results about the simplification of framed flow categories via these moves.
address:
- |
    Department of Mathematical Sciences\
    Durham University\
    United Kingdom
- |
    Department of Mathematical Sciences\
    Durham University\
    United Kingdom
- |
    Department of Mathematical Sciences\
    Durham University\
    United Kingdom
author:
- Andrew Lobb
- Patrick Orson
- Dirk Schütz
bibliography:
- 'morsebiblio.bib'
title: Framed cobordism and flow category moves
---

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Introduction
============

Roughly speaking, a *framed flow category* is a way of encoding the flow data associated to a Morse–Smale vector field or, more generally, the flow data of a Floer functional. Objects of the category correspond to critical points while the morphism set between any two objects has the structure of a framed manifold-with-corners corresponding to the space of flowlines between the objects.

A framed flow category gives rise to a CW-complex with one cell for each object of the category, and the attaching maps determined by the morphism spaces. When the framed flow category arises from a Floer functional, the (shifted) cohomology of the CW-complex agrees with the corresponding Floer cohomology. The idea is that the whole CW-complex (up to suspension and desuspension) should be considered as the Floer invariant, instead of just 