  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.
