For plain homotopy $n$-types (in ∞Grpd/Top) this is a presentation of the collection of n-groupoids. With respect to an arbitrary (∞,1)-topos it is a presentation of n-stacks.

J. Cabello, A. Garzon, Quillen’s theory for algebraic models of $n$-types, Extracta mathematica Vol. 9, Num. 1, 42-47 (1994)

J. Cabello, A. Garzon, Closed model structures for algebraic models of $n$-types, Journal of pure and applied algebra 103 (1995) 287-302

J. Cabello, Estructuras de modelos de Quillen para categorías que modelan algebraicamente tipos de homotopía de espacios, Ph.D. thesis, Universidad de Granada. (1993)