In the course of this series of notes, titled: “Examples of \(\infty\)-categories”, I will construct as many instances as possible of \(\infty\)-categories. The workhorse here will be the notion of enriched category, which permits interesting generalizations of the nerve construction, which we have now become familiar with, to more “interesting” and rich settings like that of differential graded or simplicially enriched categories.

Examples of \(\infty\)-categories

We have introduced an \(\infty\)-category as a weak Kan complex. One of the aspirations we set out for our theory of \((\infty,1)\)-categories was that it could offer a merge of ordinary category theory and algebraic topology, which we can confirm this was achieved by the fact that the following two constructions constitute examples of \(\infty\)-categories:

Ordinary categories \(\mathcal C\), via their nerve construction \(\mathsf N_\bullet (\mathcal C)\), satisfy the weak Kan extension condition (as we saw in class 2).
Homotopy types, or spaces up to homotopy, in the guise of Kan complexes, form \(\infty\)-categories. For every topological space \(X\), its \(\mathsf{Sing}_\bullet(X)\) is a Kan complex, hence in particular a weak Kan complex, and it codifies the homotopy theory of \(X\).

Following Lurie’s advice on Kerodon, it is best that we provide a word of warning about these two examples in relationship to our ultimate goal of constructing homotopically-coherent categorical structures. \(\infty\)-categories are intersting in so far as they collect not only information about the objects and morphisms, but also about the possibly interesting homotopies between the morphisms. In the case of \(\mathcal C\) being the nerve of an ordinary category, as we saw in Lecture 2, the homotopy relation amounts to the usual identity i.e. \(f,g\colon x \rightarrow y\) are homotopic iff \(f=g\), hence the possibly interesting homotopical information is just not there. On the other extreme, if \(\mathcal C = \mathsf{Sing}_\bullet (X)\), then every morphism is invertible up to homotopy, which from a categorical point-of-view is an extremely limited condition. As such, we will try to construct examples which support additional non-triviality at both ends of the spectrum.

Enriched Categories

Let \((\mathcal V,\boxtimes)\) be a monoidal category. We define a \(\mathcal V\)-enriched category \(\mathcal C\) to be the following data:

A collection \(\mathsf{Ob}(\mathcal C)\), whose elements we refer to as objects of \(\mathcal C\).
For every pair of objects \(X,Y\in \mathsf{Ob}(\mathcal C)\), an object \(\underline{\mathsf{Hom}}_{\mathcal C}(X,Y)\in \mathcal V.\)
For every triple of obejcts \(X,Y,Z\in \mathsf{Ob}(\mathcal C),\) a morphism

\[c_{Z,Y,X}\colon \underline{\mathsf{Hom}}_{\mathcal C}(Y,Z)\otimes \underline{\mathsf{Hom}}_{\mathcal C}(X,Y) \rightarrow \underline{\mathsf{Hom}}_{\mathcal C}(X,Z)\]

in the category \(\mathcal{V}\), which we will refer to as the composition law, and refer to objects in their image as \(g\circ f\) for \(g\in \underline{\mathsf{Hom}}_{\mathcal C}(Y,Z)\), \(f\in \underline{\mathsf{Hom}}_{\mathcal C}(X,Y)\). Moreover, this composition law is associative (as an exercise write down what this means with a suitable commutative diagram).

For every object \(X\in \mathcal{C}\), a morphism \(e_X\colon \mathbf{1}\rightarrow \underline{\mathsf{Hom}}_{\mathcal C}(X,X)\) which we refer to as the identity of \(X\), and as such has to satisfy an appropriate commuativity diagram (write down this diagram as an exercise).

The notion of enriched category captures simultaneously the idea that your morphisms might not be elements in some set, and that they can be elements in some set with additional structure which is compatible with composition and identity formation.

