# Limits, standard complexes and fr-codes

## TL;DR

This paper introduces a cosimplicial resolution for computing higher derived functors of limits in certain categories, with applications to K"unneth theorems and fr-code finiteness, advancing the understanding of categorical limits.

## Contribution

It develops a new cosimplicial object for strongly connected categories with coproducts, enabling computation of higher limits and exploring fr-code properties.

## Key findings

- Established a cosimplicial resolution for higher limits.
- Applied the resolution to K"unneth theorem for higher limits.
- Proved lim-finiteness of fr-codes for words of length ≤ 3.

## Abstract

For a strongly connected category $\mathcal C$ with pair-wise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to \mathrm{Ab}$. Applications involve K\"unneth theorem for higher limits and ${\sf lim}$-finiteness of ${\bf fr}$-codes. A dictionary for the ${\bf fr}$-codes with words of length $\leq 3$ is given.