A Constructive Approach to Freyd Categories

Applied Categorical Structures, vol. 29, pp. 171–211

Abstract

Abstract We discuss Peter Freyd’s universal way of equipping an additive category $$\mathbf {P}$$ P with cokernels from a constructive point of view. The so-called Freyd category $$\mathcal {A}(\mathbf {P})$$ A ( P ) is abelian if and only if $$\mathbf {P}$$ P has weak kernels. Moreover, $$\mathcal {A}(\mathbf {P})$$ A ( P ) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems $$X \cdot \alpha = \beta $$ X · α = β for morphisms $$\alpha $$ α and $$\beta $$ β in $$\mathbf {P}$$ 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 $$\mathbf {P}$$ P that helps solving linear systems in $$\mathbf {P}$$ P and even in the iterated Freyd category construction $$\mathcal {A}( \mathcal {A}(\mathbf {P})^{\mathrm {op}} )$$ A ( A ( P ) op ) , which can be identified with the category of finitely presented covariant functors on $$\mathcal {A}(\mathbf {P})$$ A ( 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.

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