The purpose of a project brief in PRINCE2 is to provide a full and firm foundation for the initiation of the project. It includes information necessary to decide whether or not to initiate the project and acts as a base document against which the project board and key stakeholders can assess project progress during its lifecycle. References: PRINCE2 Foundation Handbook, Project Brief section.

When defining risk tolerance, the factors considered include the Project Board's attitude towards risk, the organization's risk appetite, and the level of risk deemed acceptable by the Project Board. The allocated risk budget is not a direct consideration in defining risk tolerance; rather, it is a resource aspect managed within the tolerances set for the project. References: PRINCE2 Foundation Training Manual; PRINCE2 2017 Foundation handbook.

The Communication Management Approach describes the means and frequency of communication to parties both within and outside the project. It outlines how the project will engage with stakeholders, including corporate, programme management, or the customer. This approach ensures that necessary information is provided to the right stakeholders at the right time.
PRINCE2 7 Foundation Source: PRINCE2 Foundation handbook and online resources

The purpose of the Managing a Stage Boundary process is to prepare and provide the necessary information for the Project Board to make an informed decision about authorizing the next stage of the project. This includes the preparation of key documents such as the End Stage Report and the next Stage Plan. References:
PRINCE2 Foundation Training Manual; PRINCE2 2017 Foundation handbook.

The purpose of the 'Starting up a Project' process is primarily to appoint people to the major roles within the project management team. This process ensures that the right people are in place with clear responsibilities before the project kicks off, facilitating effective planning and execution from the outset. References:
PRINCE2 7 Foundation manual and the 'Starting up a Project' process documentation.