“I am not saying that I believe in the law of the excluded middle, I am just saying that it isn’t not true. ”
Kavin Satheeskumar
Recently, I’ve been reading Cisinski’s Higher Categories and Homotopical Algebra. I wanted to write down some of the things I’ve learned. In this post, we start from a review of presheaves of sets and simplicial sets, and then build towards the definition of an \(\infty\)-category. Let \(A\) be a category, recall a presheaf on \(A\) is a contravariant functor \(X:A^{\mathrm{op}}\rightarrow \mathbf{Set}\), where we denote \(X_a=X(a)\) and \(u^*:b\rightarrow a\) the induced morphism for each \(u:a\rightarrow b\). And the category of presheaves on \(A\) is denoted as \(\widehat{A}\). The category of elements \(\int_AX\) (or \(\int X\)) is the category where objects are \((a,s)\) where \(a\in A\) and \(s\in X_a\), and a morphisms \(u:(a,s)\rightarrow (b,t)\) is a morphism \(u:a\rightarrow b\) where \(u^*(t)=s\). It comes equipped with a faithful functor \(\varphi_X:\int_A X\rightarrow \widehat{A}\) given by \((a,s)\mapsto \mathscr{H}_a\) on objects and \(u\mapsto \mathscr{H}(u)\) on morphisms, where \(\mathscr{H}:A\rightarrow \widehat{A}\) is the Yoneda embedding. In this post, we ignore all set-theoretic size issues.