## 1 Introduction

It has long been understood in homotopy theory that the homotopy category is only a crude invariant of a much richer homotopy-theoretic structure. The problem of finding a suitable formalism for this additional structure, one that encodes homotopy-theoretic extensions of ordinary categorical notions, led to several foundational approaches, each with its own distinctive advantages and special characteristics. The theory of $\infty $-categories (or quasi–categories) [Reference Joyal20, Reference Lurie21, Reference Lurie22, Reference Cisinski4] is one of several successful approaches to develop useful foundations for the study of homotopy theories and has led to groundbreaking new perspectives and results in the field.

Even though passing to the homotopy category certainly neglects homotopy-theoretic information, the general problem of understanding how much information this process retains still poses interesting questions in specific contexts. This has inspired many important developments, for example, in the context of rigidity theorems for homotopy theories [Reference Franke12, Reference Schwede35, Reference Roitzheim33, Reference Patchkoria31], derived/homotopical Morita and tilting theory (see [Reference Schwede36] for a nice survey) or in connection with *K*-theory regarded as an invariant of homotopy theories [Reference Dugger and Shipley9, Reference Neeman28, Reference Schlichting34, Reference Maltsiniotis24].

The theory of derivators, first introduced and developed by Grothendieck [Reference Grothendieck16], Heller [Reference Heller19] and Franke [Reference Franke12], is a different foundational approach based on the idea of considering the homotopy categories of all diagram categories as a remedy to the defects of the homotopy category (see also [Reference Groth18]). By supplementing the homotopy category with the network of all these (homotopy) categories, it is possible to encode the collection of homotopy (co)limit functors and general homotopy Kan extensions as an enhancement of the homotopy category. This approach provides a different, lower (= 2–) categorical formalism for expressing homotopy-theoretic universal properties (see [Reference Cisinski5] and [Reference Tabuada37, Reference Cisinski and Tabuada7] for some interesting applications). Moreover, Maltsiniotis [Reference Maltsiniotis24] defined *K*-theory in the context of derivators with a view towards partially recovering Waldhausen *K*-theory from the derivator. The *K*-theory of derivators and its comparison with Waldhausen *K*-theory has been studied extensively in [Reference Cisinski and Neeman6, Reference Garkusha14, Reference Garkusha15, Reference Maltsiniotis24, Reference Muro25, Reference Muro and Raptis26, Reference Muro and Raptis27]. In the context of the theory of derivators, the question about the information retained by the homotopy category is then upgraded to the analogous question for the derivator. The theory of derivators still does not provide in general faithful representations of homotopy theories; however, it is possible in certain cases to recover in a non-canonical way the homotopy theory from the derivator (see [Reference Renaudin32]).

The purpose of this paper is to extend these ideas on the comparison between homotopy theories and homotopy categories or derivators to *n*-categories (= $(n,1)$-categories), where the ordinary homotopy category is now replaced by the homotopy *n*-category of an $\infty $-category. More specifically:

(a)

*Higher homotopy categories.*Using the definition of the higher homotopy categories by Lurie [Reference Lurie22], we consider the tower of homotopy*n*-categories $\{\mathrm {h}_n \mathscr {C}\}_{n \geq 1}$ associated to an $\infty $-category $\mathscr {C}$, together with the canonical (localization) functors $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n\mathscr {C}$, and we analyse the properties of $\mathrm {h}_n\mathscr {C}$ inherited from $\mathscr {C}$. (Sections 2–3, 6.1–6.2)(b)

*Higher derivators.*We introduce a higher categorical notion of a derivator that takes values in*n*-categories. Then we develop the basic theory of higher derivators with a special emphasis on the examples that arise from $\infty $-categories. (Section 4)(c)

*K-theory of higher derivators.*We extend the definition of derivator*K*-theory by Maltsiniotis [Reference Maltsiniotis24] and Garkusha [Reference Garkusha14, Reference Garkusha15] to*n*-derivators and study the comparison map from Waldhausen*K*-theory. Our main result shows that the comparison map is $(n+1)$-connected (Theorem 5.5). Moreover, following [Reference Muro and Raptis26], we prove that this comparison map has a universal property (Theorem 5.13).(d)

*K-theory of homotopy n-categories.*In analogy with the*K*-theory of triangulated categories [Reference Neeman28], we introduce*K*-theory for*n*-categories equipped with distinguished squares. In the case of a homotopy*n*-category, we study the comparison map from Waldhausen*K*-theory and prove that it is*n*-connected (Theorem 6.5).

(a) **Higher homotopy categories.** Every $\infty $-category $\mathscr {C}$ has an associated homotopy *n*-category $\mathrm {h}_n \mathscr {C}$ and a canonical functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$. The construction of the homotopy *n*-category and its properties are studied in [Reference Lurie22]. We will review this construction and its properties in Section 2. Intuitively, for $n \geq 1$, $\mathrm {h}_n \mathscr {C}$ is an $\infty $-category with the same objects as $\mathscr {C}$ and whose mapping spaces are the appropriate Postnikov truncations of the mapping spaces in $\mathscr {C}$. For $n=1$, the homotopy category $\mathrm {h}_1 \mathscr {C}$ is the ordinary homotopy category of $\mathscr {C}$. The collection of homotopy *n*-categories defines a tower of $\infty $-categories

that approximates $\mathscr {C}$. One of the defects of the homotopy category $\mathrm {h}_1 \mathscr {C}$, which is essentially what the theory of derivators tries to rectify, is that it does not in general inherit (co)limits from $\mathscr {C}$. As a general rule, if $\mathscr {C}$ admits (co)limits, then $\mathrm {h}_1 \mathscr {C}$ admits only weak (co)limits – which may or may not be induced from $\mathscr {C}$. (We recall that a weak colimit of a diagram is a cocone on the diagram that admits a morphism to every other such cocone, but this morphism may not be unique in general.) Do the higher homotopy categories have *better* inheritance properties for (co)limits, and in what sense? This question is closely related to the problem of understanding how much information $\mathrm {h}_n \mathscr {C}$ retains from $\mathscr {C}$. We introduce a higher categorical version of weak (co)limit in order to address this question. In analogy with ordinary weak (co)limits, a higher weak colimit is a cocone for which the mapping spaces to other cocones are highly connected but not necessarily contractible. The relative strength of the weak colimit is measured by how highly connected these mapping spaces are; the connectivity of these mapping spaces determines the *order* of the weak colimit. The properties of higher weak (co)limits will be discussed in Section 3. For any simplicial set *K*, there is a canonical functor

which is usually not an equivalence. The properties of this functor are relevant for understanding the interaction of *K*-colimits in $\mathscr {C}$ and in $\mathrm {h}_n \mathscr {C}$. One of our conclusions (Corollary 3.17) is the following:

Moreover, in connection with higher weak colimits in $\mathrm {h}_n \mathscr {C}$, we also conclude (Corollary 3.22):

These properties of homotopy *n*-categories $\mathrm {h}_n\mathscr {C}$ single out a class of *n*-categories for each $n \geq 1$, called *(finitely) weakly cocomplete n-categories* (Definition 3.23). These classes of $\infty $-categories form a sequence of refinements between ordinary categories with (finite) coproducts and weak pushouts and (finitely) cocomplete $\infty $-categories. We explore further the properties of these *n*-categories in connection with adjoint functor theorems and (higher) Brown representability in joint work with H. K. Nguyen and C. Schrade [Reference Nguyen, Raptis and Schrade30] (building on and extending our previous joint work in [Reference Nguyen, Raptis and Schrade29]).

