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By Korchmaros G.

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3 (Joyal [44]). Let C be an ∞-category and φ : ∆1 → C a morphism of C. Then φ is an equivalence if and only if, for every n ≥ 2 and every map f0 : Λn0 → C such that f0 |∆{0,1} = φ, there exists an extension of f0 to ∆n . 2. 5 ∞-Groupoids and Classical Homotopy Theory Let C be an ∞-category. We will say that C is an ∞-groupoid if the homotopy category hC is a groupoid: in other words, if every morphism in C is an equivalence. 1, we asserted that the theory of ∞-groupoids is equivalent to classical homotopy theory.

The unit map u : S → Sing |S| is always a weak homotopy equivalence but generally not a categorical equivalence. For example, if S is an ∞-category, then u is a categorical equivalence if and only if S is a Kan complex. In general, Sing |S| may be regarded as the ∞-groupoid obtained from S by freely adjoining inverses to all the morphisms in S. 6. The inclusion functor i and its homotopy-theoretic left adjoint may also be understood using the formalism of localizations of model categories. 5. These model structures have the same cofibrations (in both cases, the cofibrations are simply the monomorphisms of simplicial sets).

However, it is still not the right definition: for a pair highertopoi August 2, 2012 16 CHAPTER 1 of objects X and Y of a higher category C, the morphism space MapC (X, Y ) should itself be well-defined only up to homotopy equivalence. 2. Let C be a topological category. The homotopy category hC is defined as follows: • The objects of hC are the objects of C. • If X, Y ∈ C, then we define HomhC (X, Y ) = π0 MapC (X, Y ). • Composition of morphisms in hC is induced from the composition of morphisms in C by applying the functor π0 .

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