  Our main objective is to demonstrate how homological perturbation theory
(HPT) results over the last 40 years immediately or with little extra work give
some of the Koszul duality results that have appeared in the last decade.
  Higher homotopies typically arise when a huge object, e. g. a chain complex
defining various invariants of a certain geometric situation, is cut to a small
model, and the higher homotopies can then be dealt with concisely in the
language of sh-structures (strong homotopy structures). This amounts to precise
ways of handling the requisite additional structure encapsulating the various
coherence conditions. Given e. g. two augmented differential graded algebras A
and B, an sh-map from A to B is a twisting cochain from the reduced bar
construction of A to B and, in this manner, the class of morphisms of augmented
differential graded algebras is extended to that of sh-morphisms. In the
present paper, we explore small models for equivariant (co)homology via
differential homological algebra techniques including homological perturbation
theory which, in turn, is a standard tool to handle sh-structures.
  Koszul duality, for a finite type exterior algebra on odd positive degree
generators, then comes down to a duality between the category of sh-modules
over that algebra and that of sh-comodules over its reduced bar construction.
This kind of duality relies on the extended functoriality of the differential
graded Tor-, Ext-, Cotor-, and Coext functors, extended to the appropriate
sh-categories. We construct the small models as certain twisted tensor products
and twisted Hom-objects. These are chain and cochain models for the chains and
cochains on geometric bundles and are compatible with suitable additional
structure.
