## Appendix A.2 (model categories and their homotopy categories) ## 2. Fibrations of simplicial sets [[Fibrations of simplicial sets]] ## 1.1 (definitions of $\infty$-categories) $\infty$-categories as simplicial sets $\infty$-categories as categories enriched in 1. $sSet$ 1. $Top_CG$ ## 1.2 (basic $\infty$-category theory) [[1.2 basic infinity-category theory]] ### 1.2.3 (the homotopy category of a simplicial set) ### 1.2.4 (objects and morphisms in an $\infty$-category) ### 1.2.5 ($\infty$-groupoids) ### 1.2.6 (homotopy commutativity and homotopy coherence) ### 1.2.7 (functors between $\infty$-categories) Proposition 1.2.7.3 ### 1.2.10, 1.2.11, 1.2.16 ## 4 Limits and colimits ### 4.1 Definition 4.1: cofinal arrow Proposition 4.1.3.1: Cofinal arrows preserve colimits ### 4.2 Theorem 4.2.4.1: relation of $\infty$-categorial colimits and homotopy colimits in simplicially enriched categories. Proposition 4.2.4.4 (and Corollary 4.2.4.7)in a simplicial model category every homotopy coherent diagram is equivalent to a commutative diagram ### 4.3 (Kan extensions) ### 4.4 Examples for limits and colimits construction of colimits from basic diagrams ## 5 ### 5.1 Presheaves ### 5.2 Definition 5.2.2.1 Proposition 5.2.2.6 Proposition 5.2.2.8 Proposition 5.2.2.9 Proposition 5.2.2.12 Proposition 5.2.3.5 Adjoint functors preserve (co)limits