Examples of these objects abound. Linear categories are categories enriched over vector spaces with the usual tensor product - in particular \(\mathsf{Vect}\), the category of finite-dimensional vector spaces and linear maps, the category \(\mathsf{Rep}(G)\) of finite-dimnsional representations of some group \(G\). In the case of categories enriched over abelian groups, we usually say they are additive - examples here are for exmaple the category \(\mathsf{Mod}_R\) of left modules of some ring \(R\). On a more topologically flavoured side we have for example \(\mathsf{Top}\), the category of topological and continuous maps with the Cartesian product, then \(\mathsf{Top}\) is itself enriched over \(\mathsf{Top}\) by taking the compact-open topology, or the category of \(n\)-dimensional topological manifolds and open embeddings among them is also a topologically enriched category by taking the compact-open topolgy on the space of embeddings.

There are some details about enriched categories which I will sweep under the rug for now, or let them for your own discovery. Among these, I point to the following two:

What is the correct generalization of a \(\mathcal V\)-functor between \(\mathcal V\)-enriched categories?
Can we associate to every \(\mathcal V\)-enriched category an ordinary (enriched over sets!) category?
(If you are feeling particularly brave afer 1.) What is the correct generalizationo of \(\mathcal V\)-natural transformation?

Differential Graded Categories

We will call a category enriched in abelian groups an additive category. Let \(\mathcal A\) be an additive category, we define \(\mathsf{Ch}(\mathcal A)\) to be its associated category of chain complexes and chain maps. This category admits many additional interesting structures, some of which we will discuss in this note. One of them is a symmetric monoidal structure defined as follows: let \(\mathsf C_\ast\) and \(\mathsf D_\ast\) be chain complexes (over the integers \(\mathbb Z\)). We define a new chain complex \((\mathsf C \boxtimes \mathsf D)_n := \bigoplus_{n = i + j}\mathsf C_i \otimes \mathsf C_j\), where \(\otimes\) here is the usual tensor product of abelian groups. In order for this to be chain complex, we also require the existence of a differential

\[\begin{aligned}\partial \colon (\mathsf C\boxtimes \mathsf D )_\ast &\rightarrow (\mathsf C\boxtimes \mathsf D )_{\ast-1}\\ (x\boxtimes y) &\mapsto \partial(x)\boxtimes y + (-1)^{m}x\boxtimes \partial(y) \end{aligned}\]

where \(x\in \mathsf C_m\) and \(y\in \mathsf D_n\).

Exercise: Check that this is a differential.

The monoidal unit for this monoidal product is the complex \(\mathbb Z[0]\), which is \(\mathbb Z\) concentrated in degree 0 with trivial differential.

We make use of this monoidal structure in order to define differential graded categories as \((\mathsf{Ch}(\mathbb Z),\boxtimes)\)-enriched categories. Let us unpack what is in the definition of a \(\mathsf{Ch}(\mathbb Z)\)-enriched category, which we henceforth call a differential graded category:

A collection of objects \(\mathsf{Ob}(\mathcal C)\).
For every pair of objects \(X,Y\in\mathcal C\), a chain complex \((\mathsf{Hom}_{\mathcal C}(X,Y)_\ast,\partial).\) We call the \(n\)-chains of this complex, the morphisms of degree \(n\) from \(X\) to \(Y\).
For every triple of obejcts \(X,Y,Z\in \mathsf{Ob}(\mathcal C)\), a composition law: \(\circ \colon (\mathsf{Hom}_\mathcal{C}(Y,Z)\boxtimes\mathsf{Hom}_{\mathcal C}(X,Y))_\ast \rightarrow \mathsf{Hom}_{\mathcal C}(X,Z)_\ast\) which must satisfy an associativity rule.
For eery object \(X\), a morphism \(\mathsf{id}_X\in \mathsf{Hom}_{\mathcal C}(X,X)_0\) which we will refer to as the identity morphism.

Remarks:

Additionally, we must impose that for every \(f\colon X \rightarrow Y\) a morphism of degree \(n\), we have that \(f\circ \mathsf{id}_X=f= \mathsf{id}_Y \circ f\).
The composition maps are lienar and satisfy the Leibniz rule:

\[\partial (g\circ f)=(\partial g)\circ f + (-1)^n g\circ (\partial f)\]

where \(g\in \mathsf{Hom}_{\mathcal C}(Y,Z)_m\) and \(f\in \mathsf{Hom}_{\mathcal C}(X,Y)_n\).

