By Stouffer E. B.

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1 20 INSERTION OPERADS 33 2 = 02 1 FIGURE 14. An edge insertion. For the full fig-operad, the trees are slightly more elaborate to accommodate the cup product, but the insertion idea is the same. Physicists, too, are fond of insertions, leading to other operads and algebras over them. The o;-operations correspond to inserting one tree or more generally one graph into another, either into an edge or into a vertex, in a way quite different from that of Kontsevich and Soibelman. WARNING: Although terminology within graph theory is well established, in applications to physics and related parts of mathematics, terminology is not fixed.

Operads give rise to a special class of PROPS and serve much the same purpose for the main applications. Although the initial emphasis was on homotopy theory for topological spaces, from the very beginning there was implicit the study of homotopy invariant algebraic structures in other contexts. 1), while for homotopy theory it is that of a (closed) model category (CMC); see [Qui67]. We are content to work with (compactly generated) topological spaces or dg modules (chain complexes). Since there is a very simple operad Mon describing topological monoids, with Mon(n) = En, n > 1, (except for Mon(0) which is a singleton) and the more complicated IC for the homotopy invariant version of monoids, called Ate-spaces, an operad must have some special properties to capture an algebraic structure homotopy invariantly.

In this case there is another hexagon identity (which is redundant in the symmetric monoidal case) given by commutativity of a diagram similar to Figure 3 with 3AGB,c, SB,c and sA c replaced by scA®B, Sc B and sC A, respectively. In the following discussion we will focus on two main examples. The first example is the category Modk of k-modules for a commutative ring k and, more generally, differential graded k-modules. When k is a field, we will sometimes write Vec instead of Modk, gVec for the category of graded k-modules and dgVec for the category of differential graded k-modules.