Strictly unital A∞-algebras

Up-to-homotopy algebras with strict units. (Extended abstract version.)

“…They were still present in [May72] and [BoVo73], for instance, but after that, points or units generally disappeared and for a while people working with operads assumed as a starting point P (0) = ∅, in the topological setting, or P (0) = 0 in the algebraic one: see for instance [GK94]. This may have been caused because of the problems posed by those points (units); see [Hin03], or [Bur18] for two examples, or, more to the point, [Mar96] (as well as [MSS02]), where Markl constructs minimal models for operads of chain complexes over a field of zero characteristic, carefully excluding operads with non-trivial arity zero.…”

“…When introducing points (units) back in the theory of up-to-homotopy things, there are two main possibilities: either you consider strict ones, as in Stasheff's original papers [Sta63], or in [May72], [Fre17a], [Bur18], or you consider up-to-homotopy ones, or other relaxed versions of them: [BoVo73], [HM12], [MT14], and many others. You can even do both: [KS09].…”

“…The contribution we add to the present panorama is to prove the existence of Sullivan minimal models P ∞ for operads P on cochain complexes over a characteristic zero field k, with non-trivial arity zero in cohomology, HP (0) = k. In doing so, we extend the work of Markl [Mar96], (see also [MSS02]) which proved the existence of such models for non-unitary operads, P (0) = 0. Our models include the one of [Bur18] for the unitary associative operad Ass + . More precisely, our main result says: Theorem 1.1.…”

Transfer of A-infinity structures to projective resolutions

“…In this section we collect various definitions we need to precisely state our main results. See e.g., [19,13,14] for an introduction to, and further context on, A ∞objects, and [4] for an expanded version of the development below, using string diagrams. Throughout this section A and B denote graded modules.…”

Up to homotopy algebras with strict units

“…This may have been caused because of the problems posed by those points (units), including (1) [Hin03] had to correct his paper [Hin97] about the existence of a model structure in the category of operads of complexes over an arbitrary commutative ring, excluding the arity zero of the operads-or considering just the case of characteristic zero. (2) [Bur18] explains how the bar construction of a dg associative algebra with unit is homotopy equivalent to the trivial coalgebra, thus destroying the usual bar-cobar construction through which one usually builds minimal models for operads in Koszul duality theory.…”