There is a brief entry at bar construction together with a blog link There is some discussion of the bar-cobar adjointness as it relates to twisting cochains, at that entry. Here we will concentrate on the bar-cobar adjointness itself and start exploring the links with other parts of differential algebra. One of the earliest examples of a pair of adjoint functors studied in algebraic topology was that giving the relationship between the functors for reduced suspension and based loop space. If we consider a pointed connected topological space , then its reduced suspension is obtained by taking the cylinder and identifying the subspace to a point. (Think of crushing the two ends of the cylinder and the line through the base point to a point.) This can also be thought of as forming the smash product of the circle with . Adjoint to is the based loop space functor: is the space of pointed maps from to . This has a monoid structure (up to homotopy) given by concatenation of loops. (Back in , we have a comonoid structure with respect to the pointed coproduct as described at interval object. This in some sense is ‘subdivision as an inverse for composition’.) (perhaps: Picture to go here?) Using ordinary (co)homology to study spaces such as CW-complexes, we naturally use the complexes of (cellular) chains on spaces. The structure of chains on the suspension is easy to work out using the obvious cellular structure, but that on the loop space is much harder as is given the compact-open topology and only has the homotopy type of a CW-complex, so no nice cellular structure is given us ‘on a plate’. The idea is thus to start with a chain complex model, , for a CW-complex, , (usually the complex of cellular chains on ), and we try to construct from a ‘model’ for the chain complex of the loop space of . Adams’ cobar construction was such a method (see below). This was adjoint to a bar construction defined by Eilenberg and MacLane. Both directions use an abstract algebraic model of...