Simplicial Sets and $\infty$-Categories: Part I

“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.

Presheaves

First, we prove a variation of the Yoneda lemma (the coend calculus version).

Theorem. The cocone defined by the collection of maps \(s:\mathscr{H}_a\rightarrow X\) for \((a,s)\in \int_AX\) (identifying via Yoneda lemma) exhibits \(X\) as a colimit of \(\varphi_X\) viewed as a diagram, that is, \(X=\mathrm{colim}\ \varphi_X\).

Proof

Let \(Y\) be a presheaf on \(A\). By the Yoneda lemma \(\mathrm{Hom}_{\widehat{A}}(\mathscr{H}_a,Y)\cong Y_a\), so a cocone from \(\varphi_X\) to \(Y\) can be viewed as a collection of sections \(f_s\in Y_a\) for \((a,s)\in \int_AX\) such that \(u^*(f_t)=f_s\) for all \(u:(a,s)\rightarrow (b,t)\). This means the collection of maps \(X_a\rightarrow Y_a\) by \(s\mapsto f_s\) is a morphism of presheaves. Hence, the map \(\Phi: \mathrm{Hom}(X,Y)\rightarrow (\varphi_X\downarrow Y)\) given by composition with \(s:\mathscr{H}_a\rightarrow X\) for \((a,s)\in \int_AX\) has a two-sided inverse, therefore we have \(X=\mathrm{colim}\varphi_X\).

Let \(C\) be a category with limits. For \(u:A\rightarrow C\), we can define a functor of evaluation

\[u^*:C\rightarrow \widehat{A}\quad\quad Y\mapsto \left[a\mapsto \mathrm{Hom}_{C}(u(a),Y)\right]\]

i.e. \(u^*(Y)=\mathrm{Hom}_C(-,Y)\circ u\). By this version of Yoneda lemma, we have the following consequence.

Theorem. (Kan) The functor \(u^*:C\rightarrow \widehat{A}\) has left ajoint \(u_{!}:\widehat{A}\rightarrow C\). Moreover, there exists a natural isomorphism \(u(a)\xrightarrow{\sim} u_{!}(\mathscr{H}_a)\) for \(a\in A\), such that for any \(Y\in C\), the induced bijection \[\mathrm{Hom}_C(u(a),Y)\xrightarrow{\sim} \mathrm{Hom}_C(\mathscr{H}_a,Y)\] is the inverse of the composition \[\mathrm{Hom}_C(u(a),Y)=u^*(Y)_a\xrightarrow{\sim} \mathrm{Hom}_{\widehat{A}}(\mathscr{H}_a,u^*(Y))\xrightarrow{\sim} \mathrm{Hom}_C(u_!(\mathscr{H}_a), Y)
\] of the Yoneda equivalence with the adjunction formula.

Simplicial Sets

Let \(\Delta\) be the category where the objects are finite sets \([n]=\{0,\dots,n\}\) for \(n\in\mathbb N\), and the morphisms are order preserving functions, called the simplex category. A simplicial set is a presheaf on \(\Delta\), and we denote the category of simplicial sets by \(\mathbf{SSet}=\widehat{\Delta}\). For \(n\in\mathbb N\), denote \(\Delta^n=\mathscr{H}_n\) as the standard \(n\)-simplex. For a simplicial set \(X\), we write \(X_n=X([n])=\mathrm{Hom}(\Delta^n,X)\) the set of \(n\)-simplices of \(X\). For integer \(n\ge 1\) and \(0\le i\le n\), the map \(\partial^n_i:\Delta^{n-1}\rightarrow \Delta^n\) corresponds to the map \([n-1]\rightarrow [n]\) where the value \(i\) is not taken, and for \(n\ge 0\), the map \(\sigma^n_i:\Delta^{n+1}\rightarrow \Delta^n\) corresponds to the map \([n+1]\rightarrow [n]\) that takes the value \(i\) twice. We also write \(d^i_n=(\partial_i^n)^*:X_n\rightarrow X_{n-1}\) and \(s^i_n=(\sigma_i^n)^*:X_n\rightarrow X_{n+1}\). The category \(\Delta\) is uniquely captured by a set of identities involving these operations. There is a geometric realization functor \(|\cdot|:\mathbf{SSet}\rightarrow \mathbf{Top}\) where \(\mathbf{Top}\) is the category of compactly generated Hausdorff topological spaces, given by

