(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Classically, the theory of stacks was motivated by the study of moduli problems for which objects are classified up to isomorphism. Higher stacks are a generalization where objects are classified up to some notion of equivalence, like complexes up to quasi-isomorphism, topological spaces up to weak homotopy equivalence, or abelian categories up to equivalence of categories.
Carlos Simpson, Algebraic (geometric) $n$-stacks, 1996, arXiv:alg-geom/9609014.
André Hirschowitz, Carlos Simpson, Descente pour les n-champs (Descent for n-stacks), 1998, arXiv:math/9807049.
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