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By John B. Fraleigh

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It may be characterized by the following universal property: for any Kan complex K, the induced map HomSet∆ (K, C� ) → HomSet∆ (K, C) is a bijection. highertopoi 36 August 2, 2012 CHAPTER 1 Proof. It is straightforward to check that C� is an ∞-category. Moreover, if f is a morphism in C� , then f has a homotopy inverse g ∈ C. Since g is itself an equivalence in C, we conclude that g belongs to C� and is therefore a homotopy inverse to f in C� . 1. To prove the last assertion, we observe that if K is an ∞-category, then any map of simplicial sets φ : K → C carries equivalences in K to equivalences in C.

1. 2: we require the same extension conditions but drop the uniqueness assumption. 5. Any Kan complex is an ∞-category. 4. 6. The nerve of any category is an ∞-category. We will occasionally abuse terminology by identifying a category C with its nerve N(C); by means of this identification, we may view ordinary category theory as a special case of the study of ∞-categories. 4 leads to a very elegant and powerful version of higher category theory. 4 are called quasi-categories) and will be used throughout this book.

We will show that (2) ⇒ (3) ⇒ (4) ⇒ (1); the implications (2) ⇒ (5) ⇒ (6) ⇒ (1) follow using the same argument. To see that (2) implies (3), we note that if g is a homotopy inverse to f , then composition with g gives a map MapC (Z, Y ) → MapC (Z, X) which is homotopy inverse to composition with f . It is clear that (3) implies (4). Finally, if (4) holds, then we note that X and Y represent the same functor on hC so that f induces an isomorphism between X and Y in hC. 2. Let C be the category of CW complexes which we regard as a topological category by endowing each of the sets HomC (X, Y ) with the (compactly generated) compact open topology.

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A First Course in Abstract Algebra, 7th Edition by John B. Fraleigh

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