Topos of Music

The megabook by Guerino Mazzola is based on Grothendieck topologies.

From Chloe's Friends (improbably relating Leyton's wreath product ideas to Mazzola's topological ones):

The so-called “section” amounts to a “characteristic function” for its
subset. Reading the ordered pairs in the above example backward (“contravariant,”
as it were), the function yields “on” just for those pcs that are
in the subset, and “off” elsewhere. In category theory, it is this characteristic
function which amounts to the subobject classifier for the category
Sets (Mazzola, 1126). The definition of a Topos requires the existence of
a subobject classifier for the category (Mazzola, 1127 infra). But neither
the category of abelian groups nor the category of R-modules has a subobject
classifier, and thus these categories have no topoi (Mazzola,
1127). Mazzola wants to work with Grothendieck topologies, which
require topoi (Mazzola, 1129). Mazzola’s work-around is to translate
the category of modules into the category of presheaves over modules,
notated Mod@. This is the category Func(Modopp, Sets) of contravariant
set-valued functors on Mod, which provides the required translation
from modules into a category that has a subobject classifier, namely Set
(see Mazzola, 1126, example 97, and p. 1119, example 92).
Mazzola’s entire book, The Topos of Music, is based on this.

See also in Cool Tools for the way into Grothendieck topologies as construably basic for music theory:

Finally, we note that the underlying digraph and object-graph of any proper Lewin net
(one that does conform to Lewin’s definitions) can be understood as a restriction of
a pre-order to the particular objects and labelled arrows shown in the Lewin-net. But
this is not true for Nets. A pre-order (or quasi-order) is a relation that is reflexive and
transitive. A pre-order, P, is a category in which there is at most one arrow from any
object s to any object t [15, pp. 11]. Pre-orders include partial orders and linear orders.
To make sense of the structural order of Nets, which may have multiple arrows between
s and t, one would define a forgetful functor from Net to P that includes a forgetful
functor from Grph to P, collapsing multiple parallel arrows in the Net’s underlying
digraph to one arrow. (There is a unique arrow in P from s to t whenever there is at least
one arrow in Grph from s to t.) Alternatively, construct the underlying digraph of the
chain-hom-sets of Net. These form a pre-order induced by the structure of the Net. The
labelling arrows of the Net can then be construed as a pre-sheaf over the pre-order,
leading further into topology and into category theory [18, pp. 744ff].*