(b) **Higher derivators.** The main example of a (pre)derivator is given by the 2-functor that sends a small category *I* to the homotopy category $\mathrm {h}_1(\mathscr {C}^{N(I)})$, for suitable choices of an $\infty $-category $\mathscr {C}$. Using homotopy *n*-categories instead, we may consider more generally the example of the enriched functor that sends a simplicial set *K* to the homotopy *n*-category $\mathrm {h}_n(\mathscr {C}^K)$. Following the axiomatic definition of ordinary derivators, we introduce a definition of an *n*-derivator, which is meant to encapsulate the main abstract properties of this example. The basic definitions and properties of (left, right, pointed, stable) *n*-(pre)derivators, $1 \leq n \leq \infty $, will be discussed in Section 4. For any $\infty $-category $\mathscr {C}$, there is an associated *n*-prederivator

where $\mathsf {Dia}$ denotes a category of diagram shapes and $\mathsf {Cat_{n}}$ is the $\infty $-category of *n*-categories. These assemble to define a tower of $\infty $-prederivators

that approximates $\mathbb {D}^{(\infty )}_{\mathscr {C}} \colon K \mapsto \mathscr {C}^K$. The *n*-prederivator $\mathbb {D}_{\mathscr {C}}^{(n)}$ is an *n*-derivator if certain homotopy Kan extensions exist in $\mathscr {C}$ and the corresponding base change transformations satisfy the Beck–Chevalley condition. For any fixed $n \in \mathbb {Z}_{\geq 1} \cup \{\infty \}$, we prove the following fact, which can be used to obtain many examples of *n*-derivators (Proposition 4.17 and Theorem 4.18):

The motivation for higher derivators is to bridge the gap between $\infty $-categories and derivators by introducing a hierarchy of intermediate notions, a different one for each categorical level, starting with ordinary derivators. For any fixed $1 \leq n < \infty $, the theory of *n*-derivators is still not a faithful representation of homotopy theories; it dwells in an $(n+1)$-categorical context in the same way the classical theory of derivators is restricted to a 2-categorical context. In this respect, our approach using *n*-derivators remains close to the original idea of a derivator and differs from other recent perspectives on (pre)derivators in which (pre)derivators are reconsidered and revised into a model for the theory of $\infty $-categories [Reference Fuentes-Keuthan, Kedziorek and Rovelli13, Reference Arlin2, Reference Muro and Raptis26]. We will address the problem of comparing suitable nice classes of $\infty $-categories with *n*-derivators in future joint work with D.–C. Cisinski.

(c) **K-theory of higher derivators.** *K*-theory for (pointed, right) derivators was introduced by Maltsiniotis [Reference Maltsiniotis24] and Garkusha [Reference Garkusha14, Reference Garkusha15]. The basic feature of a derivator that allows this definition of *K*-theory is that there is a natural notion of cocartesian square for a derivator. The motivation for introducing this *K*-theory is connected with the problem of recovering Waldhausen *K*-theory from the derivator. Maltsiniotis [Reference Maltsiniotis24] conjectured that derivator *K*-theory of stable derivators satisfies *additivity* and *localization* and that it *agrees* with Quillen *K*-theory for exact categories. Cisinski and Neeman [Reference Cisinski and Neeman6] proved additivity for the derivator *K*-theory of stable derivators, and Coley [Reference Coley8] has recently extended this result to the unstable context. In joint work with Muro [Reference Muro and Raptis27], we proved that localization and agreement with Quillen *K*-theory cannot both hold. On the other hand, Muro [Reference Muro25] proved that agreement with Waldhausen *K*-theory holds for $K_0$ and $K_1$ (see also [Reference Maltsiniotis24, Section 6]). Moreover, Garkusha [Reference Garkusha15] obtained further positive results in the case of abelian categories. In Section 5, after a short review of the *K*-theory of $\infty $-categories, we will define derivator *K*-theory for general (pointed, right) $\infty $-derivators. For any pointed $\infty $-category $\mathscr {C}$ with finite colimits, there is a comparison map to derivator *K*-theory,

and these comparison maps assemble to give a tower of derivator *K*-theories and comparison maps

that approximates $K(\mathscr {C})$. Our main result on the comparison map $\mu _n$ is the following connectivity estimate (Theorem 5.5):

We believe that this connectivity estimate is best possible in general (Remarks 5.7 and 5.8). Following the ideas of [Reference Muro and Raptis26], we also consider the Waldhausen *K*-theory $K^{W, \operatorname {Ob}}(\mathbb {D})$ (and $K^W(\mathbb {D})$) of a general (pointed, right) $\infty $-derivator $\mathbb {D}$. This *K*-theory always agrees with the usual *K*-theory (Proposition 5.10); the proof is based on a version of the $\textsf {s}_{\bullet }$-construction in the $\infty $-categorical context (Proposition 5.1). However, Waldhausen *K*-theory of derivators is not invariant under equivalences of derivators in general. Similarly to the case of ordinary derivators treated in [Reference Muro and Raptis26], we prove that the comparison map to derivator *K*-theory,

identifies derivator *K*-theory in terms of a universal property (Theorem 5.13):

(d) **K-theory of homotopy** *n* **-categories**. The motivation for introducing *K*-theory for homotopy *n*-categories is to identify the part of Waldhausen *K*-theory that may be reconstructed from the homotopy *n*-category. As a basic instance of this phenomenon, we recall that $K_0(\mathscr {C})$ can be recovered from the triangulated category $\mathrm {h}_1 \mathscr {C}$ for any stable $\infty $-category $\mathscr {C}$. The main feature of the homotopy *n*-category that is needed for our definition of *K*-theory is the collection of higher weak pushouts that come from pushouts in $\mathscr {C}$. We will revisit the properties of homotopy *n*-categories in Subsections 6.1–6.2 and discuss possible axiomatisations of these properties. We will then define *K*-theory for pointed *n*-categories with distinguished squares – this is a higher categorical, but much more elementary, version of Neeman’s *K*-theory of categories with squares [Reference Neeman28]. For a pointed $\infty $-category $\mathscr {C}$ with finite colimits, we consider the *K*-theory $K(\mathrm {h}_n\mathscr {C}, \mathrm {can})$ associated to $\mathrm {h}_n\mathscr {C}$ with the canonical structure of higher weak pushouts as distinguished squares. For every $n \geq 1$, there is a comparison map

and these assemble to define a tower of *K*-theories and comparison maps

that approximates $K(\mathscr {C})$. Our main result in Section 6 on the comparison map $\rho _n$ is the following connectivity estimate (Theorem 6.5):

This connectivity estimate is best possible in general (Remark 6.10). Let $P_n X$ denote the Postnikov *n*-truncation of a topological space *X*: that is, the homotopy groups of $P_n X$ vanish in degrees $> n$, and the canonical map $X \to P_n X$ is $(n+1)$-connected. Based on the connectivity estimate above, we conclude (Corollary 6.11):

This confirms a recent conjecture of Antieau [Reference Antieau1, Conjecture 8.35] in the case of connective *K*-theory.

## 2 Higher homotopy categories

### 2.1 *n*-categories

We recall the definition and basic properties of *n*-categories following [Reference Lurie22, 2.3.4]. Let $\mathscr {C}$ be an $\infty $-category, and let $n \geq -1$ be an integer. $\mathscr {C}$ is an *n*-*category* if it satisfies the following conditions:

(1) Given a pair of maps $f,f': \Delta ^n \to \mathscr {C}$, if

*f*and $f'$ are homotopic relative to $\partial \Delta ^n$, then $f = f'$. (We recall that the notion of homotopy employed here means that the two maps are homotopic via equivalences in $\mathscr {C}$.)(2) Given a pair of maps $f, f': \Delta ^m \to \mathscr {C}$, where $m> n$, if $f_{|\partial \Delta ^m} = f^{\prime }_{|\partial \Delta ^m}$, then $f = f'$.

These conditions say that $\mathscr {C}$ has no morphisms in degrees $> n$ and any two morphisms in degree *n* agree if they are equivalent. The conditions can be equivalently expressed as follows: $\mathscr {C}$ is an *n*-category, $n \geq 1$, if for every diagram

where $m> n$ and $0 < i < m$, there exists a *unique* dotted arrow that makes the diagram commutative [Reference Lurie22, Proposition 2.3.4.9]. Using an inductive argument (see [Reference Lurie22, Proposition 2.3.4.7]), it can also be shown that the conditions (1) and (2) together are equivalent to:

(3) Given a simplicial set

*K*and maps $f,f': K \to \mathscr {C}$ such that $f_{| \mathrm {sk}_n(K)}$ and $f^{\prime }_{|\mathrm {sk}_n(K)}$ are homotopic relative to $\mathrm {sk}_{n-1}(K)$, then $f = f'$.

