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Higher Topos Theory

2021 Oct 2

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• \(K(G, 2)\) is a classifying space of \({{\mathbf{B}}G}\), which can be made a topological group. Then \(K(G, 2) \cong E/{{\mathbf{B}}G}\) for some contractible space \(E\).

• Each stalk equivalent to a classifying space \({{\mathbf{B}}G}\): gerbes.

• \(n{\hbox{-}}\)stacks of groupoids on \(X\) are like sheaves of homotopy \(n{\hbox{-}}\)types on \(X\).

• \((\infty, 1){\hbox{-}}\)cats: all \(k{\hbox{-}}\)morphisms are invertible.

• Replace \({\mathsf{Top}}(x, y)\) with \(\mathop{\mathrm{Maps}}(x, y)\) there the objects are maps \(f:x\to y\) and morphisms are homotopies.

• Morphisms between morphisms are 2-morphisms.

• \(n{\hbox{-}}\)groupoid: every \(k{\hbox{-}}\)morphism has an inverse for \(k\leq n\).

  • More generally, \((\infty, n){\hbox{-}}\)categories.
  • \(\infty{\hbox{-}}\)groupoids are \((\infty, 0)\) categories.

• There is an adjunction \begin{align*} \adjunction{{ {\left\lvert {{-}} \right\rvert} } }{{\operatorname{Sing}}({-})}{{\mathsf{Top}}}{{\mathsf{sSet}}} ,\end{align*} and the counit \({ {\left\lvert {{\operatorname{Sing}}(X)} \right\rvert} }\) is weakly equivalent to \(X\).

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