Eilenberg–Moore

Eilenberg–Moore is a term associated with foundational constructions in category theory and homological algebra introduced by Samuel Eilenberg and John Coleman Moore. The constructions connect ideas from algebraic topology, functor theory, monad theory, and homological functor methods, influencing work in Saunders Mac Lane, Alexander Grothendieck, Jean-Pierre Serre, Henri Cartan, and later in Max Kelly and F. William Lawvere. The developments have been applied in contexts ranging from Hochschild homology to stable homotopy theory.

Definition and history

The origin of this concept traces to collaborative work by Samuel Eilenberg and John Coleman Moore in the mid-20th century, building on foundations laid in Homological Algebra by Cartan–Eilenberg and the categorical perspectives advocated by Mac Lane. Early motivations involved problems in spectral sequence computations for fibrations in algebraic topology and constructions in cohomology theory linked to the work of Henri Cartan, Jean Leray, Jean-Pierre Serre, and contemporaries. Subsequent formalization connected to the abstract notion of a monad developed by Roger Godement and popularized in category theory texts by Mac Lane and Saunders Mac Lane. Influential contributions that framed the present viewpoint came from Max Kelly, G. M. Kelly, and Bill Lawvere, culminating in modern expositions used by researchers in stable homotopy theory, derived categories, André-Quillen cohomology, and operad theory.

Eilenberg–Moore category

The Eilenberg–Moore category constructs algebras for a given monad on a base category; this construction sits alongside the Kleisli category associated with the same monad and interacts with adjunctions studied by William Lawvere and G. M. Kelly. Concretely, given a monad arising from an adjunction as in classical examples studied by Eilenberg and Moore—such as those in algebraic topology involving the loop space and classifying space adjunctions—the Eilenberg–Moore category formalizes the category of Eilenberg–Moore algebras. The structural properties of this category are central to results by Barr, Beck, and J. Beck culminating in the Beck monadicity theorem, with later refinements by Street, Kelly, and Borceux. Important settings include categories used by Grothendieck in SGA and by Quillen in model category frameworks, where the Eilenberg–Moore construction is compatible with homotopical techniques developed by Daniel Quillen and J. Peter May.

Eilenberg–Moore spectral sequence

The Eilenberg–Moore spectral sequence arises in the calculation of cohomology of pullbacks and fiber squares, originating from computations by Eilenberg and Moore in the study of loop spaces and fibrations. It provides a tool comparable to the Serre spectral sequence used by Jean-Pierre Serre and the Leray–Serre spectral sequence originating in work of Jean Leray, enabling modern applications across homotopy theory and algebraic K-theory studied by Daniel Quillen and Friedhelm Waldhausen. The convergence and E2-term descriptions depend on homological algebra results from Cartan–Eilenberg and the formalism of derived functor spectral sequences developed by Grothendieck and applied by Pierre Deligne in contexts related to Hodge theory and étale cohomology. Later computational techniques incorporate inputs from Adams spectral sequence methods of J. F. Adams and innovations by Mark Hovey and Steffen Sagave in structured ring spectra.

Applications and examples

Applications span classical and modern areas: computations in loop space cohomology, analysis of Hochschild homology and cyclic homology influenced by Brylinski and Connes, and categorical algebra problems studied by Mac Lane and Lawvere. The Eilenberg–Moore construction plays a role in the formulation of operad actions found in work by Getzler and Jones, in the study of bialgebra and Hopf algebra structures examined by Sweedler and Milnor–Moore theorem contexts, and in categorical formulations used in computer science influenced by Moggi and the semantics community around Gavin Bierman and Philip Wadler. Concrete examples include the category of modules over a ring related to Emmy Noether's algebraic frameworks, comodules studied by André Joyal contexts, and algebraic models for rational homotopy theory developed by Sullivan and Quillen.

Relationship to Kleisli and monadicity theorems

The Eilenberg–Moore category complements the Kleisli category associated to a monad, both arising from an adjunction as articulated in expositions by Mac Lane and Kelly. The interplay is formalized by the Beck monadicity theorem attributed to Jonathan Beck and further clarified in categorical treatises by Borceux and Street. Applications of monadicity criteria inform reconstructions of categories in contexts studied by Grothendieck in the stacks program and by Lurie in higher category theory, connecting to developments in ∞-categories and model categories influenced by Jacob Lurie and J. Lurie. The Eilenberg–Moore perspective is essential in recognizing when a functor is monadic, with ramifications for algebraic geometry via Grothendieck's descent theory, for homotopical algebra via Quillen's model structures, and for semantics in computer science via Moggi's computational monads.

Category:Category theory