An important immediate consequence of (3) is that for every *n*-category $\mathscr {C}$, the $\infty $-category $\mathrm {Fun}(K, \mathscr {C})$ is again an *n*-category for any simplicial set *K* [Reference Lurie22, Corollary 2.3.4.8].

Example 2.1. The only $(-1)$-categories up to isomorphism are $\varnothing $ and $\Delta ^0$. An $\infty $-category is a $0$-category if and only if it is isomorphic to (the nerve of) a poset. $1$-categories are up to isomorphism (nerves of) ordinary categories. See [Reference Lurie22, Examples 2.3.4.2–2.3.4.3, Proposition 2.3.4.5].

The property of being an *n*-category is not invariant under equivalences of $\infty $-categories. The following proposition gives a characterization of the invariant property that an $\infty $-category is equivalent to an *n*-category. We recall that an $\infty $-groupoid (= Kan complex) *X* is *n-truncated*, where $n \geq -1$, if *X* has vanishing homotopy groups in degrees $> n$. We say that *X* is *(-2)-truncated* if *X* is contractible.

Proposition 2.2. Let $\mathscr {C}$ be an $\infty $-category, and let $n \geq -1$ be an integer. Then $\mathscr {C}$ is equivalent to an *n*-category if and only if $\mathrm {Map}_{\mathscr {C}}(x,y)$ is $(n-1)$-truncated for every pair of objects $x,y \in \mathscr {C}$.

Proof. See [Reference Lurie22, Proposition 2.3.4.18].

### 2.2 Homotopy *n*-categories

Let $\mathscr {C}$ be an $\infty $-category, and let $n \geq 1$ be an integer. We recall from [Reference Lurie22] the construction of the homotopy *n*-category $\mathrm {h}_n \mathscr {C}$ of $\mathscr {C}$. Given a simplicial set *K*, we denote by $[K, \mathscr {C}]_n$ the set of maps

which extend to $\mathrm {sk}_{n+1}(K)$. Two elements $f,g \in [K, \mathscr {C}]_n$ are called equivalent, denoted $f \sim g$, if the maps $f,g: \mathrm {sk}_n(K) \to \mathscr {C}$ are homotopic relative to $\mathrm {sk}_{n-1}(K)$. The equivalence classes of such maps for $K = \Delta ^m$ define the *m*-simplices of a simplicial set $\mathrm {h}_n \mathscr {C}$: that is,

Clearly an *m*-simplex of $\mathscr {C}$ defines an *m*-simplex in $\mathrm {h}_n \mathscr {C}$, so we have a canonical map $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$. Note that this map is a bijection in simplicial degrees $< n$ and surjective in degrees *n* and $n+1$.

The following proposition summarises some of the basic properties of this construction.

Proposition 2.3. Let $\mathscr {C}$ be an $\infty $-category and $n \geq 1$.

(a) The set of maps $K \to \mathrm {h}_n \mathscr {C}$ is in natural bijection with the set $[K, \mathscr {C} ]_n / \sim $.

(b) $\mathrm {h}_n \mathscr {C}$ is an

*n*-category. In particular, it is an $\infty $-category.(c) $\mathscr {C}$ is an

*n*-category if and only if the map $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ is an isomorphism.(d) Let $\mathscr {D}$ be an

*n*-category. Then the restriction functor along $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$,$$ \begin{align*}\gamma_n^* \colon \mathrm{Fun}(\mathrm{h}_n \mathscr{C}, \mathscr{D}) \to \mathrm{Fun}(\mathscr{C}, \mathscr{D}),\end{align*} $$is an isomorphism.

Proof. See [Reference Lurie22, Proposition 2.3.4.12].

Example 2.4. For $n = 1$, the $1$-category $\mathrm {h}_{1} \mathscr {C}$ is isomorphic to the (nerve of the) usual homotopy category of $\mathscr {C}$.

Remark 2.5. For an $\infty $-category $\mathscr {C}$, the homotopy $0$-category $\mathrm {h}_0 \mathscr {C}$ can be described in the following way. For $x, y \in \mathscr {C}$, we write $x \leq y$ if $\mathrm {Map}_{\mathscr {C}}(x, y)$ is non-empty. This defines a reflexive and transitive relation. We say that two objects *x* and *y* are equivalent if $x \leq y$ and $y \leq x$. Then the relation $\leq $ descends to a partial order on the set of equivalence classes of objects in $\mathscr {C}$. The homotopy $0$-category $\mathrm {h}_0 \mathscr {C}$ is isomorphic to the nerve of this poset. We will often ignore the case $n = 0$ and focus on the homotopy *n*-categories for $n \geq 1$.

Proposition 2.6. Let $\mathscr {C}$ be an $\infty $-category and $n \geq 1$. The functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ is a categorical fibration between $\infty $-categories. In addition, for every lifting problem

where $m \leq n + 1$ (respectively, $m < n$), there is a (unique) filler $\Delta ^m \to \mathscr {C}$ that makes the diagram commutative.

Proof. Clearly, for any object *c* in $\mathscr {C}$ and any equivalence $f \colon c \to c'$ in $\mathrm {h}_n \mathscr {C}$, we may find a lift $\tilde {f} \colon c \to c'$ of *f* in $\mathscr {C}$ (uniquely if $n>1$), which is again an equivalence. Then we need to show that $\gamma _n$ is an inner fibration. Consider a lifting problem

where $0 < i < m$. For $m \leq n$, there is a diagonal filler $\Delta ^m \to \mathscr {C}$ because $\gamma _n$ is a bijection in simplicial degrees $< n$ and surjective on *n*-simplices. For $m> n$, there is a map $v \colon \Delta ^m \to \mathscr {C}$ that extends *u* because $\mathscr {C}$ is an $\infty $-category. We claim that *v* defines a diagonal filler for the diagram. To see this, note that an extension of $\gamma _n u$ along *j* is unique up to homotopy relative to $\Lambda ^m_i$ ($\supseteq \mathrm {sk}_{n-1} \Delta ^m$), since $\mathrm {h}_n \mathscr {C}$ is an $\infty $-category. In particular, the maps $\sigma $ and $\gamma _n v$ are homotopic relative to $\mathrm {sk}_{n-1} \Delta ^m$. Then the result follows because the *n*-category $\mathrm {h}_n \mathscr {C}$ satisfies condition (3) (see Subsection 2.1). Therefore, $\gamma _n$ is an inner fibration, and this completes the proof of the first claim.

The second claim for $m \leq n$ follows again from the fact that $\gamma _n$ is bijective in simplicial degrees $< n$ and surjective in degree *n*. For $m = n+1$, we may find a map $\sigma ' \colon \Delta ^{n+1} \to \mathscr {C}$ such that $\gamma _n \sigma ' = \sigma $, since $\gamma _n$ is surjective on $(n+1)$-simplices. The maps

become equal after postcomposition with $\gamma _n$. By Proposition 2.3(a), this means they are homotopic (in the sense of the Joyal model category) relative to $\mathrm {sk}_{n-1}(\partial \Delta ^{n+1})$. Using that the inclusion $\partial \Delta ^{n+1} \subset \Delta ^{n+1}$ is a cofibration, this homotopy can be extended to a homotopy on $\Delta ^{n+1}$ between $\sigma '$ and a map $v \colon \Delta ^{n+1} \to \mathscr {C}$ such that $v_{| \partial \Delta ^{n+1}} = u$. This new homotopy is still constant on $\mathrm {sk}_{n-1}(\partial \Delta ^{n+1}) = \mathrm {sk}_{n-1}(\Delta ^{n+1})$. Therefore, given that $\sigma '$ and *v* are homotopic relative to $\mathrm {sk}_{n-1}(\Delta ^{n+1})$, it follows that $\gamma _n v = \gamma _n \sigma ' = \sigma $; this shows that *v* defines a diagonal filler for the diagram.

