Here's an elementary property of maps of algebras for a monad. I'm interested to know what's known about it. Let T be a monad on some category. A map of T-algebras is a commutative square TA ----> TB | | | | V V A -----> B. When is this square a pullback? I tried working this out for various examples of monads T. You recover some interesting properties, including: - for functors: the unique factorization lifting property - for natural transformations: the property of being cartesian - for maps of compact Hausdorff spaces: with a bit of a tweak, the property of being a local homeomorphism. Explanation of these and other examples is here: http://golem.ph.utexas.edu/category/2010/08/pullbackhomomorphisms.html But this property is so elementary that presumably it's been studied before. Does anyone know where? Thanks, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]