turkmath.org

If X and Y are algebraic objects like monoids, groups, rings, etc., we define bimodules to carry a left action from X and right action for Y and typically organize them in a double category. We will entertain the idea of a bimodule in the case where X and Y are simplicial sets (and general spaces or CW-complexes). We will draw the analogy from the special case of categories, where bimodules have various interpretations and homotopical meaning. In particular, representable bimodules are mapping cylinders for categories. Because of the lack of algebraic structure bimodules of spaces do not tensor. Time permitting we will discuss a remedy for this issue.