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Consider an arbitrary site (or an ∞-site) S. In fact, the constructions below only depend on the underlying topos (or ∞-topos) T of S, and not on S itself. Below “sheaf”, “∞-sheaf”, “stack”, and “∞-st …

If we now take the ∞-sheaf of sections of the resulting map $E(F)→X$ of stacks, we recover the original ∞-sheaf $F$. (There are many other constructions, of course. …

Another application of stacks is in synthetic differential geometry. … Just like for stacks on manifolds, homotopy colimits in this category have excellent geometric properties. …

Stacks are used in complex analysis, for example. …

$\underline{G}$ is the homotopy loop space of $BG$. More precisely, the two terminal maps $G\rightarrow pt$ and $G\rightarrow pt$ yield a weak equivalence $\underline{G} \rightarrow pt\times_{BG} pt …

The easiest way to see local trivializations is to compute the homotopy pullback using the local projective model structure. For differential geometry, we can $C$ to be the category of cartesian space …

I will start by explaining the easiest possible case of bundle gerbes, when the band A (alias structure group) is an abelian Lie group. A bundle n-gerbe with band A over a smooth manifold M is a prin …

Pullbacks of stacks coming from Lie groupoids are not always equivalent to Lie groupoids. Take $G=H=\mathbb{R}$. Define $F(x)=0$ if $x\leq 0$ and $F(x)=exp(−1/x^2)$ if $x>0$. …

Gerbes over stacks are classified by (∞,1)-sheaf cohomology. Concretely, one can implement it as a derived mapping space in the model category of simplicial presheaves. …

Two different cleavages produce isomorphic pseudofunctors. This follows immediately from Theorem 8.3.1 in Borceux's Handbook of Categorical Algebra 2. Specifically, part (1) of this theorem states t …