Corollary 2.7. Let $\mathscr {C}$ be an $\infty $-category, and let $n \geq 1$ be an integer. There is a (non-canonical) map

such that the following diagram commutes

where the horizontal maps are the canonical inclusions.

Proof. We have a diagram as follows:

We may choose a section $\epsilon ' \colon \mathrm {sk}_n \mathrm {h}_n \mathscr {C} \to \mathrm {sk}_n \mathscr {C}$ – uniquely up to equivalence. We claim that this section can be extended further to a section $\epsilon $ as required. Let $\sigma \colon \Delta ^{n+1} \to \mathrm {h}_n\mathscr {C}$ denote a nondegenerate $(n+1)$-simplex in $\mathrm {h}_n \mathscr {C}$. Then we consider the composite map

and this commutative diagram:

By Proposition 2.6, there is a diagonal filler $\tau \colon \Delta ^{n+1} \to \mathscr {C}$ – that is, an $(n+1)$-simplex of $\mathscr {C}$ – that makes the diagram commutative. We set $\epsilon (\sigma ) := \tau $. Repeating this process for each $\sigma $, we obtain the required extension $\epsilon \colon \mathrm {sk}_{n+1} \mathrm {h}_n \mathscr {C} \to \mathrm {sk}_{n+1} \mathscr {C}$.

Example 2.8. Let $\mathscr {C}$ be an $\infty $-category. The functor $\gamma _1 \colon \mathscr {C} \to \mathrm {h}_1 \mathscr {C}$ is bijective on objects, so there is a unique section $\mathrm {sk}_0 \mathrm {h}_1 \mathscr {C} \to \mathrm {sk}_0 \mathscr {C}$. By making choices of morphisms, one from each homotopy class, this map extends to a section $\mathrm {sk}_1 \mathrm {h}_1 \mathscr {C} \to \mathrm {sk}_1 \mathscr {C}$. The last map extends further to a section $\mathrm {sk}_2 \mathrm {h}_1 \mathscr {C} \to \mathrm {sk}_2 \mathscr {C}$ by making (non-canonical) choices of homotopies for compositions.

The functor $\mathrm {h}_n(-)$ preserves categorical equivalences of $\infty $-categories. Using Proposition 2.3, it follows that there is a tower of $\infty $-categories:

By construction, the canonical map

is an isomorphism, and by Proposition 2.6, this inverse limit defines also a homotopy limit in the Joyal model structure.

Example 2.9. As a consequence of Proposition 2.2, an $\infty $-groupoid *X* is categorically equivalent to an *n*-category if and only if it is *n*-truncated. For example, a Kan complex is equivalent to a $0$-category if and only if it is homotopically discrete and to an $1$-category if and only if it is equivalent to the nerve of a groupoid. Given an $\infty $-groupoid *X*, the canonical tower of $\infty $-groupoids

models the Postnikov tower of *X* and the map $X \to \mathrm {h}_n X$ is $(n+1)$-connected (i.e., for every $x \in X$, the map $\pi _k(X, x) \to \pi _k(\mathrm {h}_n X, x)$ is a bijection for $k \leq n$ and surjective for $k = n+1$.)

Remark 2.10. An *n*-category $\mathscr {C}$ is *n*-truncated in the $\infty $-category of $\infty $-categories: that is, the $\infty $-groupoid $\mathrm {Map}(K, \mathscr {C})$ is *n*-truncated for any $\infty $-category *K* (see [Reference Lurie22, 5.5.6] for the definition and properties of truncated objects in an $\infty $-category). To see this, recall that $\mathrm {Fun}(K, \mathscr {C})$ is again an *n*-category, and then apply Proposition 2.2. But an *n*-truncated $\infty $-category $\mathscr {C}$ is not equivalent to an *n*-category in general, so the analogue of Example 2.9 fails for general $\infty $-categories. An $\infty $-category $\mathscr {C}$ is *n*-truncated if and only if $\mathscr {C}$ is equivalent to an $(n+1)$-category and the maximal $\infty $-subgroupoid $\mathscr {C}^{\simeq } \subseteq \mathscr {C}$ is *n*-truncated. Indeed, given an *n*-truncated $\infty $-category $\mathscr {C}$, $\mathscr {C}^{\simeq } \simeq \mathrm {Map}(\Delta ^0, \mathscr {C})$ is *n*-truncated. Moreover, since *n*-truncated objects are closed under limits, it follows that $\mathrm {Map}_{\mathscr {C}}(x, y)$ is *n*-truncated for every $x, y \in \mathscr {C}$, using the fact that there is a pullback in the $\infty $-category of $\infty $-categories:

(Using the definition of *n*-truncated objects, we see that $\mathscr {C}^{\Delta ^1}$ is again *n*-truncated.) Conversely, if $\mathscr {C}$ is an $(n+1)$-category and $\mathscr {C}^{\simeq }$ is *n*-truncated, then it is possible to show that $\mathrm {Map}(\Delta ^k, \mathscr {C})$ is *n*-truncated by induction on $k \geq 0$, from which it follows that $\mathscr {C}$ is *n*-truncated. (I am grateful to Hoang Kim Nguyen for interesting discussions related to this remark.)

Let $\mathsf {Cat_{\infty }}$ denote the category of $\infty $-categories, regarded as enriched in $\infty $-categories, and let $\mathsf {Cat_{n}}$ denote the full subcategory of $\mathsf {Cat_{\infty }}$ that is spanned by *n*-categories.

Proposition 2.11. Let $\mathscr {C}$ and $\mathscr {D}$ be $\infty $-categories, and let $n \geq 1$ be an integer.

(a) The natural map $\mathrm {h}_n (\mathscr {C} \times \mathscr {D}) \xrightarrow {\cong } \mathrm {h}_n \mathscr {C} \times \mathrm {h}_n \mathscr {D}$ is an isomorphism.

(b) There is a functor

$$ \begin{align*}\mathrm{h}_{n}^{\mathscr{C}, \mathscr{D}} \colon \mathrm{Fun}(\mathscr{C}, \mathscr{D}) \to \mathrm{Fun}(\mathrm{h}_n \mathscr{C}, \mathrm{h}_n \mathscr{D})\end{align*} $$that is natural in $\mathscr {C}$ and $\mathscr {D}$. In particular, $\mathrm {h}_n \colon \mathsf {Cat_{\infty }} \to \mathsf {Cat_{n}}$ is an enriched functor.

Proof. (a) follows directly from the definition of $\mathrm {h}_n$. For (b), we define the functor $\mathrm {h}_n^{\mathscr {C}, \mathscr {D}}$ as follows: a *k*-simplex $F \colon \mathscr {C} \times \Delta ^k \to \mathscr {D}$ is sent to the composite

The functor $\mathrm {h}_n^{\mathscr {C}, \mathscr {D}}$ is natural in $\mathscr {C}$ and $\mathscr {D}$ and turns $\mathrm {h}_n$ into an enriched functor.

## 3 Higher weak colimits

### 3.1 Basic definitions and properties

It is well known that homotopy categories do not admit small (co)limits in general, even when the underlying $\infty $-category has small (co)limits. On the other hand, if the $\infty $-category $\mathscr {C}$ admits pushouts (respectively, coproducts), for example, then the homotopy category $\mathrm {h}_1 \mathscr {C}$ admits weak pushouts (respectively, coproducts), which are induced from pushouts (respectively, coproducts) in $\mathscr {C}$. Moreover, if $\mathscr {C}$ admits small colimits, then $\mathrm {h}_1 \mathscr {C}$ admits small weak colimits – which may or may not be induced from $\mathscr {C}$. These observations suggest the following questions: does $\mathrm {h}_n \mathscr {C}$, $n> 1$, have in some sense more or *better* (co)limits than the homotopy category, and how do these compare with (co)limits in $\mathscr {C}$?

In this section, we introduce and study a notion of higher weak (co)limits in the context of $\infty $-categories. This is both a higher categorical version of the classical notion of weak (co)limits and a weak version of the higher categorical notion of (co)limits. We will mainly focus on the properties of higher weak (co)limits in the context of higher homotopy categories. We also refer to [Reference Nguyen, Raptis and Schrade30] for further properties and applications of higher weak (co)limits.

