Trying to implement schemes and homotopy

Algebraic · March 6, 2021, 5:02am

Captainkk75/Schemes/blob/main/Scheme.hs


-- A scheme consists of 
    --questions:    a list of tuples of 2 integers representing the coefficients of a linear equation in 1 variable
    --answers :     a list of tuples of 1 integer representing the solution to linear equation in 1 variable
data Pool = Pool [(Int,Int)] [(Floating)]

assign :: Pool a => a -> [((Int,Int),Floating)]
-- assigns questions to answers
-- input:   a pool containing questions and answers 
-- output:  a list describing the assignment of answers to questions

transformQuestions :: Pool a => a -> ([(Int,Int)] -> [(Int, Int)]) -> a
-- transforms the questions -- for example by matrix multiplication
-- input:   a pool containing questions and answers
--          a function that transforms the questions to new questions
-- output:  a pool containing new questions and old answers

transformAnswers :: Pool a => a -> ([(Floating)] -> [(Floating)]) -> a
-- transforms the answers -- for example by matrix multiplication
-- input:   a pool containing questions and answers

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I have tried to describe an idea which includes sort of using the idea of Homotopy from Topology and Scheme from Algebraic Geometry; using only type signatures. I would like to ask if something like this can be implemented, and if anybody has done such a project.

I am quite new to Haskell. I am studying Programming with Categories by David Spivak, Brendan Fong and Bartosz Milewski. I am also studying Algebraic Geometry of Schemes. And I was wondering if something slightly abstract as such can be implemented in a useful way.