Types of quantum field thories
In the context of electromagnetism, Kirchhoff’s laws are a kind of coarse-grained form of Maxwell's equations. Where the latter deal with infinitesimal quantities, Kirchhoff’s law involve macroscopic current? and voltage? in electrical circuits?.
One speaks of two laws,
the Kirchhoff voltage law (abbreviated KVL)
the Kirchhoff current law (abbreviated KCL)
Nevertheless, Kirchhoff’s laws preserve the cohomological nature of Maxwell’s equations: where the latter involves de Rham cohomology and hence “infinitesimal cochains”, Kirchhoff’s laws can neatly be formulated in terms of cochains on finite cell complexes. (See for instance appendix B of (Frankel) or the section “Basic concepts” in (Baez)).
lecture 7 in
appendix B in
section Basic concepts in
The cohomological nature of Kirchoff’s laws was maybe first made explicit in
A textbook on electromagnetism amplifying this point of view is