We will restrict to higher weak colimits since the corresponding definitions and results about higher weak limits are obtained dually. First we recall that an $\infty $-groupoid (= Kan complex) *X* is *k*-*connected*, for some $k \geq -1$, if it is non-empty and $\pi _i(X, x) \cong 0$ for every $x \in X$ and $i \leq k$. For example, *X* is $(-1)$-connected (respectively, $0$-connected, $\infty $-connected) if it is non-empty (respectively, connected, contractible).

We begin with the definition of a weakly initial object. Fix $t \in \mathbb {Z}_{\geq 0} \cup \{\infty \}$.

Definition 3.1. An object *x* of an $\infty $-category $\mathscr {C}$ is *weakly initial of order t* if the mapping space $\mathrm {Map}_{\mathscr {C}}(x, y)$ is $(t-1)$-connected for every object $y \in \mathscr {C}$.

Example 3.2. If $\mathscr {C}$ is an ordinary category, a weakly initial object $x \in \mathscr {C}$ of order $0$ is a weakly initial object in the classical sense. For a general *n*-category $\mathscr {C}$, a weakly initial object of order *n* is an initial object.

Proposition 3.3. Let $\mathscr {C}$ be an $\infty $-category and $x \in \mathscr {C}$. The following are equivalent:

(1)

*x*is weakly initial in $\mathscr {C}$ of order*t*.(2)

*x*is weakly initial in $\mathrm {h}_n \mathscr {C}$ of order*t*for any $n> t$.(3)

*x*is initial in $\mathrm {h}_{t} \mathscr {C}$.

These imply:

(4)

*x*is initial in $\mathrm {h}_n \mathscr {C}$ for any $n < t$.

Proof. This follows from the fact that the functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ restricts to an *n*-connected map $\mathrm {Map}_{\mathscr {C}}(x, y) \to \mathrm {Map}_{\mathrm {h}_n \mathscr {C}}(x, y)$ for every $x, y \in \mathscr {C}$.

Proposition 3.4. Let $\mathscr {C}$ be an $\infty $-category, and let $t> 0$. The full subcategory $\mathscr {C}'$ of $\mathscr {C}$, which is spanned by the weakly initial objects of order *t* is either empty or a *t*-connected $\infty $-groupoid.

Proof. Suppose that the full subcategory $\mathscr {C}'$ is non-empty. Then the mapping spaces of $\mathscr {C}'$ are $(t-1)$-connected, where $t-1 \geq 0$. It follows that every morphism in $\mathscr {C}'$ is an equivalence, and therefore $\mathscr {C}'$ is an $\infty $-groupoid.

Remark 3.5. The full subcategory $\mathscr {C}'$ of weakly initial objects in $\mathscr {C}$ of order 0 is not an $\infty $-groupoid in general. In this case, we only have that $\mathrm {Map}_{\mathscr {C}}(x, y)$ is non-empty for every $x, y \in \mathscr {C}'$.

Definition 3.6. Let $\mathscr {C}$ be an $\infty $-category and *K* a simplicial set, and let $F \colon K \to \mathscr {C}$ be a *K*-diagram in $\mathscr {C}$. A weakly initial object $G \in \mathscr {C}_{F/}$ of order *t* is called a *weak colimit of F of order t*.

Example 3.7. If $\mathscr {C}$ is an *n*-category and $G \colon K^{\triangleright } \to \mathscr {C}$ is a weak colimit of $F = G_{|K} \colon K \to \mathscr {C}$ order $t \geq n$, then *G* is a colimit diagram. This follows from Example 3.2 using the fact that $\mathscr {C}_{F/}$ is again an *n*-category (see [Reference Lurie22, Corollary 2.3.4.10]). In particular, a weak colimit of order $\infty $ is a colimit diagram. If $\mathscr {C}$ is an ordinary category, then a weak colimit of order $0$ is a weak colimit diagram in the classical sense.

Remark 3.8. There is an important difference between weak colimits of order $0$ and weak colimits of order $> 0$: as a consequence of Proposition 3.4, any two weak colimits of *F* of order $> 0$ are equivalent. In particular, if $G \in \mathscr {C}_{F/}$ is a weak colimit of order $t> 0$ and $G' \in \mathscr {C}_{F/}$ is a weak colimit of order $>0$, then $G'$ is also a weak colimit of order *t*.

The following proposition gives an alternative characterization of higher weak colimits following the analogous characterization for colimits in [Reference Lurie22, Lemma 4.2.4.3].

Proposition 3.9. Let $\mathscr {C}$ be an $\infty $-category and *K* a simplicial set, and let $G \colon K^{\triangleright } \to \mathscr {C}$ be a diagram with cone object $x \in \mathscr {C}$. Then *G* is a weak colimit of $F = G_{| K}$ of order *t* if and only if the canonical restriction map

is *t*-connected for every $y \in \mathscr {C}$, where $c_y$ denotes, respectively, the constant diagram at $y \in \mathscr {C}$.

Proof. The fibre of the restriction map over $F' \colon K^{\triangleright } \to \mathscr {C}$ with cone object *y* is identified with $\mathrm {Map}_{\mathscr {C}_{F/}}(G, F')$ (see the proof of [Reference Lurie22, Lemma 4.2.4.3]).

The basic rules for the manipulation of higher weak colimits can be established similarly as for colimits. The following procedure shows that higher weak colimits can be computed iteratively, exactly like colimits, but with the difference that the order of the weak colimit may decrease with each iteration.

Proposition 3.10. Let $\mathscr {C}$ be an $\infty $-category, and let $K = K_1 \cup _{K_0} K_2$ be a simplicial set, where $K_0 \subseteq K_1$ is a simplicial subset. Let $F \colon K \to \mathscr {C}$ be a diagram, and denote its restrictions by $F_i := F_{|K_i }$, for $i = 0,1,2$. Suppose that $G_i \colon K_i^{\triangleright } \to \mathscr {C}$ is a weak colimit of $F_i$ of order $t_i \geq 0$, for $i = 0,1,2$.

(a) There are morphisms $G_0 \to G_1{}_{|K_0^{\triangleright }}$ and $G_0 \to G_2{}_{|K_0^{\triangleright }}$ in $\mathscr {C}_{F_0/}$. These together with $G_1$ and $G_2$ determine a diagram in $\mathscr {C}$ as follows:

$$ \begin{align*}H \colon K_1^{\triangleright} \cup_{K_0 \ast \Delta^{\{1\}}} (K_0 \ast \Delta^1) \cup_{K_0 \ast \Delta^{\{0\}}} (K_0 \ast \Delta^1) \cup_{K_0 \ast \Delta^{\{1\}}} K_2^{\triangleright} \to \mathscr{C}.\end{align*} $$(b) Let $H_{\ulcorner } \colon \Delta ^1 \cup _{\Delta ^0} \Delta ^1 \to \mathscr {C}$ be the restriction of

*H*to the cone objects. Suppose that $H' \colon \Delta ^1 \times \Delta ^1 \to \mathscr {C}$ is a weak pushout of $H_{\ulcorner }$ of order*k*. Then $H'$ determines a cocone $G \colon K^{\triangleright } \to \mathscr {C}$ over*F*, with the same cone object as $H'$, which is a weak colimit of*F*of order $\ell : = \mathrm {min}(k, t_1, t_0 -1, t_2)$.

