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2 Examples of $\infty $-Categories

In Chapter 1, we introduced the notion of an $\infty $-category: that is, a simplicial set which satisfies the weak Kan extension condition (Definition 1.3.0.1). The theory of $\infty $-categories can be understood as a synthesis of classical category theory and algebraic topology. This perspective is supported by the two main examples of $\infty $-categories that we have encountered so far:

  • Every ordinary category $\operatorname{\mathcal{C}}$ can be regarded as an $\infty $-category, by identifying $\operatorname{\mathcal{C}}$ with the the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ of Construction 1.2.1.1.

  • Every Kan complex is an $\infty $-category. In particular, for every topological space $X$, the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is an $\infty $-category.

Beware that, individually, both of these examples are rather special. An $\infty $-category $\operatorname{\mathcal{C}}$ can be regarded as a mathematical structure which encodes information not only about objects and morphisms (given by the vertices and edges of $\operatorname{\mathcal{C}}$, respectively), but also about homotopies between morphisms (Definition 1.3.3.1). When $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, the notion of homotopy is trivial: two morphisms in $\operatorname{\mathcal{C}}$ (having the same source and target) are homotopic if and only if they are identical. On the other hand, if $\operatorname{\mathcal{C}}$ is a Kan complex, then every morphism in $\operatorname{\mathcal{C}}$ is invertible up to homotopy (Proposition 1.3.6.12); from a category-theoretic perspective, this is a very restrictive condition.

Our goal in this chapter is to supply a larger class of examples of $\infty $-categories, which are more representative of the subject as a whole. To this end, we introduce three variants of the nerve construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which can be used to produce $\infty $-categories out of other (possibly more familiar) mathematical structures. To describe these constructions in a uniform way, it will be convenient to employ the language of enriched category theory, which we review in §2.1. Let $\operatorname{\mathcal{A}}$ be a monoidal category: that is, a category equipped with a tensor product operation $\otimes : \operatorname{\mathcal{A}}\times \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{A}}$, which is unital and associative up to (specified) isomorphisms (see Definition 2.1.2.10). A $\operatorname{\mathcal{A}}$-enriched category is a mathematical structure $\operatorname{\mathcal{C}}$ consisting of the following data (see Definition 2.1.7.1):

  • A collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$ whose elements we refer to as objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a mapping object $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \in \operatorname{\mathcal{A}}$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a composition law

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z), \]

    which we require to be unital and associative.

Taking our Examples , , and , we consider three examples of this paradigm:

  • Let $\operatorname{\mathcal{A}}= \operatorname{Set_{\Delta }}$ be the category of simplicial sets, equipped with the monoidal structure given by Cartesian product. In this case, we refer to an $\operatorname{\mathcal{A}}$-enriched category as a simplicial category (Definition 2.4.1.1). In §2.4, we associate to each simplicial category $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which we refer to as the homotopy coherent nerve of $\operatorname{\mathcal{C}}$ (Definition 2.4.3.5). Moreover, we show that if each of the simplicial sets $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ is a Kan complex, then the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1).

  • Let $\operatorname{\mathcal{A}}= \operatorname{Ch}(\operatorname{\mathbf{Z}})$ be the category of chain complexes of abelian groups, equipped with the monoidal structure given by tensor product of chain complexes. In this case, we refer to an $\operatorname{\mathcal{A}}$-enriched category as a differential graded category (Definition 2.5.2.1). In §2.5, we associate to each differential graded category $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$, which we refer to as the differential graded nerve of $\operatorname{\mathcal{C}}$ (Definition 2.5.3.7), and show that $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is always $\infty $-category (Theorem 2.5.3.10).

  • Let $\operatorname{\mathcal{A}}= \operatorname{Cat}$ be the category of (small) categories, equipped with the monoidal structure given by the Cartesian product. In this case, we refer to an $\operatorname{\mathcal{A}}$-enriched category as a strict $2$-category (Definition 2.2.0.1). This is a special case of the more general notion of $2$-category (or bicategory, in the terminology of Bénabou), which we review §2.2. In §2.3, we will associate to each $2$-category $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, which we refer to as the Duskin nerve of $\operatorname{\mathcal{C}}$ (Construction 2.3.1.1). Moreover, we show that if each of the categories $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ is a groupoid, then $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.3.2.1).

Simplicial categories, differential graded categories, and $2$-categories are ubiquitous in algebraic topology, homological algebra, and category theory, respectively. Consequently, the constructions of this section furnish a rich supply of examples of $\infty $-categories.

Structure

  • Section 2.1: Monoidal Categories
    • Subsection 2.1.1: Nonunital Monoidal Categories
    • Subsection 2.1.2: Monoidal Categories
    • Subsection 2.1.3: Examples of Monoidal Categories
    • Subsection 2.1.4: Nonunital Monoidal Functors
    • Subsection 2.1.5: Lax Monoidal Functors
    • Subsection 2.1.6: Monoidal Functors
    • Subsection 2.1.7: Enriched Category Theory
  • Section 2.2: The Theory of $2$-Categories
    • Subsection 2.2.1: $2$-Categories
    • Subsection 2.2.2: Examples of $2$-Categories
    • Subsection 2.2.3: Opposite and Conjugate $2$-Categories
    • Subsection 2.2.4: Functors of $2$-Categories
    • Subsection 2.2.5: The Category of $2$-Categories
    • Subsection 2.2.6: Isomorphisms of $2$-Categories
    • Subsection 2.2.7: Strictly Unitary $2$-Categories
  • Section 2.3: The Duskin Nerve of a $2$-Category
    • Subsection 2.3.1: The Duskin Nerve
    • Subsection 2.3.2: From $2$-Categories to $\infty $-Categories
    • Subsection 2.3.3: Thin $2$-Simplices of a Duskin Nerve
    • Subsection 2.3.4: Recovering a $2$-Category from its Duskin Nerve
    • Subsection 2.3.5: Twisted Arrows and the Nerve of $\operatorname{Corr}(\operatorname{\mathcal{C}})^{\operatorname{c}}$
    • Subsection 2.3.6: The Duskin Nerve of a Strict $2$-Category
  • Section 2.4: Simplicial Categories
    • Subsection 2.4.1: Simplicial Enrichment
    • Subsection 2.4.2: Examples of Simplicial Categories
    • Subsection 2.4.3: The Homotopy Coherent Nerve
    • Subsection 2.4.4: The Path Category of a Simplicial Set
    • Subsection 2.4.5: From Simplicial Categories to $\infty $-Categories
    • Subsection 2.4.6: The Homotopy Category of a Simplicial Category
    • Subsection 2.4.7: Example: Braid Monoids
  • Section 2.5: Differential Graded Categories
    • Subsection 2.5.1: Generalities on Chain Complexes
    • Subsection 2.5.2: Differential Graded Categories
    • Subsection 2.5.3: The Differential Graded Nerve
    • Subsection 2.5.4: The Homotopy Category of a Differential Graded Category
    • Subsection 2.5.5: Digression: The Homology of Simplicial Sets
    • Subsection 2.5.6: The Dold-Kan Correspondence
    • Subsection 2.5.7: The Shuffle Product
    • Subsection 2.5.8: The Alexander-Whitney Construction
    • Subsection 2.5.9: Comparison with the Homotopy Coherent Nerve