In a classical category, the collection of morphisms between two objects forms a set. In contrast, we have seen in Definition 1.4.17 that for \(\infty \)-categories the morphisms between objects instead form an anima. As a first approximation, we may thus think of an \(\infty \)-category as a ‘category enriched in animae’. While a formal implementation of enrichments in animae would already require a well-developed theory of \(\infty \)-categories1 , we may sidestep this problem by working instead with enrichment in the 1-category of spaces: every topological space \(X\) admits an underlying anima \(\Pi _{\infty }(X)\), and the functor \(\Pi _{\infty }(-)\colon \Top \to \An \) preserves finite products. In this case, classical enriched category theory provides a well-defined notion of a topologically enriched category (see Definition 24.3.1), of which classical topology provides plenty of examples. It was shown by Bergner (2007) that the theory of topologically enriched categories provides a reasonable model for the theory of \(\infty \)-categories.
The goal of this section is to explain how to assign an \(\infty \)-category \(\Ntop (D)\) to every topologically enriched category \(D\), called the homotopy coherent nerve of \(D\). The standard way to make this construction precise is via quasicategories, cf. Example 1.6.14. In this section, we provide an alternative approach to homotopy coherent nerves that avoids explicit models by quasicategories, fitting it in the model-agnostic approach to higher category theory taken in this book.
Definition 24.3.1. A topologically enriched category is a classical 1-category \(D\) together with a chosen topology on each of the hom sets \(\Hom _D(X,Y)\) for \(X,Y \in D\), satisfying the property that the composition maps \(\Hom _D(Y,Z) \times \Hom _D(X,Y) \to \Hom _D(X,Z)\) are continuous with respect to these topologies. We denote by \(\Homtop _D(X,Y) \in \Top \) the hom set in \(C\) equipped with the chosen topology.
Construction 24.3.2 (Homotopy coherent nerve). Consider a topologically enriched category \(D\). We will define an \(\infty \)-category \(\Ntop (D)\), called the homotopy coherent nerve of \(D\).
For every \([n] \in \simp \catop \), consider the (classical) 1-category \(D_n\) defined as follows:
- The objects of \(D_n\) are the objects of \(D\);
- The morphisms \(X \to Y\) of \(D_n\) are the continuous maps \(\abs {\Delta ^n} \to \Homtop _D(X,Y)\);
- The identity in \(D_n\) is the constant map \(\abs {\Delta ^n} \to \Homtop _D(X,X)\) with value \(\id _X\);
- The composition in \(D_n\) of morphisms \(f\colon \abs {\Delta ^n} \to \Homtop _D(X,Y)\) and \(g\colon \abs {\Delta ^n} \to \Homtop _D(Y,Z)\) is the composite \[ \abs {\Delta ^n} \xrightarrow {\Delta } \abs {\Delta ^n} \times \abs {\Delta ^n} \xrightarrow {g \times f} \Homtop _D(Y,Z) \times \Homtop _D(X,Y) \xrightarrow {\circ } \Homtop _D(X,Z). \]
Since \(\abs {\Delta ^n}\) is functorial in \([n] \in \simp \) (see Example 1.6.8) we see that the assignment \([n] \mapsto D_n\) determines a functor \[ D_{\bullet }\colon \simp \catop \to \Cat _1 \hookrightarrow \Cat _{\infty }. \] We then define the \(\infty \)-category \(\Ntop (D)\) as the colimit of this functor: \[ \Ntop (D) \quad := \quad \colim (D_{\bullet }\colon \simp \catop \to \Cat _{\infty }). \]
Note that \(D_0\) is the underlying category of \(D\). Since it admits a canonical functor to \(\Ntop (D)\), we see that every object of \(D\) gives rise to an object of \(\Ntop (D)\), and in fact all objects are of this form.
Remark 24.3.3. The fact that we refer to \(\Ntop (D)\) as the ‘homotopy coherent nerve’ is somewhat abusive: this terminology is often used instead for the simplicial set \(N^{\Delta }(\bC )\) associated to a simplicially enriched category, cf. Example 1.6.14. The two constructions are closely related: given a topologically enriched category \(D\), we may obtain a simplicially enriched category \(\bC \) by taking \(\Hom ^{\Delta }_{\bC }(X,Y) := \Sing (\Homtop _{D}(X,Y))\), i.e. we apply the singular simplicial complex construction from Example 1.6.8 to each of the hom spaces of \(D\). Then the underlying \(\infty \)-category of the quasicategory \(N^{\Delta }(\bC )\) is equivalent to \(\Ntop (D)\). The proof of this fact is quite technical: it relies on the fact that there is a model structure (due to Bergner (2007)) on the category of (small) simplicially enriched categories whose localization at the weak equivalences is \(\Cat _{\infty }\). A reference for this claim in the literature may be found in Gepner and Meier (2023), Corollary B.4.
Proposition 24.3.4. For objects \(X\) and \(Y\) of \(D\), the hom anima between \(X\) and \(Y\) in \(\Ntop (D)\) is equivalent to the underlying anima of the topological space \(\Homtop _D(X,Y)\): \[ \Hom _{\Ntop (D)}(X,Y) \simeq \Pi _{\infty }(\Homtop _D(X,Y)). \]
Proof. A proof is given in the standalone supplementary material. □
Exercises
Exercise 24.1 (Composition in a Segal anima). Let \(X\) be a Segal anima. Show that the degeneracy map \(s_0\colon X_0\to X_1\) defines identity morphisms and that the composition map \[ X_1\times _{X_0}X_1\simeq X_2\xrightarrow {d_1}X_1 \] is unital and associative.
Exercise 24.2 (Nerves and completeness).
- (1)
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Let \(C\) be an \(\infty \)-category. Show that its \(\infty \)-categorical nerve \(N(C)_n=\Map ([n],C)\) is a complete Segal anima.
- (2)
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Let \(C\) be a classical 1-category. Regard its nerve \(N(C)\in \sSet \) from Example 1.6.11 as a simplicial anima. Show that it is a Segal anima. Is it complete? Determine precisely when this holds.
Exercise 24.3 (Discrete topological enrichment). Let \(C\) be a classical 1-category and regard it as a topologically enriched category \(C^{\mathrm {disc}}\) by giving each hom set the discrete topology. Show that the canonical functor \[ C\longrightarrow \Ntop (C^{\mathrm {disc}}) \] is an equivalence of \(\infty \)-categories.
Notes
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