Proof. (a) follows directly from the properties of higher weak colimits. For (b), we first explain the construction of the cocone $G \colon K^{\triangleright } \to \mathscr {C}$. The functor $H'$ is represented by a diagram

where $x_i$ is the cone object of $G_i$ and the morphisms *u* and *v* are given, respectively, by the morphisms $G_0 \to G_1{}_{|K_0^{\triangleright }}$ and $G_0 \to G_2{}_{|K_0^{\triangleright }}$ in $\mathscr {C}_{F_0/}$. The morphisms *f* and *g* produce two new cocones (essentially uniquely) $G^{\prime }_1 \colon K_1^{\triangleright } \to \mathscr {C}$ over $F_1$ and $G^{\prime }_2 \colon K_2^{\triangleright } \to \mathscr {C}$ over $F_2$, with common cone object *y*. The restrictions $G^{\prime }_1{}_{|K_0^{\triangleright }}$ and $G^{\prime }_2{}_{|K_0^{\triangleright }}$ are equivalent as cocones over $F_0$. We may then extend $G^{\prime }_2{}_{|K_0^{\triangleright }}$ in an essentially unique way to a new cocone $G^{\prime \prime }_1 \colon K_1^{\triangleright } \to \mathscr {C}$ over $F_1$, which is equivalent to $G^{\prime }_1$. The resulting cocones

assemble to define the required cocone $G \colon K^{\triangleright } \to \mathscr {C}$. Then the claim in (b) is shown by applying Proposition 3.9, first for weak pushouts and then for $K_i$-diagrams, and using the fact that $\mathscr {C}^K$ is the homotopy pullback (in the Joyal model category) of the diagram of $\infty $-categories ($\mathscr {C}^{K_1} \to \mathscr {C}^{K_0} \leftarrow \mathscr {C}^{K_2}$), so its mapping spaces can also be identified with the corresponding (homotopy) pullbacks of mapping spaces.

Example 3.11. Let $\mathscr {C}$ be (the nerve of) an ordinary category that admits small coproducts and weak pushouts. By Proposition 3.10, every diagram $F \colon K \to \mathscr {C}$, where *K* is 1-dimensional, admits a weak colimit (of order 0). Now suppose that $F \colon I \to \mathscr {C}$ is a diagram where *I* is (the nerve of) an arbitrary ordinary small category. Since $\mathscr {C}$ is an $1$-category, a cocone $G \colon I^{\triangleright } \to \mathscr {C}$ over *F* is determined uniquely by its restriction to a cocone $G' \colon (\mathrm {sk}_1 I)^{\triangleright } \to \mathscr {C}$ over $F_{|\mathrm {sk}_1 I}$, and similarly for morphisms between cocones. As a consequence, we may deduce the well-known fact that $\mathscr {C}$ has weak *I*-colimits.

Example 3.12. Let $T \subset \Delta ^1 \times \Delta ^2$ be the full subcategory spanned by the objects $(0, i)$, for $i = 0,1,2$, and $(1,0)$. Let $\mathscr {C}$ be an $\infty $-category with weak pushouts of order *t*, and let $F \colon T \to \mathscr {C}$ be a *T*-diagram in $\mathscr {C}$. Write $T = T_1 \cup _{T_0} T_2$, where $T_1$ is spanned by $(0,i)$, $i = 0,1$ and $(1,0)$, $T_2$ is spanned by $(0, i)$, $i = 1,2$ and $T_0 = \{(0,1)\}$. Using Proposition 3.10, we may compute a weak colimit of *F* of order *t* in terms of iterated weak pushouts of order *t*.

### 3.2 Homotopy categories and (co)limits

Let $\mathscr {C}$ be an $\infty $-category, and let *K* be a simplicial set. By the universal property of $\mathrm {h}_n(-)$, the functor

which is given by composition with $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$, factors canonically through the homotopy *n*-category:

The comparison between *K*-colimits in $\mathscr {C}$ and in $\mathrm {h}_n \mathscr {C}$ is essentially a question about the properties of the functor $\Phi ^K_n$. Note that for $n=1$, $\Phi ^K_1$ is simply the canonical functor of ordinary categories: $\mathrm {h}_1(\mathscr {C}^K) \to \mathrm {h}_1(\mathscr {C})^K$.

Lemma 3.14. Let $\mathscr {C}$ be an $\infty $-category and *K* a finite dimensional simplicial set of dimension $d> 0$, and let $n \geq 1$ be an integer. The functor

satisfies the following:

(a) $\Phi ^K_n$ is a bijection in simplicial degrees $< n - d$.

(b) $\Phi ^K_n$ is surjective in simplicial degree $n-d$; it identifies $(n-d)$-simplices, which are homotopic relative to the $(n-1)$-skeleton of $\Delta ^{n-d} \times K$.

(c) $\Phi ^K_n$ is surjective in simplicial degree $n-d+1$.

Proof. The *m*-simplices of $\mathrm {Fun}(K, \mathrm {h}_n \mathscr {C})$ are equivalence classes of maps

that extend to $\mathrm {sk}_{n+1}(\Delta ^m \times K)$. On the other hand, the *m*-simplices of $\mathrm {h}_n \mathrm {Fun}(K, \mathscr {C})$ are equivalence classes of maps

that extend to $\mathrm {sk}_{n+1}(\Delta ^m) \times K$. The functor $\Phi ^K_n$ is induced by the canonical map

which is an isomorphism if $d \leq \mathrm {max}(n - m, 0)$. This shows that the map $\Phi ^K_n$ is surjective in simplicial degrees $\leq n-d + 1$. Similarly, the map

is an isomorphism if $d \leq \mathrm {max}(n - m -1, 0)$, so the two equivalence relations agree for $m < n -d$.

Remark 3.15. The case $d = 0$ is both special and essentially trivial, since the functor $\Phi ^K_n$ is an isomorphism in this case.

Proposition 3.16. Let $\mathscr {C}$ be an $\infty $-category and *K* a finite dimensional simplicial set of dimension $d> 0$, and let $n \geq 1$ be an integer. Then for every lifting problem

where $m \leq n-d+1$ (respectively, $m < n - d$), there is a (unique) filler $\Delta ^m \to \mathrm {h}_n (\mathscr {C}^K)$, which makes the diagram commutative.

Proof. The case $m < n-d + 1$ is a direct consequence of Lemma 3.14. For $m = n -d +1 \leq n$, Lemma 3.14 shows that there is a lift $\tau \colon \Delta ^m \to \mathrm {h}_n(\mathscr {C}^K)$ of $\sigma $, represented by a map $\tau ' \colon \Delta ^m \to \mathscr {C}^K$. The maps

become equal after composition with $\Phi ^K_n$. Moreover, note that *u* corresponds uniquely to a map $u \colon \partial \Delta ^m \to \mathscr {C}^K$. The maps $\Phi ^K_n \circ u$ and $\Phi ^K_n \circ (\gamma _n \tau ^{\prime }_{|\partial \Delta ^m})$ correspond to the equivalence classes of maps (using here the same notation for the adjoint maps)

so these last two maps are homotopic relative to $\mathrm {sk}_{n-1}(\partial \Delta ^m \times K)$ (see Proposition 2.3). Let

be a homotopy from $\tau ^{\prime }_{| \partial \Delta ^m \times K}$ to *u* relative to the subspace $\mathrm {sk}_{n-1}(\partial \Delta ^m \times K)$, where $J = N(0 \rightleftarrows 1)$ denotes the Joyal interval object. Using the fact that

is a cofibration, this homotopy can be extended to a homotopy

where $\tau ' = H^{\prime }_{|\Delta ^m\times K \times \{0\}}$ and $H'$ is constant on $\mathrm {sk}_{n-1}(\Delta ^m \times K)$. Then the map

extends *u*. Moreover, the (adjoint) map $\widetilde {\tau } \colon \Delta ^m \to \mathscr {C}^K$ defines an element in $\mathrm {h}_n(\mathscr {C})^K_m$, after composition with $\gamma _n$, which agrees with $\Phi ^K_n \circ \tau = \sigma $, so $\widetilde {\tau }$ represents a diagonal filler, as required.

As a consequence of Proposition 3.16, we obtain the following result about the comparison between the *n*-categories $\mathrm {h}_n(\mathscr {C}^K)$ and $\mathrm {h}_n(\mathscr {C})^K$.

