Homotopy structures for algebras over a monad
Let \({\mathbf A}\) be a category with a homotopy structure given by a cocylinder (path-endofunctor) \(P:{\mathbf A}\to {\mathbf A}\) and three natural transformations \(\partial^\varepsilon: P\to 1_{\mathbf A}\), \(e:1_{\mathbf A}\to P (\varepsilon=0,1)\) such that \(\partial^\varepsilon \circ e=1(\varepsilon=0,1)\). Let \(T\) be a monad over \({\mathbf A}\). It is shown that the homotopy structure on \({\mathbf A}\) can be lifted to the category \({\mathbf A}^T\) of Eilenberg-Moore algebras over \(T\) provided that \(P\) is consistent with \(T\), i.e. equipped with a natural transformation \(\lambda:TP\to PT\) satisfying some obvious axioms. Furthermore, homotopy operations such as faces, degeneracies, connections, reversion, interchange, etc. together with their axioms can be lifted from \({\mathbf A}\) to \({\mathbf A}^T\) if they are consistent with \(\lambda\). The wide range of examples includes the transition from topological spaces to fibrewise homotopy and to equivariant homotopy.
scientific article; zbMATH DE number 1384513