Let me begin by noting that I know quite little about category theory. So forgive me if the title is too vague, if the question is trivial, and if the question is written poorly.


Let $\mathcal{C}$ and $\mathcal{D}$ be categories. Say that a functor $T: \mathcal{C} \to \mathcal{D}$ has property X (maybe there is a real name for this property?) if a morphism $f: A \to B$ between objects of $\mathcal{C}$ is an isomorphism whenever $T(f): T(A) \to T(B)$ is an isomorphism. For example, the obvious forgetful functor $CH \to Set$ where $CH$ is the category of compact Hausdorff spaces has property X because a continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism.


Here is my question. Is there a (nontrivial) functor $T: LCH \to \mathcal{D}$ from the category of locally compact Hausdorff spaces to some category $\mathcal{D}$ with property X? Even better, can we assume that $LCH$ is a subcategory of $\mathcal{D}$ and that $T$ is a forgetful functor?


I don't care to specify what I mean by "nontrivial", except that the "identity" functor from $LCH$ to itself doesn't count. I want it to be genuinely easier to decide whether or not a morphism is an isomorphism in $\mathcal{D}$. If there happen to be lots of ways to do this, perhaps it will help to know that my interest comes from some problems in analysis.


Thanks in advance!



Category Theory / Topology Question
The property $X$, as you call it, is well-known. A functor with this propery is said to "reflect isomorphisms". Another example of such a functor is the geometric realization functor from simplicial sets to compactly generated Hausdorff spaces. There are all sorts of ways of building a category $D$ and a functor $T$ with the properties you want, however, depending on what you want to do, different answers can be more suiting. For example, you can let $D$ be the category $Sh(CH)$ of sheaves on the site of compact Hausdorff spaces and $T$, be "Yoneda": if $Y$ is a $LCH$ space, then $T(Y)$ is the sheaf that assigns each compact Hausdorff space $X$ the set $Hom(X,Y)$. Then, it is a simple exercise to verify that $T$ is fully-faithful and that $T(f)$ is an isomorphism implies that $f$ is (SINCE every locally compact Hausdorff space is compactly generated).