One can show, using the Leibniz rule, that \(\mathsf{id}_X\) is a 0-cycle, and that if \(f,g\) are \(n\)- and \(m\)-cycles respectively, then \(g\circ f\) is an \(n+m\)-cycle.

The Differential Graded Nerve

The goal in this section is to expand on a construction \(\mathsf N_\bullet ^{\mathsf {dg}}(\mathcal C)\) which we will call the differential graded nerve. It associates an \(\infty\)-category to every differential graded category \(\mathcal C\).

Construction: Let \(\mathcal C\) be a differential graded category. For \(n\geq 0\), we define \(\mathsf N_n ^{\mathsf{dg}}(\mathcal C)\) as the collection of all ordered pairs \((\{X_i\}_{0\leq i\leq n},\{f_{I}\})\), where:

Each \(X_i\) is an object of the differential graded category \(\mathcal C\).
For every subset \(I=\{i_0 > i_1 > \cdots > i_k\}\subseteq [n]\) having at least two elements, \(f_I\) is an element of the abelian group \(\mathsf{Hom}_{\mathcal{C}}(X_{i_k},X_{i_0})_{k-1}\) which satisfies the following identity

\[\partial f_{I} = \sum_{a=1}^{k-1}(-1)^{a}(f_{i_0 > i_1 > \cdots > i_a}\circ f_{i_a > \cdots > i_k}-f_{I\setminus \{i_a\}})\]

This looks awakward until we start unwinding what it means in the lower dimensional cases. Let us begin with the case \(n=0\):

For \(n=0\), we have that the 0-cells correspond to ordered pairs \((\{X\},\varnothing)\) i.e. it corresponds to the set of objects of \(\mathcal C\). In summary \(\mathsf N_{0}^{\mathsf{dg}}(\mathcal C) = \mathsf{Ob}(\mathcal C).\)
For \(n=1\), as there is a unique subset \(I\subseteq [1]\) with at least two elements, the 1-cells correspond to triples \((\{X_0,X_1\},f)\) composed of a pair of objects \(X_0,X_1\in \mathcal C\) and a 0-chain \(f\in \mathsf{Hom}_{\mathcal C}(X_0,X_1)_0\) satisfying \(\partial f=0\) i.e. such that \(f\) is a 0-cycle. As we saw before, these correspond to the morphisms in the underlying category \(\overset{\circ}{\mathcal C}\). In summary, \(\mathsf{N}_{1}^{\mathsf{dg}}(\mathcal C) = \mathsf{Mor}(\overset{\circ}{\mathcal C})\).
For \(n=2\), it is relevant to start by noting that there are four subsets of \([2]\) with at least two elements: \(\{0,1\},\{1,2\},\{0,2\},\{0,1,2\} \subseteq [2]\). As such we have that a 2-cell is specified by:
- A triple of objects \(X_0,X_1,X_2\in \mathcal C\)
- A triple of 0-cycles
\[f_{01}\in \mathsf{Hom}_{\mathcal C}(X_0,X_1)_0, \quad f_{12}\in \mathsf{Hom}_{\mathcal C}(X_1,X_2)_0, \quad f_{02}\in \mathsf{Hom}_{\mathcal C}(X_0,X_2)_0\]
- A 1-chain \(f_{012}\in \mathsf{Hom}_{\mathcal C}(X_0,X_2)_1\) satisfying the following condition
\[\partial(f_{012}) = f_{02}-(f_{21}\circ f_{01}).\]
Here we think of the 1-chain as a witness to the fact that \(f_{02}\) and \(f_{12}\circ f_{01}\) are homolgous 0-cycles i.e. represent the same class in \(\mathsf H_0(\mathsf{Hom}_{\mathcal{C}}(X_0,X_2)_\bullet)\)

We want these sets \(\mathsf{N}_{\bullet}^{\mathsf{dg}}(\mathcal C)\) to assemble into a simplicial set, and for that we have to define a coherent action by simplicial operators. In order not to overbudern the exposition, we will only give a hint on how to prove the following result, and leave it as an exercise.

