Doesn't the preservation of limits follow from the fact that the forgetful functor is right adjoint to the free functor? A left adjoint functor is always cocontinuous and a right adjoint functor is always continuous.
Looking at examples of "algebras of a monad on set", I see groups and monoids. Does this explain why:
- The trivial group corresponds to a singleton set?
- The direct product of two groups G and H corresponds to their Cartesian products as sets? And so on to direct products of infinitely many groups?
- An equaliser of G under homs f and g has as its underlying set {x \in G : f(x) = g(x)}?
And so on, I see that the same must be true for any other algebraic structure which is a "category of algebras of a monad on Set". Or have I misunderstood?
[edit]
I also see that because the free functor is cocontinuous (because it's left adjoint to the forgetful functor), the free group generated by the union of some sets is the same as the free product (not direct sum) of their individual free groups (because the free product is a coproduct but the direct sum is not). If we want a correspondence with direct sum instead, we need to generate free abelian groups instead of free groups.
Yes, all of this is true. Other examples of 'categories of algebras of a monad on Set' include vector spaces, rings, modules over a given ring, representations of a given group, and Lie algebras.
So if we could somehow teach category theory first (which would be hard because students would have no examples) we could shave a few lectures off each of those courses.
Looking at examples of "algebras of a monad on set", I see groups and monoids. Does this explain why:
- The trivial group corresponds to a singleton set?
- The direct product of two groups G and H corresponds to their Cartesian products as sets? And so on to direct products of infinitely many groups?
- An equaliser of G under homs f and g has as its underlying set {x \in G : f(x) = g(x)}?
And so on, I see that the same must be true for any other algebraic structure which is a "category of algebras of a monad on Set". Or have I misunderstood?
[edit]
I also see that because the free functor is cocontinuous (because it's left adjoint to the forgetful functor), the free group generated by the union of some sets is the same as the free product (not direct sum) of their individual free groups (because the free product is a coproduct but the direct sum is not). If we want a correspondence with direct sum instead, we need to generate free abelian groups instead of free groups.