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Andrius Kulikauskas

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Software

See: Yoneda Lemma, Topos, Sets, Logic

Study:

  • Applied Category Theory Course, Discussion and Textbook
  • 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.
  • 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-identityf of an M-category? with copies of an M-category? Perhaps by embedding it in a bigger system?

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

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.

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.

Adjoint functors

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).

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.

CategoryTheory


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