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Many initiatives aim at answering the challenges of the fourth industrial revolution (Industry 4.0). A key aspect of these initiatives is the connectivity of the different industrial entities. Communication and information exchange between entities usually flows in two dimensions; a vertical one, that spans the different levels of abstraction within the company, and a horizontal one connecting the different entities in the supply chain. In contrast, the Internet of Production (IoP) presents the World Wide Lab (WWL) allowing an extra dimension of communication, which we call the cross-company dimension. Hence, the WWL integrates the different entities within and across potentially competing labs and companies. The WWL leverages and aggregates accessible production data fostering an exchange of information, that prevents redundant experimentation and increases quality of data-driven models by including data from diverse manufacturing settings. This paper discusses work done towards this vision and technical, economic, social, legal, and ethical challenges the WWL faces and provides a roadmap towards infrastructuring the WWL.