# Book: CategoryTheory

Study:

Higher Topos Theory

• Kerodon: An online resource for homotopy-coherent mathematics
• Lurie: Higher Topos Theory
• Riehl and Dominic Verity. Model independent higher category theory.
• Describe the mathematics of names (terms, symbols, labels).
• natural contradiction inherent in the approach of defining categories by starting with a category of categories and then taking a category to be its object
• is there a category of universal properties?
• Understand limits in terms of least upper bounds and greatest lower bounds, as with adjunct functors.
• Is it possible to categorify everything, that is, to understand all inner properties of a system in terms of external relationships?
• How does categorification relate to internalization, as with the representations of the sixsome?
• In what sense are q-analogues the opposite of categorification?

Natural transformations

• If a functor takes us from a syntactic category to a semantic category, then what does the adjoint functor mean?
• How is "extending the domain" related to adjunction?
• How do the six operations match the six criteria?
• How to deal with self-identity or non-identity f of an M-category? with copies of an M-category? Perhaps by embedding it in a bigger system?

Equality

• At what level is equality defined in defining a category? Equality is needed for the properties of identity and associativity. But is it the same identity as the identity for other morphisms?

Hidden assumptions

• Is it possible to show that category theory presumes the Axiom of Choice?

I'm trying to learn category theory because it is relevant for homotopy type theory and other areas of mathematics. But also I may be able to use it to model parts of my philosophy.

Category theory concepts such as adjoints (least upper bounds, greatest lower bounds) and limits-colimits are actually concepts of analysis.

Category theory externalizes the internal structure of mathematical objects. It reexpresses that internal structures in terms of constraints on external relationships between structures.

Category theory relates God's outer perspective (on the general, external "black box" relationships) and our inner perspective, within the system, in terms of the properties of our particular system. The question of God's necessity and nature includes the relationship between God and human's perspectives.

Category theory expresses a duality between objects and mappings. Thus functors likewise have this dual nature, and so do natural transformations.

Community

Category Theory Videos

Category Theory Books and Articles

Ideas

• Categorification (making math explicit) vs. Decategorification (making math implicit). Algebraic combinatorics is the concrete flip-side of the abstractness of category theory. But algebraic combinatorics comes with implicit interpretation whereas category theory comes with explicit notation.
• Categorical models for psychological consciousness. Sheaf theory - consciousness.
• 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.

For each object {$x$}, the identity morphism {${id}_x$} must be unique. Because consider {${id1}_x \circ {id2}_x$}. The identity is whichever disappears.

I am thinking that categories should be considered on three levels:

• Objects (of being - what is)
• Arrows (of doing)
• Equations (of reflecting) that relate arrows (or objects), especially in composition.

The composition of the arrows always seems to me underexplained. And that is where the different levels of equivalence become relevant. Also to be considered is whether an object should be thought of as an arrow to itself.

Functors

• A functor is defined by what it does on a composition triangle of morphisms, and what it does on the identity morphism: {$F(a \overset{f}{\rightarrow} b \overset{g}{\rightarrow} c) \Rightarrow F(a) \overset{F(f)}{\rightarrow} F(b) \overset{F(g)}{\rightarrow} F(c)$}
• A functor is an interpretation that takes us from a syntax category to a semantics category.
• Milewski: A functor embeds one category in another.
• Milewski: A functor may collapse multiple objects/functions into one, but it never breaks connections.

Natural transformations

• Given functors F and G, both from C to D, a natural transformation {$eta$} maps every particular object X in C to a particular morphism {$eta_X$} from {$F(x)$} to {$G(x)$}. In this sense, the object is why (as a generalization of how) and the morphism is whether (as a generalization of what). Why and whether hold beyond circumstances (the functor), whereas how and what make sense within circumstances (the functor).
• Natural transformations don't depend on the structure internal to the objects, but only on their external relationships, as expressed by the category.
• The components of natural transformations depend only on the objects. If you know these components, then the morphisms carry over automatically.
• Natural transformations say that the trivially existing bijection (between FA and GA, FB and GB) is actually a morphism in the category D.

