Homomorphisms that are pullbacks

Homomorphisms that are pullbacks

[Tom Leinster]

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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Re: Homomorphisms that are pullbacks

[Fred E.J. Linton]

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



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