Are Banach spaces monadic?

Abstract

We will show that Banach spaces are monadic over complete metric spaces via the unit ball functor. For the forgetful functor, one should take complete pointed metric spaces.

Metric monads

“…(1) Let Norm be the category of generalized normed spaces (i.e., norm ∞ is allowed) and linear maps of norm ≤ 1. Analogously as in [32] 2.2, we show that the forgetful functor V : Norm → Met is monadic. In fact, normed spaces are monoids in Met equipped with unary operations c • −, for |c| ≤ 1, satisfying the appropriate axioms.…”

“…(2) Let Ban be the category of Banach spaces and linear maps of norm ≤ 1. Following [32], the unit ball functor U : Ban → Met is monadic. Since U is an enriched functor, we get a finitary enriched monad on Met (see [32 / / 2 ε is an ε-pushout for every ε > 0 and ε-pushouts are weighted colimits (see [6] 4.1), F 2 ε is an ε-pushout…”

“…Consider the ⊗-theory from 5.10(1)over CMet given by addition + : Z ⊗ Z → Z and scalar multiplications c • − : Z → Z for c ∈ C with |c| ≤ 1 . Following[32] 2.2, Alg(T ) = Ban ∞ is the category of generalized Banach spaces with the forgetful functor U T : Ban ∞ → CMet. The corresponding monad on CMet preserves sifted colimits.…”