Closing the category of finitely presented functors under images made constructive

TL;DR

This paper constructs a category that effectively models finitely presented functors and their images in additive categories, generalizing computational data structures for modules and functors.

Contribution

It provides an explicit construction of a category capturing images of morphisms in additive categories without quotients, extending computational models for finitely presented modules and functors.

Findings

Effective calculation within the constructed category is possible with algorithmic handling of syzygies.

The category is equivalent to a subcategory of finitely presented functors closed under images.

The category is abelian if and only if the underlying category has weak kernels.

Abstract

For an additive category $\mathbf{P}$ we provide an explict construction of a category $\mathcal{Q}( \mathbf{P} )$ whose objects can be thought of as formally representing $\frac{\mathrm{im}( \gamma )}{\mathrm{im}( \rho ) \cap \mathrm{im}( \gamma )}$ for given morphisms $\gamma: A \rightarrow B$ and $\rho: C \rightarrow B$ in $\mathbf{P}$, even though $\mathbf{P}$ does not need to admit quotients or images. We show how it is possible to calculate effectively within $\mathcal{Q}( \mathbf{P} )$, provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of $\mathcal{Q}( \mathbf{P} )$ with the subcategory of the category of contravariant functors from $\mathbf{P}$ to the category of abelian groups $\mathbf{Ab}$ which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: $\mathcal{Q}( \mathbf{P} )$ is abelian if and only if it is equivalent to $\mathrm{fp}( \mathbf{P}^{\mathrm{op}}, \mathbf{Ab} )$, the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if $\mathbf{P}$ has weak kernels. The category $\mathcal{Q}( \mathbf{P} )$ is a categorical abstraction of the data structure for finitely presented $R$-modules employed by the computer algebra system Macaulay2, where $R$ is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring.