# On the functoriality of sl(2) tangle homology

## TL;DR

This paper establishes a functorial framework for sl(2) tangle homology by constructing explicit equivalences between web categories and cobordism categories, enabling strictly functorial link invariants.

## Contribution

It introduces web versions of arc algebras and their covers, providing a functorial approach to link homology theories factoring through the Bar-Natan category.

## Key findings

- Constructed an explicit equivalence between web and cobordism categories.
- Defined web versions of arc algebras with quasi-hereditary covers.
- Achieved a strictly functorial version of annular link homology.

## Abstract

We construct an explicit equivalence between the (bi)category of gl(2) webs and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasi-hereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley-Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova-Putyra-Wehrli quantization of the annular link homology.