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Andrius Kulikauskas

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  • 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?

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 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).

Yoneda lemma and the Foursome

  • Whether: object A (Accordion)
  • What: image F(A)
  • How: morphism from A: A->
  • Why: all morphisms to A: ->A

X (Xylophone)

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

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

Significant=unencompassable.

Covering=encompassing=Why.

CategoryTheory


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Puslapis paskutinį kartą pakeistas 2017 spalio 31 d., 10:50
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