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As my math research has opened new major areas of mathematics, I suggest to add certain new categories to classification schemes.
This classifier
also contains links to materials relevant for
the suggested categories.
Name: Theory of Dependencies or Dependencies Theory.
Because currently the main interest of researching dependencies are the categories of dependencies (and related categories of elements), the category Theory of Dependencies should be a subcategory of the category Category Theory.
Alternatively Theory of Dependencies can be classified as a subcategory of Model Theory because theory of dependencies is a model and is used to model such objects as expressions and algebras. (This variant has the advantage that then if the theory of formulas will be also classified as a subcategory of Model Theory then these two related categories will be near (siblings) in the classifier.)
Theory of dependencies should have a special category because of its special relations with the rest of category theory as well as with universal algebra and model theory in general, and also its role for the theory of formulas (see below).
Related materials
Name: Theory of Formulas or Formulas Theory or Theory of Expressions or Expressions Theory.
Note that theory of expressions was existing before my research but it was not algebraic and so not enough advanced for a separate category.
Because formulas clearly belong to foundations of mathematics (see also this article), this category should be a subcategory of Foundations of Mathematics.
Alternatively Formulas Theory may be classified as a subcategory of Algebraic Logic because the theory of formulas is algebraic and currently its main purpose is for math logic.
Alternatively Formulas Theory may be classified as a subcategory of Model Theory because the theory of formulas is used to model math logics and other objects. (This variant has the above mentioned advantage of being placed near to the related category Theory of Dependencies.) In certain classification schemes (such as ODP) Model Theory is a subcategory of Logic and Foundations, in such classifiers Theory of Dependencies should be a subcategory of Model Theory.
I do not recommend Formulas Theory or a similar category to be a subcategory of Theory of Dependencies because we cannot have two different categories for algebraic and old theory of formulas and so should classify Formulas Theory by its purpose (that is logic and foundations) rather than by the used means (which is theory of dependencies for new algebraic theory and is indefinite for the old theory of expressions).
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Name: Algebraic General Topology.
Note that this category also to contain Math Synthesis (in the sense of applying Algebraic General Topology to Math Analysis) information.
Because the main purpose of Algebraic General Topology is researching of general (and even more general) topological objects, this category should be a subcategory of General Topology.
Additional note: In my opinion General Topology is wrongly classified as a subcategory of Topology, I deem this a rather historical and naming issue instead of real classification reflecting structure of mathematical knowledge.
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