Homomorphisms that are pullbacks

[Tom Leinster <tl@maths.gla.ac.uk>]

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


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