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/ ]
On Mon, 02 Aug 2010 08:36:21 AM EDT, Tom Leinster <tl@maths.gla.ac.uk> asked,
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?
One very simple instance, in Banach spaces (with norm-non-increasing linear mappings), with T the double-dualization monad: if B is reflexive, then such a square is a pullback iff A is reflexive, too. The superficial similarity with the finite/discrete case of Tom's local homeomorphism remark in the instance of compact Hausdorff spaces may, with luck, be more than just coincidental ... (the case I mean is that if B is a finite discrete space, then the square (in KT_2 spaces, with T the Stone-Cech compactification monad) is a pullback iff A is finite discrete as well). Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Fred E.J. Linton -
Tom Leinster