Examples

• Z ir R. Include Z in R. Get floor of element in R to go back.
• Return and extract in Haskell.
• Expanding (replacing Identities with L*R) and collapsing (replacing L*R with Identity).
• See Wikipedia: Currying {$B \mapsto B\times C$} is left adjoint to {$A \mapsto A^C$}. This grounds the equation {$A^{B\times C}\cong (A^C)^B$}.

Relate id-A to consciousness, to constancy of attention, recurring attention.

Significant=unencompassable.

Covering=encompassing=Why.

Computation trinitarianism takes as equivalent a proof of a proposition, a program with output of some type, and a generalized element of some object.

• Category theory models perspectives and attention shifting. (Or thoughts as objects?)
• Category of perspectives: stepping-in and stepping-out as adjoints? there exists vs. for all?
• Eugenia Cheng: Mathematics is the logical study of how logical things work. Abstraction is what we need for logical study. Category theory is the the math of math, thus the logical study of the logical study of how logical things work.
• Category theory shines light on the big picture. Perspectives shine light on the big picture (God's) or the local picture (human's).
• Lygmuo Kodėl viską išsako ryšiais. O tas ryšys yra tarpas, kuriuo išsakomas Kitas. Kategorijų teorijoje panašiai, tikslas yra pereiti iš narių (objektų) nagrinėjimą į ryšių (morfizmų) nagrinėjimą.
• The theory of quasi-categories II
• Higher Algebra
• Groth, M., A Short Course on ∞-categories
• Cisinski's notes
• http://plato.stanford.edu/entries/category-theory/
• Emily Riehl, [http://www.math.jhu.edu/~eriehl/ssets.pdf | A leisurely introduction to simplicial sets]]
• Categorical Logic lecture notes by Steve Awodey

Notes

• Kategorijų teorijoje: nagrinėti įvairių lygmenų klaidas.
• Adjunction is like extension of the domain, but in terms of structure: extension of structure. Polymorphism. Try to relate Z and Q and R and other examples.
• Network theory (wiki) and Network theory (blog) by John Baez
• What can graph theory (for example, random graphs, or random order) say about category theory?
• In the category of Sets, is there any way to distinguish between the integers and the reals? Are all infinities the same?
• In the category Set, how can you distinguish between a countable and uncountable set?
• Kromer, R.: Tool and Object: A History and Philosophy of Category Theory. Birkhauser (2007)
• Fong, B., Spivak, D.I.: Seven Sketches in Compositionality: An Invitation to Applied Category Theory. Cambridge (2019),http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf
• Lawvere, W., Schanuel, S.: Conceptual Mathematics: A first introduction to categories. Cambridge (1997)
• Emily Riehl. Category Theory in Context
• In category theory, what is the relationship between structure preservation of the objects, internally, and their external relationships?
• Axiom of forgetfullness.
• Internal discussion with oneself vs. external discussion with others (Vygotsky) is the distinction that category theory makes between internal structure and external relationships.
• In most every category, can we (arbitrarily) define (uniquely) distinguished "generic objects" or "canonical objects", which are the generic equivalents for all objects that are equivalent to each other? For example, in the category of sets, the generic set of size one.
• In functional programming with monoids and monads, can we think of each function as taking us from a question type to an answer type? In general, in category theory, can we think of each morphism as taking us from a question to an answer?
• Category theory for me: distinguishing what observations are nontrivial - intrinsic to a subject - and what are observations are content-wise trivial or universal - not related to the subject, but simply an aspect of abstraction.
• In product, the information from A and B is stored externally in A x B. In coproduct, the information from A and B is stored internally in A union B (A+B). Note: multiplication is external, and addition is internal.
• In a diagram, we have a map from shape J (the index category) to the category C. Note that the index diagram is How.
• There are always dual categories {$C$} and {$C^{op}$}.