  The paper is the survey of the modern results and applications of the theory
of homotopes. The notion of a well-tempered element in an associative algebra
is introduced and it is proven that the category of representations of the
homotope constructed by a well-tempered element is the heart of a suitably
glued t-structure. Hochschild and global dimensions of the homotopes are
calculated. The homotopes constructed by generalized Lapalce operators in
Poincare groupoids of graphs are studied. It is shown that they are quotients
of Temperley-Lieb algebras of general graphs. The perverse sheaves on a
punctured disc and on a 2 dimensional sphere with a double point are identified
with representations of suitable homotopes. Relations of the theory to
orthogonal decompositions of sl(n, C) into the sum of Cartan subalgebras, to
classifications of configurations of lines, to mutually unbiased bases, to
quantum protocols, to generalized Hadamard matrices are discussed.
