ScalingStacks

The notion of a $d$-category was introduced by Lurie in [Lur09, 2.3.4], as a strict model for what we call an essentially $d$-category; An $\infty$-category all of whose mapping spaces are homotopically $(d-1)$-truncated. With any $\infty$-category $\mathcal{C}$, Lurie associates a $d$-category $h_{d}\mathcal{C}$, which we call the $d$-homotopy category of $\mathcal{C}$. While this $d$-category is shown to be universal in the 1-categorical (simplicially enriched) sense among $d$-categories that $\mathcal{C}$ is mapped to [Lur09, 2.3.4.12], the question of how does $h_{d}\mathcal{C}$ relate to $\mathcal{C}$ as an $\infty$-category, is left unaddressed. The goal of this note is to fill this gap and to give an analogous treatment for operads.

In section 2, we begin by showing that the right mapping spaces of $h_{d}\mathcal{C}$ are given, upto isomorphism, by applying $h_{d-1}$ to the right mapping spaces of $\mathcal{C}$ (2.13). This is the main technical result of this note, the proof of which goes through the comparison with the “middle mapping spaces”. From this we deduce that $h_{d}\mathcal{C}$ is obtained from $\mathcal{C}$ by $(d-1)$-truncation of the mapping spaces. More precisely, we show that $h_{d}$ can be promoted to a functor of $\infty$-categories, which is left adjoint to the inclusion of the full subcategory spanned by essentially $d$-categories into $\mathbf{Cat}_{\infty}$. And furthermore, that the unit map of this adjunction is essentially surjective and is given on mapping spaces by the $(d-1)$-truncation map (2.15).

In section 3, we develop a parallel theory for operads. We call an $\infty$-operad an essentially $d$-operad if all of its multi-mapping spaces are $(d-1)$-truncated. We begin by defining a notion of a $d$-operad (3.4) that relates to essentially $d$-operads in the same way that $d$-categories relate to essentially $d$-categories. We then define the $d$-homotopy operad functor (3.6), again by analogy with (and by means of) the $d$-homotopy category functor. This is achieved by analyzing the behavior of the $d$-homotopy category functor on inner and coCartesian fibrations (3.3). Finally, we bootstrap the results of section 2, to obtain analogues results for (essentially) $d$-operads (3.12) and some corollaries.

This work grew out of a project whose goal is to generalize the classical Eckmann-Hilton argument to the $\infty$-categorical setting. This application, which motivated the general theory we present here, will appear elsewhere.

We work in the setting of $\infty$-categories (a.k.a. quasi-categories) and $\infty$-operads, relying heavily on the results of [Lur09] and [Lur]. Since we have numerous references to these two foundational works, references to [Lur09] are abbreviated as T.? and those to [Lur] as A.?. As a rule, we follow the notation of [Lur09] and [Lur] whenever possible. However, we supplement this notation and deviate from it in several cases in which we believe this enhances readability:

1. 1.

   We abuse notation by identifying an ordinary category $\mathcal{C}$ with its nerve $N\left(\mathcal{C}\right)$.

2. 2.

   We abbreviate the data of an $\infty$-operad $p\colon\mathcal{O}^{\otimes}\to\mathbf{Fin}_{*}$ by $\mathcal{O}$ and reserve the notation $\mathcal{O}^{\otimes}$ for the $\infty$-category that is the source of $p$. Similarly, given two $\infty$-operads $\mathcal{O}$ and $\mathcal{U}$, we write $f\colon\mathcal{O}\to\mathcal{U}$ for a map of $\infty$-operads from $\mathcal{O}$ to $\mathcal{U}$. The underlying $\infty$-category of $\mathcal{O}$, which in [Lur] is denoted by $\mathcal{O}_{\left\langle 1\right\rangle}^{\otimes}$, is here denoted by $\underline{\mathcal{O}}$.

3. 3.

   Given two $\infty$-operads $\mathcal{O}$ and $\mathcal{U}$, we denote by $\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{U}\right)$ the $\infty$-operad $\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{U}\right)^{\otimes}\to\mathbf{Fin}_{*}$ from Example A.3.2.4.4. This is the internal mapping object induced from the closed symmetric monoidal structure on $\mathbf{Op}_{\infty}$ (see A.2.2.5.13). The underlying $\infty$-category $\underline{\operatorname{Alg}}_{\mathcal{O}}\left(\mathcal{U}\right)$ is the usual $\infty$-category of $\mathcal{O}$-algebras in $\mathcal{U}$ (which in [Lur] is denoted by $\operatorname{Alg}_{\mathcal{O}}\left(\mathcal{U}\right)$). Moreover, the maximal Kan sub-complex $\underline{\operatorname{Alg}}_{\mathcal{O}}\left(\mathcal{U}\right)^{\simeq}$ is the space of morphisms $\operatorname{Map}_{\mathbf{Op}_{\infty}}\left(\mathcal{O},\mathcal{U}\right)$ from $\mathcal{O}$ to $\mathcal{U}$ as objects of the $\infty$-category $\mathbf{Op}_{\infty}$.