Corollary 3.17. Let $\mathscr {C}$ be an $\infty $-category and *K* a finite dimensional simplicial set of dimension $d> 0$, and let $n \geq d$ be an integer. The functor $\Phi ^K_n \colon \mathrm {h}_n(\mathscr {C}^K) \to (\mathrm {h}_n \mathscr {C})^K$ is essentially surjective, and for every pair of objects $F, G$ in $\mathrm {h}_n(\mathscr {C}^K)$, the induced map between mapping spaces

is $(n-d)$-connected. As a consequence, $\Phi ^K_n$ induces an equivalence:

Now assume that $\mathscr {C}$ is an $\infty $-category that admits *K*-colimits, where *K* is a simplicial set. We have colimit-functors:

Assuming also that $\mathscr {C}$ and *K* are as in Corollary 3.17, and passing to the homotopy $(n-d)$-categories as in (3.18), we obtain the following corollary.

Corollary 3.19. Let $\mathscr {C}$ be an $\infty $-category and *K* a finite dimensional simplicial set of dimension $d> 0$, and let $n \geq d$ be an integer. Suppose that $\mathscr {C}$ admits *K*-colimits. Then there is an adjoint pair

where *c* denotes the constant *K*-diagram functor.

Proposition 3.20. Let $\mathscr {C}$ be an $\infty $-category with weak *K*-colimits of order *k*, where *K* is a simplicial set of dimension $d> 0$, and let $n \geq 1$ be an integer. Then $\mathrm {h}_n \mathscr {C}$ has weak *K*-colimits of order $\ell = \mathrm {min}(n-d, k)$. Moreover, the functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ sends weak *K*-colimits of order *k* to weak *K*-colimits of order $\ell $.

Proof. We may assume that $n \geq d$, and therefore the functor $\mathscr {C}^K \to (\mathrm {h}_n \mathscr {C})^K$ is surjective on objects. Then it suffices to prove the second claim. Let $G \colon K^{\triangleright } \to \mathscr {C}$ be a weak colimit of $F = G_{|K}$ of order *k* with cone object $x \in \mathscr {C}$. By Proposition 3.9, it suffices to prove that the canonical map

is $\ell $-connected for every $y \in \mathscr {C}$. Note that there is a *k*-connected map:

Hence it suffices to show that the canonical map

is $(n-d)$-connected. This follows from Corollary 3.17. (Alternatively, note that the last map is identified with the canonical map from the $(n-1)$-truncation of a $K^{\operatorname {op}}$-limit of $\infty $-groupoids to the $K^{\operatorname {op}}$-limit of the $(n-1)$-truncations of the $\infty $-groupoids:

An inductive argument on *d* shows that the map is $(n-d)$-connected.)

Remark 3.21. A different proof of Proposition 3.20 is also possible using elementary lifting arguments based on Proposition 3.16.

Corollary 3.22. Let $\mathscr {C}$ be an $\infty $-category that admits finite (respectively, small) colimits.

(a) The homotopy

*n*-category $\mathrm {h}_n \mathscr {C}$ admits finite (respectively, small) coproducts and weak pushouts of order $n-1$. Moreover, the functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ preserves finite (respectively, small) coproducts and sends pushouts in $\mathscr {C}$ to weak pushouts of order $n-1$.(b) Suppose that $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ preserves finite colimits. Then $\mathscr {C}$ is equivalent to an

*n*-category.

Proof. (a) $\mathrm {h}_n \mathscr {C}$ admits finite coproducts by Remark 3.15. The existence and preservation of higher weak pushouts is a consequence of Proposition 3.20. (b) is a consequence of [Reference Nguyen, Raptis and Schrade29, Corollary 3.3.5].

These properties of homotopy *n*-categories of (finitely) cocomplete $\infty $-categories suggest considering the following class of *n*-categories as a convenient general context for the study of these *n*-categories.

Definition 3.23. Let $n \geq 1$ be an integer or $n=\infty $. A *(finitely) weakly cocomplete n-category* is an *n*-category $\mathscr {C}$ that admits small (finite) coproducts and weak pushouts of order $n-1$.

These *n*-categories are studied further in [Reference Nguyen, Raptis and Schrade30] in connection with adjoint functor theorems and Brown representability theorems for *n*-categories (extending the results of [Reference Nguyen, Raptis and Schrade29]). Also, in Section 6 (which can be read independently of Sections 4 and 5), we will return to the properties of weak (co)limits in higher homotopy categories, especially, for pointed and stable $\infty $-categories, and use higher weak colimits in order to define *K*-theory for higher homotopy categories.

Remark 3.24. Corollary 3.19 produces a *truncated K-colimit functor* for $\mathrm {h}_n \mathscr {C}$:

According to Proposition 3.16 (compare to Proposition 2.6 and Corollary 2.7), there is a (non-canonical) section

(Note that the target is closely related to $\mathrm {sk}_{n-d+1}(\mathscr {C}^K).$) Then we may define the *partial K-colimit functor* for $\mathrm {h}_n (\mathscr {C})$ (which depends on $\mathscr {C}$ and the section $\epsilon $)

as the composition

Example 3.25. Let $\mathscr {C}$ be an $\infty $-category that has pushouts, and let *K* denote the (nerve of the) ‘corner’ category $\ulcorner : = \Delta ^1 \cup _{\Delta ^0} \Delta ^1$. The functor

is surjective on objects and full. By Corollary 3.17, the pushout-functor on $\mathscr {C}$ induces a truncated pushout-functor

which is left adjoint to the constant diagram functor. Furthermore, we have a map as follows

which sends $F \colon K \to \mathrm {h}_1 \mathscr {C}$ to the pushout of a choice of a lift $\widetilde {F} \colon K \to \mathscr {C}$. This is simply regarded as a map from the *set* of $0$-simplices. Moreover, (3.26) extends further to the $1$-skeleton, but this involves non-canonical choices that are not unique even up to homotopy. As explained in Remark 3.24, an extension of this type can be obtained from a section $\epsilon \colon \mathrm {sk}_1 \big (\mathrm {Fun}(K, \mathrm {h}_1\mathscr {C})\big ) \to \mathrm {sk}_1\big (\mathrm {h}_1(\mathscr {C}^K)\big ).$ The fact that this process cannot be continued to higher-dimensional skeleta is related to the non-functoriality of weak pushouts in the homotopy category.

More generally, for $n \geq 1$, there is a left adjoint truncated pushout-functor

and partial pushout-functors

that define weak pushouts in $\mathrm {h}_n(\mathscr {C})$ of order $n-1$.

## 4 Higher derivators

### 4.1 Basic definitions and properties

We recall that $\mathsf {Cat_{\infty }}$ denotes the category of $\infty $-categories, regarded as enriched in $\infty $-categories. Let $\mathsf {Dia}$ denote a full subcategory of $\mathsf {Cat_{\infty }}$ that has the following properties:

(Dia 0) $\mathsf {Dia}$ contains the (nerves of) finite posets.

(Dia 1) $\mathsf {Dia}$ is closed under finite coproducts and under pullbacks along an inner fibration.

(Dia 2) For every $\mathsf {X} \in \mathsf {Dia}$ and $x \in \mathsf {X}$, the $\infty $-category $\mathsf {X}_{/x}$ is in $\mathsf {Dia}$.

(Dia 3) $\mathsf {Dia}$ is closed under passing to the opposite $\infty $-category.

The main examples of such subcategories of $\mathsf {Cat_{\infty }}$ are the following:

(i) The full subcategory of (nerves of) finite posets.

(ii) The full subcategory of (ordinary) finite direct categories $\mathcal {D}ir_{f}\kern-1pt$. (We recall that an ordinary category $\mathsf {C}$ is called

*finite direct*if its nerve is a finite simplicial set.)(iii) The full subcategory $\mathsf {Cat_{n}} \subset \mathsf {Cat_{\infty }}$ of

*n*-categories for any $n \geq 1$.(iv) $\mathsf {Cat_{\infty }}$.

