The associated homotopy category is defined by localizing "C" with respect to the weak equivalences. The homotopy category Ho("M") is then the localization with respect to the weak equivalences. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following logical equivalences demonstrate that associativity is a property of particular connectives. The following logical equivalences demonstrate that distributivity is a property of particular connectives. Biadjacency matrices may be used to describe equivalences between bipartite graphs, hypergraphs, and directed graphs. Dually, the injective model structure is similar with cofibrations and weak equivalences instead. There are numerous examples of categorical equivalences from many areas of mathematics. Asymptotic equivalences, formula 107, are true if formula 108. Any set of automorphic equivalences are also regular equivalences. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. In other tuning systems, such enharmonic equivalences in general do not exist. In any other tuning system, such enharmonic equivalences in general do not exist. Many identities can be proved, which establish equivalences between particular formulas. In logic, many common logical equivalences exist and are often listed as laws or properties.