10 JUSTIN R. SMITH I are both chain-maps. This implies that the boundary of EC is — t °dc ° I and the boundary of E _ 1 C is — j ode ° T- 1.2.5. We will use the symbol T to denote the transposition operator for tensor products of chain-complexes T:C ® D - • D ® C, where T(c ® d) = (_i)dim(c)dim(d)d 0 C/ and ceC,d£ D. 2. Formal coalgebras In this section we will develop an algebraic construct called a formal coalgebra. Formal coalgebras model some of the formal properties of coalgebras and facilitate the definition of coherent m-coalgebras. In order to motivate the concept, we will begin with an example: Let C be a DGA-module with augmentation e: C — 7L, and with the property that Co = Z. We want to model "coalgebra structures" on C. This immediately brings to mind a map A: C —• C2, where C2 — C ®s C. We will be interested in coalgebra structures that are not necessarily co-associative. This means that we must consider composites { A , ( l ® A ) o A , ( A ® l ) o A , . . . } . It follows that we must be concerned with maps C —• Cn for all 0 n oo. Here Cn — C ® • • • ® (7^ and by convention, n factors we will regard C° = Z, concentrated in dimension 0. It follows, that if we want to describe the set of generalized coalgebra structures on the chain-complex C, we must consider: • Hom^C, Cn) for all 0 n oo. Note that each element in Homz(C, Cn) has two numbers associated with it: its rank, which simply records the fact that it is a map to Cn (i.e., it has n copies of C in its target), and its dimension, which is equal to its degree as a map of chain-complexes — see 1.1 on page 9. • given Hom^(C, Cn) and Hom^(C, Cm), we must consider the ways of com- posing maps contained in these modules. Composition defines bilinear pair- ings o,:Homz(C,JCfn) ® Homz(C, Cm) - • Homz(C, C n + m - 1 ) , where o?- represents the composition of maps that plugs a map of Hom^(C5 Cn) into the ith factor of the target of a map of Hom^(C, Cm). In other words, if / G Hom^C, Cm) and g E Hom^(C, Cn), / o8- g is the composition: c l c m = c1-1 ®c®cm~{ ® • • • 1 ® flf ® 1 ® • • • 1 c i _ 1 ® cn ® c m _ i cfn+m_1 Clearly, z can only run from 1 to m. Careful consideration of how the compositions are defined implies that these oper- ations satisfy the following "associativity" conditions: • (u Oj- v) Oj W — U 0(+j_i (u Oj w)', • if j i then u o , ^ ^ ) . ! (v oj w) = (-i^egCtiJ-deg^)^ 0 i (w 0 . w ) ( b y t h e Koszul convention).

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