Proposition: Let \(\mathcal C\) be a differential graded catgeory. Let \(\alpha \colon [n] \rightarrow [m]\) be a simplicial operator, then we have that

\[(\{X_i\}_{i\in [m]},\{f_I\})\mapsto (\{X_{\alpha(j)}\}_{j\in [n]},\{g_J\})\]

where

\[g_J = \begin{cases} f_{\alpha(J)} & \text{ if } \alpha_{|J} \text{ is injective} \\ \mathsf{id}_{X_{i}} & \text{ if } J = \{j_0 < j_1\} \text{ and } \alpha(j_0)=i=\alpha(j_1) \\ 0 & \text{ otherwise.} \end{cases}\]

Hint for the proof: The proof amounts to showing that

\[\partial g_J= \sum_{a=1}^{k-1}(-1)^{a}(g_{i_0 > i_1 > \cdots > i_a}\circ g_{i_a > \cdots > i_k}-g_{I\setminus \{i_a\}})\]

so condsider three separate cases

consider the case where \(J\) is such that \(\alpha\) is injective whenr restricted to it,
the case where \(J=\{j_0 > j_1\}\) and it is mapped to the same value \(i\),
the case where neither of the above apply, and split this one further into three cases corresponding to the position of the elements of \(J\) which have the same image under \(\alpha\).

With the result above we have then managed to define a simplicial set \(\mathsf{N}_{\bullet}^{\mathsf{dg}}(\mathcal C)\) for every differential graded category \(\mathcal C\). The next theorem shows that this is in fact an \(\infty\)-category.

Theorem: Let \(\mathcal C\) be a differential graded category. Then the differential graded nerve

\[\mathsf{N}_{\bullet}^{\mathsf{dg}}(\mathcal C)\colon \Delta^{\mathsf{op}} \rightarrow \mathsf{Set}\]

is an \(\infty\)-category i.e. it is a weak Kan complex.

Before delving into the proof, it is quite helpful to first establish the following observation.

Observation: Let \(K_\bullet\) be a simplicial set and let \(\mathcal C\) be a differential graded category. To give a map of simplicial sets \(f\colon K_\bullet \rightarrow \mathsf{N}_{\bullet}^{\mathsf{dg}}(\mathcal C)\), one must supply the following data:

For each vertex \(x\in K_\bullet\), an object \(f(x)\) of the differential graded category.
For every \(\sigma\in K_k\), a \(k\)-cell in \(K_\bullet\), such that \(\sigma_0 = x\) is the initial vertex, and \(\sigma_k=y\) is the final vertex, a \((k-1)\)-chain

\[f(\sigma)\in \mathsf{Hom}_{\mathcal C}(f(x),f(y))_{k-1}.\]

Moreover, this data must satisfy the following conditions:

If \(e\) is a degenerate edge of \(K_\bullet\) connecting a vertex \(x\) to itself, then \(f(e)\) is the identity morphism \(\mathsf{id}_{f(x)}\in \mathsf{Hom}_{\mathcal C}(X,X)_0.\)
If \(\sigma\) is a degenerate simplex of \(K_\bullet\) having dimension \(\geq 2\), then \(f(\sigma)=0.\)
Let \(k>0\) and let \(\sigma\colon \Delta^k \rightarrow K_\bullet\) be a \(k\)-simplex of \(K_\bullet\). For \(0<b<k\), let \(\sigma_{\leq b} \Delta^b \hookrightarrow K_\bullet\) denote the composition of \(\sigma\) with the inclusion map \(\Delta^b \hookrightarrow \Delta^k\) (which is the identity on vertices), and let \(\sigma_{\geq}\colon \Delta^{k-b}\hookrightarrow K_\bullet\) denote the composition of \(\sigma\) with the map \(\Delta^{k-b}\hookrightarrow \Delta^k\) given on vetices by \(i\mapsto i+b\). Then we have

\[\partial f(\sigma) = \sum_{b=1}^{k-1}(-1)^{k-b}(f(\sigma_{\geq b})\circ f(\sigma_{\leq b})-f(d_{b}\sigma)).\]

Proof: To be announced…