homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
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Given a site $C$ equipped with an interval object ${*}\amalg {*} \stackrel{[i_0, i_1]}{\to}I$ the homotopy localization of an (∞,1)-category of (∞,1)-sheaves $Sh_\infty(C)$ on $C$ is the (∞,1)-categorical localization of $Sh_\infty(C)$ at the morphisms of the form
Taking $C =$ Top and the interval object $I$ to be the standard topological interval $I = [0,1]$, the homotopy localization of $\infty$-stacks on $Top$ is equivalent to the (∞,1)-category Top itself again. For more on this see the discussion and references at topological ∞-groupoid.
A homotopy localization of the (∞,1)-topos of ∞-stacks on the Nisnevich site is used in motivic homotopy theory. See there for more details.