TITLE: Occurrences of (co)homology in other disciplines and/or nature
QUESTION [140 upvotes]: I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics.  The idea of a family of groups linked by a series of arrows such that the composition of consecutive arrows is zero seems like a fairly general notion, but I have not come across it in fields like biology, economics, etc.  Are there examples of non-trivial (co)homology appearing outside of pure mathematics?  
I think Hatcher has a couple illustrations of homology in his textbook involving electric circuits.  This is the type of thing I'm looking for, but it still feels like topology since it is about closed loops.  Since the relation $d^2=0$ seems so simple to state, I would imagine this setup to be ubiquitous.  Is it?  And if not, why is it so special to topology and related fields?

REPLY [9 votes]: To add a touch of very belated whimsy, cohomology has manifestations in art. Here are two:

A Penrose triangle (an "impossible figure") can be viewed as a Čech $1$-cycle associated to an open covering of the image plane, taking values in the sheaf of positive functions under pointwise multiplication.
In more detail, place your eye at the origin of Cartesian space, let $P$ be a region in some plane (the screen) not through the origin, and let $C \simeq P \times (0, \infty)$ (for cone) be the union of open rays from your eye to a point of $P$.
A spatial scene in $C$ is rendered in the plane region $P$ by radial projection, discarding depth (distance) information. Recovering a spatial scene from its rendering on paper amounts to assigning a depth to each point of $P$, i.e., to choosing a section of the projection $C \to P$.
Roger Penrose's three-page article On the Cohomology of Impossible Figures describes in detail (copiously illustrated with Penrose's inimitable drawings) how a two-dimensional picture can be "locally consistent" (each sufficiently small piece is the projection of a visually-plausible spatial object) yet "globally inconsistent" (the entire picture has no visually-plausible interpretation as a projection of a spatial object).
For example, each corner of a Penrose triangle has a standard interpretation as a plane projection of an L-shaped object with square cross-section, but the resulting local depth data cannot be merged consistently in a way conforming to visual expectations.
Of course, the depth data can be merged consistently (from a carefully-selected origin!) by circumventing visual expectations. There are at least two "natural" approaches: use curved sides, or break one vertex.
M. C. Escher's Print Gallery is a planar, "self-including" rendition of the complex exponential map, viewed as an infinite cyclic covering of the punctured plane. (Contrary to mathematical convention, traveling clockwise around the center in Escher's print "goes up one level".) I know of no better reference that Hendrik Lenstra's and Bart de Smit's 2003 analysis of the Droste effect.