The notion of operads was introduced by Peter May
[
May72
]
in order to study
iterated loop spaces
, although the idea is based the work
[
BV73
]
of Boardman and Vogt, which in turn is based on the
study of “higher homotopies” appeared in
problems in algebraic topology. Huebschmann’s
[
Hue
]
ranges from the study of higher homotopies in the
preoperad era to the modern applications.
Nowadays, the notion of operads are used in
algebra
and
physics
. The study of operads is much more active in these
areas than in algebraic topology. According to
Maninand Borisov
[
BM
]
, there are three interpretations of operads.

a tool to study algebraic structures on a set
together with operations satisfying certain
conditions

an example of a
categorification
of
graph theory

abstraction of computational processes in tensor
networks and quantum computations
There used to be only two choices when one wanted to
study operads, i.e. the book by Boardman and
Vog
[
BV73
]
and the book
[
May72
]
by Peter May. Now there are many choices, thanks to
the popularity of operads.
The book
[
MSS02
]
by Markl, Shnider, and Stasheff contains a lot of
topics. It might be a good idea to browse this book
first in order to grasp the whole subject.
Markl’s
[
Mar
]
serves as a shorter introduction. It contains a good
list of references. Another choice is the book by Kriz
and May
[
KM95
]
. One of the most concise expositions is
Stasheff’s
“WHAT IS...an Operad”
in the Notices of the A.M.S. vol. 51 No. 6.
Ionescu’s
[
Ion
]
contains
PROPs
.
It might be interesting to take a look at the
proceedings of the workshop at Osnabrk in 1998, which
can be available from
the preprint server SFB 343 in Bielefeld University
.
[BM]
D. Borisov and Yu. I. Manin. Internal
cohomomorhisms for operads,
arXiv:math.CT/0609748
.
[BV73]
J. M. Boardman and R. M. Vogt.
Homotopy invariant algebraic
structures on topological spaces
. SpringerVerlag, Berlin, 1973.
[Hue]
Johannes Huebschmann. Origins and breadth of the
theory of higher homotopies,
arXiv:0710.2645
.
[Ion]
Lucian M Ionescu. From operads and PROPs to
Feynman Processes,
arXiv:math/0701299
.
[KM95]
Igor Kř#382; and J. P. May. Operads,
algebras, modules and motives.
Ast
é§»isque
, (233):iv 145pp, 1995.
[Mar]
Martin Markl. Operads and PROPs,
arXiv:math.AT/0601129
.
[May72]
J. P. May.
The geometry of iterated loop spaces
. SpringerVerlag, Berlin, 1972.
[MSS02]
Martin Markl, Steve Shnider, and Jim Stasheff.
Operads in
algebra, topology and physics
, volume 96 of
Mathematical Surveys and
Monographs
. American Mathematical Society, Providence, RI,
2002.