Universal monoid actions: The power of freedom

Motivated by the recent work of Balcerzak and Kania [Proc. Amer. Math. Soc. 151 (2023) pp. 3737–3742], we show that every countable monoid has a universal action on the free object over a countable infinite set. This is a general result concerning concrete categories with a left adjoint (free) functor. On the way, we introduce an abstract concept of “being generated by a set”. At the same time we obtain a simpler proof of the result of Darji and Matheron [Proc. Amer. Math. Soc. 145 (2017), pp, 251–265] concerning a surjectively universal operator on the classical Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l 1"> <mml:semantics> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\ell _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of summable sequences.