Iterated algebraic injectivity and the faithfulness conjecture

Abstract

We introduce iterated algebraic injectivity and show how to describe Grothendieck ∞-groupoids as iterated algebraic injectives. Our approach is then used to prove the faithfulness conjecture of Maltsiniotis.in which Θ 0 is the initial globular theory, and in which each J n n+1 : T n → T n+1 is obtained by freely adjoining fillers -see Section 2.4. In [19] Maltsiniotis conjectured that each functor J n m : T n → T m for n < m defining a cellular globular theory is faithful. Assuming this conjecture Ara [2] established a sharp correspondence between the weak ω-categories introduced by Maltsiniotis and those of Batanin/Leinster [4,18]. However the faithfulness conjecture, on which the correspondence depends, was left unproven. Using the framework of iterated algebraic injectivity, we prove it in Theorem 5.3.Let us now describe the structure of the paper. In Section 2 we describe some background on globular sets, globular theories and the faithfulness conjecture. In Section 3 we recall algebraic injectivity, introduce iterated algebraic injectivity and the important class of (A, B)-iterated algebraic injectives, which are parametrised by a set of objects A and family of maps B in the base category. In Theorem 3.12 we show that, for a suitable choice of A and B, these capture the models of cellular globular theories. In Section 4.1 we abstract results of Nikolaus [20] relating to the construction of free algebraic injectives, and extend them to our iterated setting. Using these results, we prove the faithfulness conjecture in Section 5. In Section 6, using the framework of [11], we generalise some of the results in Section 3 away from the globular setting -in particular, we show that in general (A, B)-iterated algebraic injectives are the categories of algebras for cellular monads/theories with respect to a class of maps determined by A and B.

Modalities in homotopy type theory

“…In fact, parts of this work have already appeared as [40, sec 7. [6][7]. We generalize much of this section 7.6 to general modalities in our §1, which also sharpens the results in [40, sec 7.7].…”