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Why Category theory

Category theory lets you restate internal algebraic structure in terms of external relationships.

• Thus it offers a change in perspective.
• Limits lead us from internal structure to external relations.
• Colimits lead us from external relations to internal structure.
• To understand Hom(A,A) is to understand how composition maps all of the external walks from A to A onto the (internal) morphisms from A to A.
• It does not apply to analysis, where the internal structure is not discrete.
• It does not apply to Turing machines, where the structure keeps changing.

Category theory provides a framework for mathematical theories

• It is a language for math rather than a mathematical object of study.
• It reflects the kinds of problems that naturally arose in algebraic topology and other abstract fields.
• Lets you let go of the bottom level grounding, the points, the elements.
• Lets you define ideas from the top down.
• This gives broader, more powerful results that don't get hung up on the peculiarities coming from the points.
• As with Grothendieck.

Category theory describes richer mappings

• A functor is much more than just a morphism.
• A functor F maps a triple f:A->B to a triple in a way that honors composition though may lose information.

Category theory lets you reuse abstractions

• Shared notions such as adjunction, natural transformation, universality, limits and colimits, etc.
• Abstract set ups, which are hard to understand, don't have to be constructed from scratch.
• Often the details of the set ups can be not worried about.
• Commonality between top down issues in various branches of math.

Category theory lets you distinguish types of mathematical objects

• Distinguish categories, objects, morphisms, functors, etc.
• Can have type checking.
• Facilitates proof search.
• Facilitates proof checking.

Category theory is not hermetically sealed

• Thus, in foundational questions, category theory is less sealed off from the real world.
• Mathematics based on set theory is typically (as with ZFC) based on rather well defined notions (the empty set). But this seals it off from the real world.
• We can't define a set that consists of a cow, horse and a pig. We could write a code for such a set. But there would be no mathematical map to the cow, horse and pig.
• Category theory leaves this kind of question open. Thus there may be an object that is "the cow, horse and pig". And that object can belong to the category Set, so long as the morphisms work out correctly.
• Thus here the validity of a set depends on the coherence of the morphisms, and not any notion of elementhood.
• Thus category theory may be more amenable to supporting and manifesting cognitive concepts relevant for the real world.

How Category theory

Generic objects

• Studies all objects at once rather than a particular object.
• Relates the generic objects, for example, groups (studies the maps between them).
• Rather than particular objects, such as particular groups.
• Rather than study the elements of particular objects, the internal structure.
• Polymorphism works differently. In most of math, a structure like Z5 could be considered an example of a finite group, a subgroup of the integers, a group, a field, and so on. In category theory, this works differently, based not on membership, but on mappings, such as forgetful functors.

Building up the richness of an external mapping

• Trying to show that an external mapping can carry all kinds of constraints on the coherence of a network.
• Thus a natural tendency towards higher level category theory.

Understanding a category from top down

• The purpose of a category is typically to support natural transformations and other such higher level concepts.

Category theory is laborious, thus not spelled out

• "The functor" - be able to reconstruct what that must mean when they give just a few details.

Defining a category from objects on up

• You can define first objects, morphisms, composition and identity maps. Category is a 7-uple.
• Type theory used to define the parts of a category.

Linking top down and bottom up views

• Understand the meaning of commutative diagrams as holistic units of study. These are the external diagrams. Think of their meaning top down.
• But to understand what they actualy do, you have to draw internal diagrams that explain the maps which make the diagrams work, pulling together the various pieces.

Need to understand another subject

• So as to have "content" to make sense of it.
• Like algebraic topology or algebraic geometry.
• Need to have a context where this comes up. Otherwise, there are no "problems" to solve and assimilate, and there is no context to gain practice on what is a natural transformation, why are they relevant, etc.

Read category theory statements on different levels

• toy structures, finite structures, infinite structures, etc.
• Foundation: Objects are strings of characters. Prefer to work with finite objects.
• think of partial orders
• planar Heyting algebras

Category theory "cheats" by referencing internal structure nevertheless

• Yoneda Lemma references Set.
• Enriched categories reference Abelian groups.

What Category theory

Endless definitions that reorganize all of math

• Not organized or justified in any way.
• Just restating everything in math.
• But actually these can be organized...

No theorems to learn.

• No mountains to orient around.
• No Fundamental Theorem.

Doesn't give motivating examples

• Complicated relationship with the motivating examples. Sometimes they are too obvious. Typically trying to abstract away from them. Looking for generality which may or may not be warranted.
• Manifest concepts concretely - the point of learning more sophisticated concepts.
• Need toy examples.
• Can search for these examples.

Whether Category theory: What is not Category theory

Categories are not mathematical structures

• Not about categories as such.
• Categories are never studied as such, without reference to their content.
• Nobody studies arbitrary categories, as such, just to explore their structural possibilities, without reference to any content. Unlike groups, which are interesting structurally for their own sake.
• Just like English teachers don't study words made up of random letters or sounds.
• Counting arguments aren't used.
• Bounds or measures aren't established.
• You don't decompose them.
• The concept of a subcategory exists but is not very important. Whereas subgroups of groups are extremely important.
• You don't classify them.
• Thus they don't have to be well defined.
• You can't and don't ask mathematical questions about them.

Not about objects

• (Is there a need to define objects?) It's about objects, morphisms, composition rules. (Objects are not building blocks.)
• Its about constraints on a network imposed by associativity of composition. A coherent network of morphisms.
• Identity maps exist, but objects are superfluous.
• The idealist's point of view: "Whether" is superfluous, "Why" is real.
• Objects are external - like a vector external to a matrix.
• You know about objects from another source of knowledge.

Category is not a diagram

• Generally not emphasized that the morphisms of a category is closed under composition. Thus not every diagram can be understood as a category. If A->B and B->C then there must be an arrow A->C. And if f:I->I, then {$f^2$} must equal something, which could be {$1_I$} or {$f$} or something distinct, and likewise we have to consider {$f^n$} for all {$n>0$}.
• Diagrams with rules for equivalences (for composition). Equivalences are not mentioned or defined!

Category is not a skeleton.

• Category may typically be populated with many objects that are isomorphic to each other.
• Why not? Perhaps because typically we want to make use of the fact that if there exists some mathematical structure, then it exists as an object or morphism in some category, not isomorphically, but as a member (of, say, Set) directly participating in the diagrams.
• So what does equality mean?
• What distinguishes an object from itself, from identical or equivalent or isomorphic objects?

Categories are not well defined

• They do not specify any foundations.
• In the category of sets, what are the possible elements?
• In the category of groups, what are the possible cardinalities?

Yet a category is similar to a matrix

• This is ignored.
• Associativity = matrix multiplication.
• No morphisms from one object to another = 0.
• Matrices without an order (can it be chosen?) and without cardinality.
• And the possible links are added together commutatively. But we need a set of equivalences, how a morphism can be written.
• Thus a category could be studied.
• And there may be variant concepts of matrix or category that are metaphysically more fundamental.