Standard Definition of Ramified (or Branched) Cover of Topological 3-Manifolds My work with polyhedral 3-manifolds requires me to come up with a robust definition of a ramified cover in that setting. However, I want to be sure that my definition fits into the wider scheme. Therefore I would like a standard definition of a ramified (or branched) cover of a topological 3-manifold, from a book (or if absolutely necessary a paper). This question is of course very closely related but not the same, as I would like a specific literature reference; and also I want an answer specific to topological 3-manifolds, that doesn't rely on any other structure, and which can be regarded as standard.
It is very common for topologists to talk about covers of $S^3$ branching along a link. Note also the following from Manifold Atlas:

Tietze remarked that lens spaces can also be constructed by identifying the surfaces of two solid tori (as Dyck did it) and as branched covers of $S^3$ (with ramification points). This idea is attributed to W. Wirtinger, his teacher; traces of it can be found in Heegard's dissertation (1898).

The fact that the idea of branched covers of 3-manifolds traces back at least to Heegard leaves me in no doubt that there must be a proper definition somewhere in the literature.
The definition should allow for ramification along an embedded graph, not just a disjoint union of circles, and outside of the ramification locus the map should of course just be a topological covering map (probably of finite degree).

EDIT – It seems I was not clear enough originally. I'm looking for a definition of a ramified (or branched) covering map. The definition should go something like this:

Let $N$ be a 3-manifold and $M$ be a topological space. A ramified (or branched) cover of $N$ is a continuous surjection $f:M\to N$ such that there is an embedded graph $\Gamma\in N$—not necessarily connected, possibly empty, not containing isolated vertices, etc. etc.—such that the restriction $f:M\setminus f^{-1}(\Gamma)\to N\setminus\Gamma$ is a standard covering map (probably of finite degree?); $f^{-1}(\Gamma)$ is homeomorphic to a graph, and locally around any point $x\in f^{-1}(\Gamma)$, $f$ has a particularly nice form (i.e. it somehow looks like sheets of a covering coming together).

So the ramified cover is primarily a map onto a manifold. I'm looking for a reference to a standard definition, like the above, which comes from a  book (or a paper if necessary). Sorry if my original terminology was confusing. I believe, however, that it was no more confusing than the standard ambiguous terminology of covering spaces.

FINAL EDIT – Just to clarify, I cannot assume that either space is an orbifold, nor do I think that the definition can be stated in full generality using the language of orbifold projections. Here is what may be considered a prototypical example of a ramified cover (topologically $S^3\to S^3$):

There is a Euclidean simplex—call it $\Delta$—whose dihedral angles are $\pi/4,\pi/3,\pi/2,\pi/2,\pi/2,\pi/4$. If we take the double of $\Delta$—that is, we take a reverse-orientation copy of it and identify their corresponding faces by isometries—we get a 'polyhedral structure' on $S^3$: call this $N$. This shown in the figure on the left in the above image (arrows/colours denote face identifications). The singular locus of $N$ is the wireframe of the simplex.
Using $\Delta$ and its reverse-orientation copy, we can build another polyhedral structure on $S^3$, which is shown on the right (arrows/letters denote face identifications): call this $M$. The singular locus of $M$ is the union of the red, blue and black edges, with the appropriate identifications. Note that $M$ is not and orbifold, as the angle around the central red edge is $3\pi/2$. There is then an obvious map $f:M\to N$, which just sends each simplex of $M$ to the relevant simplex of $N$, depending on the orientation. Outside of the singular locus of $M$, $f$ is a covering map of degree 6, and one can very easily describe how the ramification of $f$ looks along each singular edge. For me, this map captures the essence of a ramified cover of 3-manifolds.
If we were, as Lee Mosher has suggested, to view $M$ as an orbifold with empty singular locus and $f$ as an orbifold covering projection, then we'd have to change the angles on $N$ to: red - $\pi/3$, blue - $\pi/4$, black - $2\pi$, and grey the same. But I don't think such an orbifold structure is possible on $S^3$; for example, the vertex common to the blue and black edges now only has 2 singular edges coming out of it, with angles $2\pi/3$ and $\pi$.
 A: To augment the answer of @PaulPlummer, you will find "covering orbifold" defined in Thurston's book The geometry and topology of 3-manifolds, in Chapter 13, Definition 13.2.2, on page 303. And in that definition you will find a "projection $p$" which, while no terminology is offered there, we can refer to as the "orbifold covering projection".
For example, one can deduce the following from the linked answer of @MoisheKahan:

Theorem: Given two Riemann surfaces $R,S$, a holomorphic map $f : R \to S$ is a branched covering if there exists an orbifold structure on $S$ with discrete singular locus such that $f$ is an orbifold covering projection.

Using this concept, one can formulate a similar definition which answers your question, as follows:

Definition: A map $f : M \to N$ is a ramified cover if there exists an orbifold structure on $N$ with singular locus $\Gamma$ such that $f$ is an orbifold covering projection.

Keep in mind, in this statement the domain manifold $M$ is being regarded as an orbifold with empty singular locus, as you will see explained on the bottom of the previous page 302.
A: I think you want the notion of orbifold as mentioned by Moishe Kohan in the answer you link too. You can google for orbifold and that probably is discussed in any 3-manifold book post Thurston. A good place to look is chapter 13 of Geometry and topology of three-manifolds . Another reference is Peter Scott's The geometries of 3-manifolds.

I don't really deal much  with orbifolds, but I see a potential problem (maybe it actually isn't a problem) with what you are asking for. Normally polyhedra have a geometric structure, with angles and such, and the finite groups come from this geometry. The problem is not all angles can come from finite groups. As a simple example, consider a triangle where one of the angles is 1 radian. The group you would want at that point is a dihedral group action, but rotation of plane by 1 radian will not give a finite group.
