# Blueprint for the Liquid Tensor Experiment

## 1.2 Variants of normed groups

Normed groups are well-studied objects. In this text it will be helpful to work with the more general notion of semi-normed group. This drops the separation axiom $$\| x\| = 0 \iff x = 0$$ but is otherwise the same as a normed group.

The main difference is that this includes “uglier” objects, but creates a “nicer” category: semi-normed groups need not be Hausdorff, but quotients by arbitrary (possibly non-closed) subgroups are naturally semi-normed groups.

Nevertheless, there is the occasional use for the more restrictive notion of normed group, when we come to polyhedral lattices below (see Section 1.6).

In this text, a morphism of (semi)-normed groups will always be bouned. If the morphism is supposed to be norm-nonincreasing, this will be mentioned explicitly.

Definition 1.2.1
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Let $$r {\gt} 0$$ be a real number. An $$r$$-normed $$\mathbb Z[T^{\pm 1}]$$-module is a semi-normed group $$V$$ endowed with an automorphism $$T \colon V \to V$$ such that for all $$v \in V$$ we have $$\| T(v)\| = r\| v\|$$.

The remainder of this subsection sets up some algebraic variants of semi-normed groups.

Definition 1.2.2
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A pseudo-normed group is an abelian group $$(M,+)$$, together with an increasing filtration $$M_c \subseteq M$$ of subsets $$M_c$$ indexed by $$\mathbb R_{\ge 0}$$, such that each $$M_c$$ contains $$0$$, is closed under negation, and $$M_{c_1} + M_{c_2} \subseteq M_{c_1 + c_2}$$. An example would be $$M=\mathbb {R}$$ or $$M=\mathbb {Q}_p$$ with $$M_c :=\{ x\, :\, |x|\leq c\}$$.

A pseudo-normed group $$M$$ is exhaustive if $$\varinjlim _c M_c = M$$.

All pseudo-normed groups that we consider will have a topology on the filtration sets $$M_c$$. The most general variant is the following notion.

Definition 1.2.3
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A pseudo-normed group $$M$$ is CH-filtered if each of the sets $$M_c$$ is endowed with a topological space structure making it a compact Hausdorff space, such that following maps are all continuous:

• the inclusion $$M_{c_1} \to M_{c_2}$$ (for $$c_1 \le c_2$$);

• the negation $$M_c \to M_c$$;

• the addition $$M_{c_1} \times M_{c_2} \to M_{c_1 + c_2}$$.

The pseudo-normed group $$M$$ is profinitely filtered if moreover the filtration sets $$M_c$$ are totally disconnected, making them profinite sets.

Remark 1.2.4

The topologies on the filtration sets $$M_c$$ will induce a topology on $$M$$: the colimit topology. If $$M$$ is some sort of normed group, then this topology is typically genuinely different from the norm topology.

Definition 1.2.5
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A morphism of CH-filtered pseudo-normed groups $$M \to N$$ is a group homomorphism $$f \colon M \to N$$ that is

• bounded: there is a constant $$C$$ such that $$x \in M_c$$ implies $$f(x) \in N_{Cc}$$;

• continuous: for one (or equivalently all) constants $$C$$ as above, the induced map $$M_c \to N_{Cc}$$ is a morphism of profinite sets, i.e. continuous.

The reason the two definitions of continuity are equivalent is that a continuous injection from a compact space to a Hausdorff space must be a topological embedding.

A morphism $$f \colon M \to N$$ is strict if $$x \in M_c$$ implies $$f(x) \in N_c$$ (in other words, if we can take $$C = 1$$ in the boundedness condition above).

We will also consider the analogue of an $$r$$-normed $$\mathbb Z[T^{-1}]$$-module in the pseudo-normed setting.

Definition 1.2.6
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Let $$r'$$ be a positive real number. A CH-filtered pseudo-normed group $$M$$ has an $$r'$$-action of $$T^{-1}$$ if it comes endowed with a distinguished morphism of CH-filtered pseudo-normed groups $$T^{-1} \colon M \to M$$ that is bounded by $$r'^{-1}$$: if $$x \in M_c$$ then $$T^{-1}x \in M_{c/r'}$$.

A morphism of CH-filtered pseudo-normed groups with $$r'$$-action of $$T^{-1}$$ is a morphism $$f \colon M \to N$$ of CH-filtered pseudo-normed groups that commutes with the action of $$T^{-1}$$.