Notes on Yoneda Groupoids, the Construction of their Higher Quotients, and the Root of Equality

These notes are an attempt to structure the author’s thoughts and conjectures related to higher relations and their quotients. We define the notion of a Yoneda Groupoid formally, the name of which is inspired by the relation to the Yoneda lemma, and show how a weak ω groupoid structure can be extracted. We also prove that, in the presence of bracket types (in the sense of Awodey & Bauer [3]), every Yoneda Groupoid gives rise to a higher quotient. All of this is done purely syntactically, thereby making Yoneda Groupoids a very powerful concept inside the theory itself and completely independent of the Meta theory. The question whether and in which way a Yoneda Groupoid is a stronger structure than an ordinary weak ω groupoid leads to the notion of the Root of Equality, giving rise to a problem in (∞,∞)category theory. This question seems to be fundamental but has, to the best of our knowledge, not been considered so far and is therefore an open problem.