Competition today has become a central policy imperative in higher education. Connected to resource efficiency and scarcity, it remains closely attached to the idea of the market but reaches beyond when related to positional or status orders. In the higher education literature such varieties of competition -- as distinct social processes -- are still under-investigated. In drawing on common references to competition in the literature and empirical examples the paper identifies and outlines two separable forms: market and field competition. The distinction shows that there are fundamental differences in the way competition in higher education plays out which should not be viewed merely as variations of market settings. To outline the differences among the two forms the paper follows a two-step iterative process. The paper first highlights a time-bound analytical problem in common market-references. Drawing on empirical work by Michel Callon it specifies market competition as the construction, singularisation and detachment of a good. As empirical examples from the German higher education case show, such singularisation and detachment are rarely complete, competition rather operates through the attachment of universities to status-consecrating intermediaries, as analysed in the status competition literature. In a second step, the paper outlines the main premises of status competition but also its lack of a comprehensive frame that could qualify it as a distinct form of competition. The 'field' fills this void by providing a selective relational space on which universities can act and observe themselves competing, reinforced by multiple intermediaries.