Norm

A norm or logical model defines the points of a space.

Norms map a set of primitive objects to true, i.e. that takes some structured set of objects---a structured content, as in a diagram---and reduces that to a structureless input. Formally, \[ R_0^\infty : C \to J \] where $R_0^\infty$ may be thought of as `losing' the diagram $D$ of $J$ an index category in $C$. On the other hand, we may think of a norm $\mathcal{L}$ as just a particular usage of an index $I$, without the additional categorical structure.

The norm defines things such that we can retain the difference between points. So the `new' can actually be perceived as new relative to the old, and primitive objects can be used to generate new objects (for example, from composition).

Intuitively, if implications $R_0$ operate directly on our assumptions, then norms are the mechanisms by which we assume ``nothing'' about a given object---by which we reject and forget our various biases and intuitions. Recall what happens when we normalize a database: we reject or erase the irregularities.

In the provisional formalism, a norm is written $$R_0^\infty: \{F: \downarrow \to \uparrow\} \to \{F: \uparrow \to \downarrow\}$$.

Relationship to indices
See the article on domains.

Often we appeal to an index set or index category in formulating an induction, construction, diagram, or universal property— this is especially so when we use dependent types, where almost every construction or operation is "controlled" by an index. One way of thinking about this phenomenon is to regard indices as equivalent with domains; an index, like a domain, is a collection that we "talk about" or "act upon". They serve as contexts for some hypothesis or operator. The only difference is that the word index tends to be used with hypotheses rather than operators.

Commonly we default to the natural numbers when we use an index...

Do we need to construct an index, or does the index arrive automatically on appeal? I.e. to make an appeal is the same thing as to construct an index?

A norm ...

As a logic
A logic, in the informal sense, is just a method of (valid) reasoning. More generally, a logic is some object to which we appeal when we state an argument or proof. Here we observe two things:

(1) We often associate any given logic with a formalism or symbolic representation. For example, first-order logic may be represented in classical, algebraic, proof-theoretic, and so on. The precise choice of formalism is not importan t here, but it is important to distinguish the logic from its formalism.

(2) Recall that a model is the means by which we associate properties and structures to a theory.

Every particular logic is a point and the logical model is "what defines those points".