Yes. The category of topological spaces over the category of sets is not algebraic and therefore not a category of models over the category of sets. It is a category fibered over the category of sets. The adjunction beteween the forgetful functor and its left adjoint is not monadic because the left adjoint produces discrete spaces. That is, the category of models over the monad induced by the adjunction is not equivalent to Top.

Tom Leinster's book Higher Operads, Higher Categories has a description of this phenomenon on page 8. He also gives a reference to Mac Lane's categories for the working mathematician, but you can find that yourself.

Edit: To answer your third question, the reason that you don't hear people talk about models in the way you're asking is because they're called algebras over a monad or more generally algebras over an operad.

Yes. The category of topological spaces over the category of sets is not algebraic and therefore not a category of models over the category of sets. It is a category fibered over the category of sets. The adjunction beteween the forgetful functor and its left adjoint is not monadic because the left adjoint produces discrete spaces. That is, the category of models over the monad induced by the adjunction is not equivalent to Top.

Tom Leinster's book Higher Operads, Higher Categories has a description of this phenomenon on page 8. He also gives a reference to Mac Lane's categories for the working mathematician, but you can find that yourself.

Edit: To answer your third second question, the reason that you don't hear people talk about models in the way you're asking is because they're called algebras over a monad or more generally algebras over an operad.

Yes. The category of topological spaces over the category of sets is not algebraic and therefore not a category of models over the category of sets. It is a category fibered over the category of sets. The adjunction beteween the forgetful functor and its left adjoint is not monadic because the left adjoint produces discrete spaces. That is, the category of models over the monad induced by the adjunction is not equivalent to Top.

Tom Leinster's book Higher Operads, Higher Categories has a description of this phenomenon on page 8. He also gives a reference to Mac Lane's categories for the working mathematician, but you can find that yourself.

Yes. The category of topological spaces over the category of sets is not algebraic and therefore not a category of models over the category of sets. It is a category fibered over the category of sets. The adjunction beteween the forgetful functor and its left adjoint is not monadic because the left adjoint produces discrete spaces. That is, the category of models over the monad induced by the adjunction is not equivalent to Top.

Tom Leinster's book Higher Operads, Higher Categories has a description of this phenomenon on page 8. He also gives a reference to Mac Lane's categories for the working mathematician, but you can find that yourself.

Edit: To answer your third question, the reason that you don't hear people talk about models in the way you're asking is because they're called algebras over a monad or more generally algebras over an operad.