(Q1) Given a topological space X, what should we mean by a “sheaf of n-types” on X?

(Q3) What special features (if any) does Shv_{≤n}(X) possess?

Our answers to questions (Q2) and (Q3) may be summarized as follows:

(A2) The collection Shv_{≤n}(X) has the structure of an ∞-category.

Grothendieck’s vision has been realized in various ways, thanks to the work of a number of mathematicians (most notably * Jardine*), and their work can also be used to provide answers to questions (Q1) and (Q2). Question (Q3) has also been addressed (at least in limiting case n = ∞) by

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**Toën and Vezzosi**