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Let me restate the corollary in question, for convenience:
Corollary $2.4.4.4$:
Suppose we have functors of $\infty$-categories: $$\mathcal{C}\overset{p}{\longrightarrow}\mathcal{D}\overset{q}{\longrightarrow}\mathcal{E}$$such that both $q$ and $q\circ p$ are locally Cartesian fibrations, and that $p$ maps locally $(q\circ p)$-Cartesian...

As stated; my professor was unsure on analytically finding the second solution of xcos(x) - sin(x) = 0 within [0,2π].More generally, how would you go about finding the general form for roots in (-inf, inf)?

Let $M_R$ be a right $R$-module with $E=\text{End}(M)$. Let $\bigoplus_{i\in I} e_iM$ be a summand of $M_R$ where $e_i^2 = e_i \in E$. Without getting involved in the details (which we don't need by the way), it is true that the sum $\sum_{i\in I}e_i...

Warning: the definitions used in this question are not standard!
Given a tuple (i.e., finite sequence) $(a_1,\dots,a_n)$ of positive real numbers, we say that it's $\varepsilon$-sparse if$$\#\Big\{i\,:\,a_i\ge\varepsilon\frac{a_1+\dots+a_n}{n}\Big\} \le \varepsilon n.$$Intuitively, this says that only a fraction ...