## 1.2.1 the opposite of an $\infty$-category For topological and simplicial categories the definition of the opposite category is the same as the notion from classical category theory. For a simplicial set we obtain the opposite simplicial set by component- wise reversing the order of the ordinal. +-- {: .num_defn} ###### Definition Let $S$ be a simplicial set. Let $J$ be a linear ordered set. Then the face and degeneracy maps on $S^{op}$ are given by. $$(d_i:S_n^{op}\to S_{n-1}^{op})=(d_{n-i}:S_n\to S_{n-1})$$ $$(s_i:S_n^{op}\to S_{n+1}^{op})=(s_{n-i}:S_n\to S_{n+1})$$ =-- ## 1.2.2 mapping spaces in higher category theory +-- {: .num_defn} ###### Definition Let $S$ be a simplicial set. Let $x,y\in S$ be vertices. Then the *simplicial mapping space* is defined by $$Map_S (x,y):=Map_{|S|} (x,y)$$ where $|-|:sSet Cat\to s Set$ denotes the adjoint of the [[nLab:homotopy coherent nerve]]: the [[homotopy coherent realization]]. We have $$|-|=Lan_y \mathfrak{C}$$ where $y:\Delta\hookrightarrow [\Delta^{op},Set]$ denotes the Yoneda embedding and $\mathfrak{C}: \Delta\to sSet Cat$ denotes the [[cosimplicial-thickening functor]]. We think of $\mathfrak{C}$ as assigning to an ordinal $[n]$ (considered as a category) a simplicially-enriched category which is thickened. =-- ## 1.2.3 the homotopy category ## 1.2.4 objects morphisms and equivalences ## 1.2.5 groupoids and classical homotopy theory ## 1.2.6 homotopy commutativity versus homotopy coherence ## 1.2.7 functors between higher categories ## 1.2.8 joins of $\infty$-categories ## 1.2.9 overcategories and undercategories ## 1.2.10 fully faithful and essentially surjective functors ## 1.2.11 subcategories of $\infty$-categories ## 1.2.12 initial and final objects ## 1.2.13 limits and colimits ## 1.2.14 presentations of $\infty$-categories ## 1.2.15 Set-theoretic technicalties ## 1.2.16 the $\infty$-category of spaces