A little searching turned up:

**Ring epimorphisms and C(X)** by Michael Barr, W.D. Burgess and R. Raphael (article).

They consider this question for rings of the form of continuous functions on a topological space. They quote the following characterisation of epimorphisms in the category of commutative rings:

**Proposition**: A homomorphism f : A → B is an epimorphism if and only if for all b ∈ B there exist matrices C, D, E of sizes 1 × n, n × n, and n × 1 respectively, where *(i)* C and E have entries in B, *(ii)* D has entries in f(A), *(iii)* the entries of CD and of DE are elements of f(A) and *(iv)* b = CDE. (Such a triple is called a *zig-zag* for b.)

This seems a little more complicated than localisation, though I haven't checked the details.

They then go on to prove that

**2.12**: A subspace Y of a perfectly normal first countable space X induces an epimorphism if and only if it is locally closed.

If I understand all the terminology correctly, then this implies that

C([0,1],ℝ) → C((0,1),ℝ)

is an epimorphism.

There are plenty more references in that article, and it would be nice to have an actual zig-zag for this situation. But in the spirit of open-source mathematics, I thought I'd post this and see if someone (possibly me later on) can fill in the details.

**Added Later:** The example I gave: C([0,1],ℝ) → C((0,1),ℝ) is a localisation. It is obtained by inverting all functions in C([0,1],ℝ) which are zero **only** at the end-points. Given a function f ∈ C((0,1),ℝ), there will be a function g ∈ C([0,1],ℝ) which is non-zero apart from at 0 and 1 and which goes to 0 at 0 and 1 faster enough that the product g f also goes to 0 at the end-points. Then g f is (the restriction of something in) C([0,1],ℝ) and g becomes invertible in C((0,1),ℝ). So f = g^{-1} (g f) is in the specified localisation of C([0,1],ℝ).

Indeed, the Barr et. al. paper comments on the fact that in all the examples they consider (function rings), the zig-zag has length 1. I conjecture that if the zig-zags always have length 1 (for a particular function f: A → B), then B is formed by a localisation on A. A possibly stronger version of this conjecture would be that this is an if-and-only-if. In which case, finding a counter-example to Anton's conjecture would involve finding a case where there was a zig-zag of length 2. I suspect that a universal construction would be the best approach to finding one.

In the spirit of wiki-ness and only doing a little at a time, I'll leave this here.

**Added Even Later:** (Should I timestamp these? I know that the system does so, but is it useful to embed them in the edit?)

Here's one direction for my conjecture above.

If B = S^{-1}A, then for b ∈ B, we have b = s^{-1}a for some s ∈ S and a ∈ A. Then we put C = s^{-1}, D = s, E = b = s^{-1} a. Then CD = 1, DE = a, D ∈ f(A), and CDE = b. So in a localisation, zig-zags have length 1.

dolie over surjective functions (see en.wikipedia.org/wiki/Epimorphism). Another way to see this is to note that the forgetful functor $\mathbf{Top}\to\mathbf{Set}$ has a right adjoint (indiscrete topology), and therefore preserves colimits, and in particular pushouts (see also Ex. 4, p. 72 of Mac Lane) $\endgroup$4more comments