# Categorification of invariants in gauge theory and sypmplectic geometry

## TL;DR

This paper surveys and advances the categorification of invariants in gauge theory and symplectic geometry, combining analytic and algebraic methods to improve understanding of Floer homologies and their relations.

## Contribution

It introduces a new algebraic approach to categorify Floer theories, replacing complex analytic methods with cobordism and homological algebra techniques.

## Key findings

- Resolved difficulties in analytic approaches to Floer homology
- Simplified proofs of categorification results
- Connected gauge theory invariants with symplectic geometry via algebraic methods

## Abstract

This is a mixture of survey article and research anouncement. We discuss Instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998-2012, those problems have been studied emphasising the ideas from analysis such as degeneration and adiabatic limit (Instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs.