If we have intuitive ideas of a kind of systems that we want to study, but we do not know which language (list of structures) would be good to formalize them, then how can we still find a method to "anyway simply formalize it no matter what a subtle kind of system it is" ?

Simple examples of kinds of systems that look natural to formalize, are- Vector spaces
- Affine spaces

There is a range of such possible kinds of "spaces":

- Topological spaces
- Topological manifolds
- Lipschitz structures on topological manifolds,
- Differential manifolds (with whatever degree of smoothness you choose).

But, as surprising as it may seem, there is a very general method, with such a wide range of uses, and still rather simple.

First step is to have an idea of which functions between spaces will be accepted as "morphisms" (that, depending on the kind of systems, will mean to qualify functions as : continuous, differentiable, etc), so that any composite of morphisms will be a morphism. For example, no matter how topology may be formalized, we expect the concept of "continuous function" between spaces to make sense, and that a composite of continuous functions will be continuous. We have a clear intuition what it means for a function to be continuous, so there should exist a definite set of all continuous functions from a space to another. But to write a definition for it, would depend on the choice of formalization of the topological structure that we want these functions to preserve. Never mind, we will define this later.

So in the first step, let us admit as primitive data, the sets of intended morphisms between systems. This way, the intended kind of "spaces" are seen as the objects of a concrete category

First we need to pick up a fixed object (system)

A fixed

In the representation
theorem for categories, the identification
between sets of endomorphisms in the initial
category and in the category of typed algebras thus constructed,
suggests that if the objects we started with were already some
systems and thus we want to see types as similar systems, then the
initial object-systems will be mutually constructible with the new
systems (sets of morphisms from types to algebras) finally built in
the role of new interpretations of objects. For this, types need to
be "blocked" by additional structures to not have any remaining
internal morphisms (like constructions
do not add more automorphisms, but in a sense adapted to the use of
morphisms instead of isomorphisms), i.e. to be seen as if it was constructed
ex nihilo (a type of constants). (The representation theorem could
succeed by its opportunity to choose an
action that makes it always work, but now the action is fixed so we
have to adapt to it).

Anyway, whatever additional
structures on types will keep unaffected (preserve) the action of
morphisms between objects. For example if objects were vector spaces
then the role of "types" (as systems whose morphisms to the "object"
systems play the roles of the new elements of the objects), can be
played by vector spaces with choices of basis, that is some ℝ^{n}.

∀*u* ∈*M*^{B},∃!*f*∈Mor(*K*,*M*),
*f*_{|B}=*u*

In other words, (Denoting

∀*k*∈*K*,∀*u* ∈*M*^{B},
*k*_{M}(*u*)=*f*_{M,u}(*k*)

The below is a draft

This condition on

(to be completed)

This can define some more rigid structures than would be possible by the first method. For example, morphisms so defined between infinite dimensional topological affine spaces are automatically "continuous", in a sense of "continuity" that is specific to infinite-dimensional spaces, a condition which the algebraic definition by barycenters does not ensure (while both concepts of morphism are equivalent in finite dimensional spaces).

It is very simple to introduce the notion of measure on a topological manifold M : take M* the vector space of continuous functions with real values, then the space of measures on M is the vector space of linear forms on M* that is "generated by M", i.e. the set of limits of sequences of linear combinations of elements of M in the dual of M*. Now taking as M a differential manifold and M* the set of smooth functions on M, then what we get in this construction (closed vector space generated by M) is the space of distributions on M.

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