morphism in a category $\mathcal{C}$ being bijective then we have a functor $U : \mathcal{C} \to \mathrm{Set}$, and if $U$ is monadic then any bijective morphism in $\mathcal{C}$ is iso.  Moreover, any monadic functor is faithful, so we get one nice connection between monos/epis and injections/surjections.  I think this is a nice sort of explanation for why bijections do coincide with homeomorphisms when we're talking about compact Hausdorff spaces, even though they don't coincide for general topological spaces.  That said, checking whether a functor reflects isomorphisms is usually easier than checking whether it's monadic—and indeed, it's usually part of checking that it's monadic.
This definitely isn't the only class of categories in which every bijection is an isomorphism.  In fact, let $\mathcal{C}$ be a category that is locally small—that is, such that for all objects $x$ and $y$, the class $\mathcal{C}(x, y)$ of morphisms from $x$ to $y$ is a mere set, rather than a proper class.  Then there is a functor $U : \mathcal{C} \to \mathrm{Set}$ such that if $Uf : Ux \to Uy$ is a bijection, then $f : x \to y$ is an isomorphism.  This illustrates one reason to be careful about how you're interpreting a morphism in $\mathcal{C}$ as a set function: it is very often possible to find some functor $\mathcal{C} \to \mathrm{Set}$ that makes isomorphisms the same thing as bijections.
