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

We already sketched how first-order structures can be defined from this. But we only mentioned there about relations. What about operations ? In fact, they may come as particular cases of these relations, that is, when the relations so defined happen to be functional:

∀*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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