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Foursome

Understanding the Yoneda Lemma in terms of the foursome: Whether, What, How, Why

The Yoneda Lemma seems to relate the following levels of knowledge:

• A functor F takes us from a category C [How] (especially the morphisms) to a category F(C) [What] (especially the objects).
• [Whether] describes the external relationships in F(C). (The categorical outlook.)
• [Why] describes the internal structure native to C.

Relate to ways of figuring things out:

• Whether: level and metalevel are equated: proof by contradiction
• What: model
• How: working backwards
• Why: variables

Why expresses taking a stand with regard to everything. And so it is related to taking a stand with regard to one's self, what permeates one's inside entirely, as with a group's internal consistency, whereby each element multiplies to yield a distinct products. So consider how Why comes from one's (universal) relationship with one's self. All four levels (Whether, What, How, Why) express one's relationships with one's self, and the implications.

Note that a Turing machine is built on different kinds of variables, which may imply that it can't be categorified.

• Whether: object A (Accordion)
• What: image F(A)
• How: morphism from A: A->
• Why: all morphisms to A: ->A
• What: Target category. Set of properties. Why: Set of relationships as dictaded by its relationships but especially its relationship with itself (its essence).
• How: Source category. The source category of a functor is a model or blueprint for the target category of a functor.

Yoneda embedding: What=Why defines "meaning". What about other five qualities of signs?

A perspective is defined as a fixed point, namely, the identity morphism, with which we identify ourselves. This is Whether. It is the heart of the Yoneda theorem.

Natural transformation has four levels:

• Why: f:x->y in C
• How: F(f):F(x)->F(y) in F(C)
• What: G(f):G(x)->G(y) in G(C)
• Whether: F(x)->G(y)

Objectification of morphisms

• center - whether - identity morphism
• balance - what - "from and to"
• set - how - *elements* in a set
• list - why - *list* of items

Are these all internal structures? But what about external relationships? Are they given by analysis?

• B_>C ..... How->What
• External relations -> Internal logic .... (Not What=Why) Hom C -> Hom B (Not How=Whether)
• Eduardo's Yoneda Lemma diagram is the foursome.
• Loss of info from How to What is equal to the Loss of info from "Why for What" to "Why for How".
• How: inner logic. What: external view.
• Whether (objects), what (morphisms), how (functors), why (natural transformations). Important for defining the same thing, equivalence. If they satisfy the same reason why, then they are the same.
• The Yoneda Lemma: the Why of the external relationships leads to the Whether of the objects in the set. The latter are considered as a set of truth statements.
• Moving from How (arrows) to What (objects).
• Daiktai (objektai) yra visaip pramanyti. Pramanome jų žinojimo lygmenis: Ar, koks, kaip, kodėl. Nupaišyti Wenbo piešinėlį nusakantį santykius tarp smegenų (pasąmonės) kur yra tiktai pokyčiai, proto (sąmonės) kur pramonem objektus (būsenas), ir kalbos kur išmąstome objektų (būsenų) santykius. Piešinys parodo jog išmąstydami santykius, jais suprantame pokyčius. (Yoneda lema) Pokyčiai auga kvadratiniu būdu, tad yra lengviau nusakyti būsenas augančias tiesiniu būdu. Tam reikia žinoti kaip būsena susijusi su visomis kitomis būsenomis. Tuomet tokią būseną galima suvokti kaip sąvoką objektą ir santvarka tampa ja paprastesnė.
• The four levels Whether, What, How, Why differ in how they focus on objects or morphisms.

Four programming paradigms:

• Procedural (imperative, top-down) - Fortran, C, Cobol. Statements are structured into procedures, subroutines, functions, they make procedure calls.
• Logical (declarative) - Prolog. Statements express facts (assertions), rules (inferences) and queries about problems within a system. Rules are logical clauses with a head and a body. Expresses knowledge independently of implementation. Knowledge separated from use. Programs can be more flexible, compressed, understandable.
• Functional - Haskell, Lisp. Computation is an evaluation of mathematical functions. Pure functions are those that take an argument list as input and output a return value. They do not depend on global data or class member's data. No side effects. Input not affected. A function can recursively call itself. Referentially transparent expressions can be replaced by their value without changing the program's behavior. Functions are first-class, they can be treated as any other variable: they can be used as input, returned as output, or be assigned to a variable. Variables are immutable, they can't be modified after initialization.
• Object-oriented - Ruby, Java, C++, Python. Real world entities are represented by Classes. Objects are instances of classes such that each object encapsulates a state (fields, attributes) and behavior (methods, what you do with the object). Objects interact with each other by passing messages.
  • Encapsulation: Classes bundle the data and methods, hide the internal representation, and provide a simple and clear interface.
  • Inheritance: A hierarchy of classes by which one class derives from another class.
  • Data abstraction: Interface shows essential information, hides details.
  • Polymorphism: A variable, function or object may take on multiple forms.

Levels of knowledge

• In the mathematical ways of figuring things out: Multiset is What, Set is How, List is Why. The reason that Set Theory works is that it is based on How, which is the level for all answers.
• In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.
• Significant=unencompassable. Covering=encompassing=Why.

Peirce's four kinds of sign. An object C can be understood as:

• Thing. The object itself.
• Index:

In the Yoneda Lemma, a functor F takes us from How to What. And an identity functor I takes us from Why (all morphisms) to Whether (the identity (do nothing) morphism which is "ownership" of the system). Going from a functor F to the identity functor I lets us go from the relative to the absolute.

Whether-What-How-Why

• Why: Set of relationships between A and A (or A and B)
• Whether: Set of identity morphisms between A and A (or A and B)
• How and What? not sets?
• The foursome defines Not as when it relates the levels of the foursome with the negation of the four representations of the nullsome.
• Things become entangled? Why (disparate reasons) -> whether (entangled consequences).

In what sense is the foursome given by the classification of topological surfaces in two-dimensions?

• Sphere: Why - orientable, without obstruction
• Torus: How - orientable, with obstruction
• Klein bottle: What - unorientable, with obstruction
• Projective plane: Whether - unorientable, without obstruction

Essentially surjective

Emily Riehl, Dominic Verity. Elements of ∞-Category Theory

Foursome

• https://ncatlab.org/nlab/show/k-surjective+functor

For C and D categories we have

• f is (essentially) 0-surjective {$⇔$} f is (essentially) surjective on objects;
• f is (essentially) 1-surjective {$⇔$} f is full;
• f is (essentially) 2-surjective {$⇔$} f is faithful;
• f is always 3-surjective.

Foursome

• https://ncatlab.org/nlab/show/stuff%2C+structure%2C+property essentially k surjective

A functor between ordinary categories (1-categories) can be:

• essentially surjective ≃ essentially 0-surjective
• full ≃ essentially 1-surjective
• faithful ≃ essentially 2-surjective
• Every 1-functor is essentially k-surjective for all k≥3.

A functor {$F:C→D$} is essentially surjective if it is surjective on objects “up to isomorphism”: If for every object {$y$} of {$D$}, there exists an object {$x$} of {$C$} and an isomorphism {$F(x)≅y$} in D.

A functor F:C→D can be:

This formalism captures the intuition of how “stuff”, “structure”, and “properties” are expected to be related:

• stuff may be equipped with structure;
• structure may have (be equipped with) properties.

Yoneda Lemma. Infinite regress should end at the 3rd level of abstraction. And that should relate to the Yates Index Theorem.

• The third level of abstraction would be 2^3=8 as with the eight-cycle of divisions, collapsing down to zero.