Abstract
We discuss Peter Freyd’s universal way of equipping an additive category mathbf {P} with cokernels from a constructive point of view. The so-called Freyd category mathcal {A}(mathbf {P}) is abelian if and only if mathbf {P} has weak kernels. Moreover, mathcal {A}(mathbf {P}) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems X cdot alpha = beta for morphisms alpha and beta in mathbf {P}. We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for mathbf {P} that helps solving linear systems in mathbf {P} and even in the iterated Freyd category construction mathcal {A}( mathcal {A}(mathbf {P})^{mathrm {op}} ), which can be identified with the category of finitely presented covariant functors on mathcal {A}(mathbf {P}). The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra.
Highlights
With this paper, we hope to convince the reader that important parts of category theory such as the theory of Freyd categories are inherently algorithmic
We make this hidden role played by Freyd categories explicit and push it forward to reach new applications for computer algebra like the following computations with finitely presented functors, i.e., functors that are given as a cokernel of a natural transformation between representable functors:
This paper aims at appealing both to practitioners of computer algebra and pure mathematicians with a background in abelian categories
Summary
We hope to convince the reader that important parts of category theory such as the theory of Freyd categories are inherently algorithmic. Performing operations like taking kernels in terms of this data structure can be seen as a special instance of performing those operations within a Freyd category, namely A(RowsR), the Freyd category associated to the additive category of row modules We make this hidden role played by Freyd categories explicit and push it forward to reach new applications for computer algebra like the following computations with finitely presented functors, i.e., functors that are given as a cokernel of a natural transformation between representable functors:. 3 we give a constructive proof of the main theorem by Freyd [13] in a way such that a direct computer implementation becomes possible: Given an additive category P, the category A(P) is abelian if and only if the category P has weak kernels To this end, we provide explicit constructions for sufficiently many operations in A(P) like taking (co)kernels and computing lifts/colifts along monomorphisms/epimorphisms. We prefer writing α · β : A → C to β ◦ α for the composition of morphisms α : A → B and β : B → C, since this matches the row convention in a way that composition of morphisms is given by matrix multiplication
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