Exercises for Lecture 3

Exercises for Lecture 3

Exercise 1: Check that if \(\mathcal{C},\mathcal{D}\) are ordinary categories, then \(\mathsf{N}_\bullet(\mathcal{C}\times\mathcal{D})\simeq \mathsf{N}_\bullet(\mathcal{C})\times \mathsf{N}_\bullet(\mathcal{D})\), hence establishing that the product of \(\infty\)-categories generalizes that of products of categories.

Exercise 4: (\(\infty\)-subcategories are determined by their morphisms) Let \(\mathcal C\) be an \(\infty\)-category and consider a collection \(S\subset \mathcal C_1\) of morphisms in \(\mathcal C\). Define \(\mathcal C_n':=\{a\in\mathcal C_n\colon\,a_{ij}\in S \text{ for all }0\leq i \leq j\leq n\}.\) Now show that the \(\mathcal C_n ‘s\) describe a subsimplicial set \(\mathcal C'\) of \(\mathcal C\) if and only if \(f_{00},f_{11}\in S\) for all \(f\in S\).

Exercise 4’: Show that furthermore \(\mathcal C'\) is an \(\infty\)-subcategory if and only if, in addition, for all \(u\in \mathcal C_2\) we have that \(u_{01},u_{12}\in S\) implies \(u_{02}\in S\).

Exercise 5: Show that \(\mathsf{N}_\bullet (\mathcal C)^{\mathsf{op}}\simeq \mathsf N_\bullet(\mathcal C^{\mathsf {op}}).\) Use it to convince yourself that precomposing with the \(\mathsf{op}\) involution is a generalization from the usual construction for ordinary categories.

Exercise 6: Show that \(\alpha^{\ast}\colon \mathsf{Hom}(\mathsf N_{\bullet}(h X),\mathsf{N}_{\bullet}(\mathcal C))\rightarrow \mathsf{Hom}(X,\mathsf{N}_{\bullet}(\mathcal C))\) is a bijection.

Exercise 7: Show that if \(\mathcal C\) is an \(\infty\)-category, there is an isomorphism \(\mathsf{h}(\mathcal C)^{\simeq}\simeq \mathsf{h}(\mathcal C^{\simeq})\).

Exercise 9: Show that, for a collection \(\{\mathcal C_{i}\}\) of \(\infty\)-categories, the follwoing holds

\[h(\prod \mathcal C_i)\simeq \prod h(\mathcal C_i).\]

In other word, the homotopy category functor preserves products.

Exercise 9: Show that the function complex defines a functor

\[\mathsf{Fun}\colon \mathsf{sSet}^{\mathsf{op}}\times \mathsf{sSet} \rightarrow \mathsf{sSet}.\]