This is a relevant comment on an answer given in this thread, which the comment boxes cannot conveniently accommodate.

The summary of the 'current' given after "These days" in an answer in this thread seems *technically wrong* (to me): if by 'planar graphs' is meant (which is quite usual) 'planar undirected simple finite graph-theoretic graph', then this is *not* how 'currents' are most usually formalized these days: your definition of the 'conservation condition as ' $\forall v\ \sum_{v'\sim v} f(vv')=0$ ', with $f$ having been defined **as a function on a set of unordered 2-sets** is precisely the central definition of the field of **zero-sum flows**, which is a, admittedly deceptively similar-looking-to-the-casual-observer, interesting niche-subject mostly practiced in Central Asia; but this is *not* what Weyl was investigating.

The free-abelian-group-that-is-modelled-by-said-zero-sum-flows is also (as yet) rather *irrelevant to topology*, as it still 'awaits its categorification' (this might be a gap in my knowledge, but to my knowledge the class function $\mathsf{Graphs}\xrightarrow[]{\mathrm{ZeroSumFlowGroup}}\mathsf{FreeAbelianGroups}$ is *is-not/has-not-emphasized-to-be a functor for any noteworthy category of graphs* (though I don't doubt it can, rather easily, I am simply trying to substantiate that this definition is rather irrelevant to the OP proper). The main point is that with the definition in the answer, the edges are not oriented, so a natural interpretation of a direction in which the quantity flows is lacking, the *possibility of having negative values on an edge non-withstanding*. Again: what you defined are *0-sum-flows*. To add context and a reference, let me mention that a recommendable, light-on-graph-theory-focused-on-the-linear-algebra relevant article is S. Akbari, M. Kano, S. Zare: *A generalization of 0-sum flows in graphs*. Linear Algebra and its Applications 438 (2013) 3629–3634, from whose abstract I take the following excerpt:

about which one should note that (0) calling an *arbitrary* function from the (unordered-edge)-set to the group-of-coefficients is a bit unusual, (1) the heavily-yellowed equation is essentially *exactly* your definition of the 'currents'. **This is quite a different theory from classical simplicial homology/currents-à-la-Weyl's-Spanish-paper, and rather irrelevant to the OP's question.** Please consider the crucial part of Weyl's 1923 paper:

whose first sentence I now translate, for convenience of readers:

To address the problem of the distribution of current, we assume that **each segment is given a sense of traversal [i.e.: Weyl unambiguously does first orient the lines, only later defines a function thereon; this is exactly good old simplicial homology, not zero-sum-flows]**

It will be necessary for me to interrupt now, but let me summarize that

- currently the mathematical part of the answer in this thread which starts with 'The Spanish paper of Weyl' is at least a little misleading,
**since the crucial step in the formalization, usual since the 1920s, to first orient the line segment, and only then define a function on the oriented segments, is not sufficiently emphasized**. I am not sure what the best correction is. Maybe this extended footnote is more informative than a slick corrected version. Finally, to my mind the most usual treatment of simplicial homology is of course via exterior algebra: the boundary operator in question is, needless to say, the usual derivation $\partial_2\colon\bigwedge^2\bigoplus_{v\in V}\mathbb{Z}v\longrightarrow\bigoplus_{v\in V}\mathbb{Z}v$, where $\mathbb{Z}v$ denotes the free $\mathbb{Z}$-module (=abelian group) on the set $\{v\}$.

I am aware that Baez does it, and does it correctly, with quivers, and hence has 'orientations built-in'.

I am also aware that perhaps you simply intended your $G=(V,E)$ to denote a quiver; **my little-point-to-be-made is only that this distinction will be lost on the majority of readers, and perhaps also be lost on the OP**. The orientation-issue is underemphasized currently. (Whether it *should* be emphasized I am not sure, as the OP is asking quite a different thing from being served with a summary of simplicial homology. The OP is focusing on historical questions.)

Will not be able to respond until tomorrow.

data points' seem appropriate to add; they give some sort of weight to answering the second ?-postended sentence with 'Yes, they 'just' disappeared from the mainstream of mathematical research.'ifthe question is strictly construed as asking for this specific paperbeing cited often(needless to say: [...] $\endgroup$ – Peter Heinig Oct 3 '17 at 7:18the standard 1-dimensional cycle group of a simplicial complex, which inmodel-dependentand 'metrical' considerations is better called the lattice of integral flows on the graph), and, as such, implicitly is citedcountlesslyoften, ust not explicitly with reference tothisSpanish paper; end.of.the.'needless to say bracket); now the data points: Weyl's 1923 paper does (0) not figure in [History of Topology. Elsever 1999], (1) [...] $\endgroup$ – Peter Heinig Oct 3 '17 at 7:38A History of Algebraic and Differential Topology, 1900 - 1960. Birkhäuser], (2) does not figure in Erhard Scholz:Hausdorffs Blick auf die entstehende algebraische Topologie, the latter two authors being professional historians of mathematics, (3) does not figure in [P. Hilton:A Brief, Subjective History of Homology and Homotopy Theory in This Century. Mathematics Magazine 1988], which is unsurprising in view of the title of that paper; however, notably, [...] $\endgroup$ – Peter Heinig Oct 3 '17 at 7:486more comments