nLab proxi-Banach ring

Idea and motivations

The notion of proxi-Banach ring is a generalization of the notion of Banach ring, essentially obtained by replacing the usual triangular inequality $|a+b|\leq |a|+|b|$ that appears in the theory of metric spaces by the more natural (in a categorical sense) weak triangular inequality $|a+b|\leq C\cdot\max(|a|,|b|)$ that appears in the theory of proxi-metric spaces.

Their use allows the definition of an $\mathbb{R}_+^*$-action on various categories of global analytic spaces.

They are also the natural objects that appear as $\pi_0$ of spectral rings in spectral global analytic geometry.

Definition

A pseudo-Banach ring is a ring object in the rig category $(\mathbb{R}_{+\leq}^\mathrm{Sets},\oplus_\infty,\otimes_m)$ of $\mathbb{R}_+$-graded sets (with bounded maps between them). This means a set $R$ with a norm $|\cdot|_R:R\to \mathbb{R}_+$ such that there exists $C$, $D$ and $E$, $F$ with

• $|a+b|\leq C\cdot \max(|a|,|b|)$,

• $|ab|\leq D\cdot|a|\cdot|b|$,

• $|-a|\leq E|a|$,

• $|0|=0$,

• $|1|\leq F\cdot 1$ (one often supposes additionally that $|1|=1$).

The free module

If $X$ is an $\mathbb{R}_+$-graded set, and $R$ is a pseudo-Banach ring, one may define an associated pseudo-normed free module $R^{(X)}$ by putting on the usual free module the following convenient grading (not given by the usual $\ell^1$ grading): if $\sum a_x\{x\}$ is an element with support $supp(a)$, one may write it $\sum a_i\{x_i\}$, and parenthesize it in binary terms. In the case of a support of cardinal three, we may write for example

$$P(a_1\{x_1\}+a_2\{x_2\}+a_3\{x_3\})=((a_1\{x_1\}+a_2\{x_2\})+a_3\{x_3\}).$$

Each parenthesis contains only two terms, and there is a finite set of choices for the position of parenthesis, denoted $\mathcal{P}a(supp(a))$. We then define by induction ($C$ denotes the norm, i.e., infimum of the $C$ constants for the addition map), for $P$ an element of this set,

$$|(u)+(v)|_{C,P}:=C\cdot \max(|u|_{C,Pu},|v|_{C,Pv}).$$

One then takes the maximum of all these expressions to define a natural pseudo-norm on $R^{(X)}$ by

$$\left|\sum a_x\{x\}\right|_{\infty,C}:=\max_{P\in \mathcal{P}a(supp(a))}\left|P(\sum a_x\{x\})\right|_{C,P}.$$

If $C=1$, i.e., in the non-archimedean case, this gives back the usual non-archimedean (maximum) seminorm on the free module.

Modules and their operations

Using the free module, and the coproduct and product of $\mathbb{R}_+$-graded sets, one defines direct sums and tensor products of modules, and throught the symmetric algebra construction, the free $\mathbb{R}_+$-graded algebra (algebra of convergent “power series” with coefficients in $R$ and polyradii in $X$).

One may also work with the category of ind-pseudo-Banach ring, and of ind-pseudo-Banach modules over them, to develop an overconvergent version of the theory.

A natural flow

The action of $\mathbb{R}_+^*$ on all these objects is simply given by acting on the grading through

$$|\cdot|\mapsto |\cdot|^t:=e^{t\mathrm{ln}|\cdot|}.$$

Related subjects

global analytic geometry

overconvergent global analytic geometry

spectral global analytic geometry

Reference

For the category of $\mathbb{R}_+$-graded sets:

• Frédéric PaugamOverconvergent global analytic geometry (arXiv:1410.7971)