# Methods of constructive category theory

## TL;DR

This paper introduces constructive methods in category theory, focusing on computing natural transformations between functors and determining spectral sequence differentials using explicit, axiom-based calculations.

## Contribution

It presents new constructive techniques for explicit calculations in category theory, including category constructors and diagram chases within abelian categories.

## Key findings

- Category constructors enable explicit computation of natural transformations.
- Constructive diagram chases determine spectral sequence differentials.
- Methods are applicable to finitely presented functors and filtered cochain complexes.

## Abstract

We give an introduction to constructive category theory by answering two guiding computational questions. The first question is: how do we compute the set of all natural transformations between two finitely presented functors like $\mathrm{Ext}$ and $\mathrm{Tor}$ over a commutative coherent ring $R$? We give an answer by introducing category constructors that enable us to build up a category which is both suited for performing explicit calculations and equivalent to the category of all finitely presented functors. The second question is: how do we determine the differentials on the pages of a spectral sequence associated to a filtered cochain complex only in terms of operations directly provided by the axioms of an abelian category? Its answer relies on a constructive method for performing diagram chases based on a calculus of relations within an arbitrary abelian category.