andorp / category-theory

A formalization of category theory in Coq for personal study

Welcome to the Coq formalization of Edward Kmett's Hask library.

A PDF book documenting this development is here:

http://ftp.newartisans.com/pub/hasq/Hasq.pdf

HTML documentation is here:

http://ftp.newartisans.com/pub/hasq

• What are we trying to accomplish?

Proving that a view of Haskell based on curried bifunctors leads to a unification of many concepts which are presently separate in the common Haskell idiom.

• Future directions

• see how we can use univalence to avoid predicative extensionality
• start using -indices-matter (forces eq to be in Type)
• use Set Primitive Projects
• automatic dualization
• adjunctions
• limits
• trivial categories
  • product
  • sum
  • category Two
  • binary products (using limits), C^2
• category of spans
• equalizers
• pullbacks
• arbitrary products (indexing set does not have to be a number)
• copowersn
• dependent products (pi-types, exponentials)
• ends and co-ends
• dinatural transformations
• comma categories
• Grothendieck through commas
• setting up limits and colimits in terms of adjoints to the diagonal functor
• monoidal categories
• day convolution
• monoidal functors
• monads as monoids
• profunctor composition
• prearrows as monoids of profunctor composition
• a generalized tambara module that I use for lens
• free freyd categories and can freely adjoin
• 'profunctor strength' to any profunctor
• model monads and comonads on the category of profunctors
• since the Tambara construction is a comonad
• the left adjoint to the construction is a monad
• ... whose monad algebras are the strong profunctors
• once we have that we can start talking about the lens plumbing