User:Wlod/Matcat

A general introduction

The branches of mathematics are divided (classified) first of all as algebra,geometry, and mathematical analysis. Then finer classification follows. (Actually, there is more than one classification of mathematics around which already has its sceptic meaning). The classification is useful for the bookkeeping while conceptually it has only limited value. Indeed, number theory splits into elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and combinatorial number theory, to say the least. We also have general topology, algebraic topology, geometric topology, topological algebra. Every larger theory has significant portions which virtually belong to chapters different from the chapter to which the given theory is assigned.

However, as I said also on and before sci.math,

Mathematics - the art of thinking

There doesn't exist any profound classification of the branches of mathematics. Instead, there is a profound classification of mathematical thinking into three styles of thinking:

• algebraic -- concerned with the structure;
• geometric -- concerned with the symmetry;
• analytic -- concerned with infinity.

Obviously, every larger mathematical theory thinks in all three styles. This is actually a definition of a complete theory:

A theory is complete $\ \Leftarrow :\Rightarrow \$ it involves all three styles of thinking.

This understanding should result in a long-range seamless development of mathematics.

A specific introduction

These days, for each major theory, one or more categories are introduced. This involves a lot of forgetting (of forgetful functors) and a plethora of other functors, perhaps a lot more than necessary. Ideally, the whole mathematics should be done in just one categories. Of course, there will be extensions, variations, generalizations. Nevertheless, it'd be nice to do the total of about all necessary mathematics within one category (and if one must do other things, thus proving that there is more--fine).

The notions of mono- and epimorphism already are examples of what is needed. A more advanced notions are the universal morphism and the fixed morphism property. There is already a not quite obvious theorem about the universality of the direct product and composition of morphisms. One would like to devolop the whole basic mathematics along such lines.

REMARK   We don't have to be pure, i.e. avoiding all undefined notions of present mathematics. Possibly, this may come much later when the unified approach is already advanced. Thus there is no reason to avoid the notion of a set, and similar, at this stage. In particular, we may consider a small category $\ {\mathcal {C}}\$ as a start point.

Epiplumorphism and monoplumorphism

Let $\ {\mathcal {C}}\$ be a small category. Let $\ A\in {\mathcal {Ob(C)}}.\$

Let $\ {\mathcal {Arg}}(A)\$ be the set of all morphisms $\ f\in {\mathcal {Mor(C)}}\$ for which $\ A\$ is the argument-object of $\ f.\$

Let $\ {\mathcal {Trg}}(A)\$ be the set of all morphisms $\ f\in {\mathcal {Mor(C)}}\$ for which $\ A\$ is the target of $\ f.\$

Let $\ \emptyset \neq \mathbf {X} \subseteq {\mathcal {Mor(C)}}.\$ Two notions, epi- and monoplumorphism are defined as follows:

• Let $\ \mathbf {X} \subseteq {\mathcal {Trg}}(A).\$ Then $\ \mathbf {X} \$ is an epiplumorphism $\ \Leftarrow :\Rightarrow \ \forall _{f\ g\ \in \ {\mathcal {Arg}}(A)}\ ((\forall _{h\in \mathbf {X} }f\circ h=g\circ h)\ \Rightarrow \ f=g)$

• Let $\ \mathbf {X} \subseteq {\mathcal {Arg}}(A).\$ Then $\ \mathbf {X} \$ is a monoplumorphism $\ \Leftarrow :\Rightarrow \ \forall _{f\ g\ \in \ {\mathcal {Trg}}(A)}\ ((\forall _{h\in \mathbf {X} }h\circ f=h\circ g)\ \Rightarrow \ f=g)$