\[|\Delta^n|=\left\{(x_j)_{0\le j\le n}\in\mathbb R^{n+1}_{\ge 0}: \sum_{j=0}^n x_j= 1\right\}\] and \(|X|=\mathrm{colim}_{\Delta^n\rightarrow X}|\Delta^n|\) for a simplicial set \(X\). For each \(f:[m]\rightarrow [n]\), we get an associated continuous map \(|f|:|\Delta^m|\rightarrow |\Delta^n|\), defined by \(|f|(x_0,\dots,x_m)=(y_0,\dots,y_n)\) where \(y_j=\sum_{f(i)=j}x_i\). By virtue of the preceeding theorem, we have singular complex functor \(\mathrm{Sing}:\mathbf{Top}\rightarrow\mathbf{SSet}\), given by \(Y\mapsto \left[[n]\mapsto \mathrm{Hom}(|\Delta^n|,Y)\right]\) which is right adjoint to the geometric realization functor, i.e. \(|\cdot|\dashv \mathrm{Sing}\)

Definition. An Eilenberg-Zilber category is a quadruple \((A,A_{+},A_{-},d)\) where \(A\) is a category, \(A_{+},A_{-}\) subcategories, and \(d:A\rightarrow\mathbb N\) a set-function, such that

1. An isomorphism of \(A\) is an isomorphism of \(A_{+}\) and \(A_{-}\)
2. If a morphism \(a\rightarrow b\) in \(A_{+}\) (resp. \(A_{-}\)) is not the identity then \(d(a)<d(b)\) (resp. \(d(b)<d(a)\))
3. Any morphism \(u:a\rightarrow b\) in \(A\) factors uniquely as \(u=ip\) where \(p:a\rightarrow c\) in \(A_{-}\) and \(i:c\rightarrow b\) in \(A_{+}\)
4. For a morphism \(\pi:a\rightarrow b\) in \(A_{-}\), there exists \(\sigma:b\rightarrow a \) in \(A\) such that \(\pi\sigma=\mathbf{1}_b\).
5. For \(\pi,\tau:a\rightarrow b\) in \(A_{-}\), if \(\pi\) and \(\tau\) have the same set of sections then \(\pi=\tau\).

We say an object \(a\in A\) has dimension \(n\) if \(d(a)=n\).

The category \(\Delta\) is Eilenberg-Zilber with \(\Delta_{+}\) (resp. \(\Delta_{-}\)) the subcategory of monos (resp. epis), and \(d([n])=n\). Let \(X\) be a presheaf on an Eilenberg-Zilber category \(A\). For \(a\in A\), we say \(x\in X_a\) is degenerate if there is a map \(\sigma:a\rightarrow b\) in \(A\) with \(d(b)n\) any section of \(\mathrm{Sk}_n(X)\) over an object \(a\) of dimension \(m\) is degenerate, that is \(\mathrm{Sk}_n(X)\) restricts the sections to dimensions \(\le n\). We can easily make the construction \(\mathrm{Sk}_n\) functorial, so it can be viewed as a functor.

Nerves

Every poset \(E\) can be viewed as a category where objects are elements, and there is a unique morphism \(x\rightarrow y\) if \(x\le y\) and none otherwise. Let \(i:\Delta\rightarrow\mathbf{Cat}\) the inclusion functor, then the nerve functor \(N=i^*:\mathbf{Cat}\rightarrow\mathbf{SSet}\) is given by \(C\mapsto [[n]\mapsto \mathrm{Hom}_{\mathbf{Cat}}([n],C)]\). Thus the \(n\)-simplex of \(N(C)\) is a string of arrows of length \(n\) in \(C\). By the preceeding theorem, the nerve functor has a left adjoint \(\tau=i_{!}:\mathbf{SSet}\rightarrow\mathbf{Cat}\). We’ll stop here and continue more in Part II.