We denote by $\mathsf {Dia}^{\operatorname {op}}$ the opposite category taken 1-categorically: that is, the enrichment of $\mathsf {Dia}^{\operatorname {op}}$ is given by

Definition 4.1. An $\infty $-*prederivator* with domain $\mathsf {Dia}$ is an enriched functor

An $\infty $-prederivator $\mathbb {D}$ with domain $\mathsf {Dia}$ is an *n*-*prederivator* if it factors through the inclusion $\mathsf {Cat_{n}} \subset \mathsf {Cat_{\infty }}$: that is, $\mathbb {D}$ is an enriched functor

A *strict* morphism of $\infty $-prederivators is a natural transformation $F \colon \mathbb {D} \to \mathbb {D}'$ between enriched functors. Thus we obtain a category of $\infty $-prederivators, denoted by $\mathsf {PreDer}_{\infty }$, which is enriched in $\infty $-categories. For any $n \geq 1$, there is a full subcategory $\mathsf {PreDer}_n \subset \mathsf {PreDer}_{\infty }$ spanned by the *n*-prederivators. In the same way that the classical theory of (pre)derivators is founded on a ((2,2)=)2-categorical context, the general theory of $\infty $-prederivators involves an $(\infty , 2)$-categorical context. We point out that it would be natural to also consider non-strict morphisms (aka pseudonatural transformations) between $\infty $-prederivators, but these will not be needed in this paper.

**Notation.** Let $\mathbb {D}$ be an $\infty $-prederivator with domain $\mathsf {Dia}$, and let $u \colon \mathsf {X} \to \mathsf {Y}$ be a functor in $\mathsf {Dia}$. We will often denote the induced functor $\mathbb {D}(u) \colon \mathbb {D}(\mathsf {Y}) \to \mathbb {D}(\mathsf {X})$ by $u^*$. Moreover, if $i_{\mathsf {Y}, y} \colon \Delta ^0 \to \mathsf {Y}$ is the inclusion of the object $y \in \mathsf {Y}$ and $F \in \mathbb {D}(\mathsf {Y})$, we will often denote the object $\mathbb {D}(i_{\mathsf {Y}, y})(F)$ in $\mathbb {D}(\Delta ^0)$ by $F_{y}$.

Example 4.2. A 1-prederivator with domain $\mathsf {Cat_1}$ is a prederivator in the usual sense [Reference Maltsiniotis24, Reference Groth18].

Example 4.3. Let $\mathscr {C}$ be an $\infty $-category. There is an associated $\infty $-prederivator (with domain $\mathsf {Dia}$) defined by $\mathbb {D}^{(\infty )}_{\mathscr {C}} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}, \ \mathsf {X} \mapsto \mathrm {Fun}(\mathsf {X}, \mathscr {C})$. Moreover, for any $n \geq 1$,

defines an *n*-prederivator.

Definition 4.4. An $\infty $-prederivator $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ is a *right* $\infty $-*derivator* if it satisfies the following properties:

(Der 1) For every pair of $\infty $-categories $\mathsf {X}$ and $\mathsf {Y}$ in $\mathsf {Dia}$, the functor induced by the inclusions of the factors to the coproduct $\mathsf {X} \sqcup \mathsf {Y}$,

$$ \begin{align*}\mathbb{D}(\mathsf{X} \sqcup \mathsf{Y})\rightarrow \mathbb{D}(\mathsf{X}) \times \mathbb{D}(\mathsf{Y})\end{align*} $$is an equivalence. Moreover, $\mathbb {D}(\varnothing )$ is equivalent to the final $\infty $-category $\Delta ^0$.(Der 2) For every $\infty $-category $\mathsf {X}$ in $\mathsf {Dia}$, the functor

$$ \begin{align*}(i_{\mathsf{X}, x}^{*} = \mathbb{D}(i_{\mathsf{X}, x}))_{x \in \mathsf{X}} \colon \mathbb{D}(\mathsf{X}) \rightarrow \prod_{x \in \mathsf{X}} \mathbb{D}(\Delta^0)\end{align*} $$is conservative: that is, it detects equivalences. We recall that $i_{\mathsf {X}, x} \colon \Delta ^0 \to \mathsf {X}$ is the functor that corresponds to the object $x \in \mathsf {X}$.(Der 3) For every morphism $u \colon \mathsf {X} \rightarrow \mathsf {Y}$ in $\mathsf {Dia}$, the functor $u^{*} = \mathbb {D}(u) \colon \mathbb {D}(\mathsf {Y})\rightarrow \mathbb {D}(\mathsf {X})$ is a right adjoint. We denote a left adjoint of $u^*$ by

$$ \begin{align*}u_{!}\colon \mathbb{D}(\mathsf{X})\rightarrow\mathbb{D}(\mathsf{Y}).\end{align*} $$(Der 4) Given $u \colon \mathsf {X} \rightarrow \mathsf {Y}$ in $\mathsf {Dia}$ and $y \in \mathsf {Y}$, consider the following pullback diagram in $\mathsf {Dia}$:

Then the canonical base change natural transformation$$ \begin{align*}c_{u,y}\colon (p_{u/y})_{!} j^{*}_{u/y} \longrightarrow q^{*}_{\mathsf{Y} /y }u_{!}\end{align*} $$is a natural equivalence of functors.

We define left $\infty $-derivators dually.

Definition 4.5. An $\infty $-prederivator $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ is a *left* $\infty $-*derivator* if it satisfies (Der1)–(Der2) as stated above together with the following dual versions of (Der3)–(Der4):

(Der 3)* For every morphism $u \colon \mathsf {X} \rightarrow \mathsf {Y}$ in $\mathsf {Dia}$, the functor $u^{*} = \mathbb {D}(u) \colon \mathbb {D}(\mathsf {Y})\rightarrow \mathbb {D}(\mathsf {X})$ is a left adjoint. We denote a right adjoint of $u^*$ by

$$ \begin{align*}u_*\colon \mathbb{D}(\mathsf{X}) \to \mathbb{D}(\mathsf{Y}).\end{align*} $$(Der 4)* Given $u \colon \mathsf {X} \rightarrow \mathsf {Y}$ in $\mathsf {Dia}$ and $y \in \mathsf {Y}$, consider the following pullback diagram in $\mathsf {Dia}$:

Then the canonical base change natural transformation$$ \begin{align*}c^{\prime}_{u,y} \colon q^{*}_{y / \mathsf{Y} }u_{*} \longrightarrow (p_{y/u})_{*} j^{*}_{y/u}\end{align*} $$is a natural equivalence of functors.

Example 4.6. Let $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ be a left $\infty $-derivator. Then the $\infty $-prederivator

is a right $\infty $-derivator.

Definition 4.7. An $\infty $-prederivator $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ is an $\infty $-*derivator* if it is both a left and a right $\infty $-derivator.

We also specialise these definitions to *n*-prederivators as follows.

Definition 4.8. An $\infty $-prederivator $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ is a (left, right) *n*-*derivator* if it is an *n*-prederivator and a (left, right) $\infty $-derivator.

Example 4.9. A (left, right) 1-derivator with domain $\mathsf {Cat_1}$ is a (left, right) derivator in the usual sense [Reference Maltsiniotis24, Reference Groth18]. Indeed, (Der 1)–(Der 3) are completely analogous to the usual axioms. (Der 4) is a convenient variation of the usual axiom for ordinary derivators; the equivalence of the definitions is based on known properties of homotopy exact squares in the context of ordinary derivators (see [Reference Groth18, Reference Maltsiniotis23]).

Example 4.10. Let $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{n}}$ be an *n*-prederivator, where $n \in \mathbb {Z}_{\geq 1} \cup \{\infty \}$. For any $k < n$, there is an associated *k*-prederivator:

If $\mathbb {D}$ is a left (right) *n*-derivator, then $\mathrm {h}_k \mathbb {D}$ is a left (right) *k*-derivator.

As a consequence of the axioms of Definition 4.4, we have the following useful strong version of (Der 4) and (Der 4)*, which identifies a larger class of squares for which the base change transformations are equivalences. Similar results are known for $\infty $-categories and ordinary ($1$-)derivators (see, for example, [Reference Cisinski4] and [Reference Maltsiniotis23, Reference Groth18]).

Proposition 4.11. Let $\mathbb {D}$ be an $\infty $-derivator with domain $\mathsf {Dia}$. Consider a pullback square in $\mathsf {Dia}$:

(1) The canonical base change transformation