Operads are powerful tools, and this is the
book in which to read about them.
—Bulletin of the London Mathematical
Society
Operads are mathematical devices that describe algebraic structures
of many varieties and in various categories. Operads are particularly
important in categories with a good notion of “homotopy”,
where they play a key role in organizing hierarchies of higher
homotopies. Significant examples from algebraic topology first
appeared in the sixties, although the formal definition and
appropriate generality were not forged until the seventies. In the
nineties, a renaissance and further development of the theory were
inspired by the discovery of new relationships with graph cohomology,
representation theory, algebraic geometry, derived categories, Morse
theory, symplectic and contact geometry, combinatorics, knot theory,
moduli spaces, cyclic cohomology, and, last but not least, theoretical
physics, especially string field theory and deformation
quantization.
The book contains a detailed and comprehensive historical
introduction describing the development of operad theory from the
initial period when it was a rather specialized tool in homotopy
theory to the present when operads have a wide range of applications
in algebra, topology, and mathematical physics. Many results and
applications currently scattered in the literature are brought
together here along with new results and insights. The basic
definitions and constructions are carefully explained and include many
details not found in any of the standard literature.
Readership
Graduate students, research mathematicians, and mathematical
physicists interested in homotopy theory, gauge theory, and string
theory.