Iterated algebraic injectivity and the faithfulness conjecture

Algebraic injectivity was introduced to capture homotopical structures like algebraic Kan complexes. But at a much simpler level, it allows one to describe sets with operations subject to no equations. If one wishes to add equations (or operations of greater complexity) then it is natural to consider iterated algebraic injectives, which we introduce and study in the present paper. Our main application concerns Grothendieck's weak $\omega$-groupoids, introduced in Pursuing Stacks, and the closely related definition of weak $\omega$-category due to Maltsiniotis. Using $\omega$ iterations we describe these as iterated algebraic injectives and, via this correspondence, prove the faithfulness conjecture of Maltsiniotis. Through work of Ara, this implies a tight correspondence between the weak $\omega$-categories of Maltsiniotis and those of Batanin/Leinster.