Australian Category Seminar

A normed abelian group $(A, |\ |)$ is an abelian group A together with a monoidal functor $|\ | \colon A \to R^+$ (see Lawvere's Metric Spaces paper) (ie $|a + b| \leq |a| + |b|$, $|0| \leq 0$) satisfying further $|-a| = |a|$. Morphisms of these are norm-reducing group homomorphisms. The category is called Nab. Theorem 1: There is a forgetful fusctor $U : Nab - Met$ (= Metric spaces) which is monadic. Proof: $F(x,d)=(F'X, |\ |)$ where $F'X$ is the free abelian group in X. The norm is generated by $|x-y| = d(xy)$. □ A normed R-algebra is an R-algebra R together with an abelian group norm satisfying $|xy| \leq |x| . |y|$, $|rx| \leq |r| . |x|$ $r \in R$, $|1| \leq 1$. The morphisms are norm-reducing R-algebra homomorphisms; the category is $NR^+$alg. Theorem 2: $NR$-alg → Met is monadic. By cauchy-completing we get the corollary that Banach algebras are monadic over complete metric spaces.