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. (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. (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:

1. (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].

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.

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

$$ \begin{align*}\mathrm{sk}_n(K) \to \mathscr{C},\end{align*} $$

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,

$$ \begin{align*}(\mathrm{h}_n \mathscr{C})_m := [\Delta^m, \mathscr{C}]_n / \sim.\end{align*} $$

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 2.6. 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

$$ \begin{align*}u, \sigma^{\prime}_{| \partial \Delta^{n+1}} \colon \partial \Delta^{n+1} \to \mathscr{C}\end{align*} $$

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

$$ \begin{align*}\epsilon \colon \mathrm{sk}_{n+1} \mathrm{h}_n \mathscr{C} \to \mathrm{sk}_{n+1} \mathscr{C}\end{align*} $$

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

$$ \begin{align*}u \colon \partial \Delta^{n+1} = \mathrm{sk}_n\Delta^{n+1} \xrightarrow{\mathrm{sk}_n(\sigma)} \mathrm{sk}_n \mathrm{h}_n \mathscr{C} \xrightarrow{\epsilon'} \mathrm{sk}_n \mathscr{C} \to \mathscr{C}\end{align*} $$

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 $\epsilon \colon \mathrm {sk}_{n+1} \mathrm {h}_n \mathscr {C} \to \mathrm {sk}_{n+1} \mathscr {C}$.

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:

$$ \begin{align*}\mathscr{C} \to \cdots \to \mathrm{h}_n \mathscr{C} \to \mathrm{h}_{n-1} \mathscr{C} \to \cdots \to \mathrm{h}_{1} \mathscr{C}.\end{align*} $$

By construction, the canonical map

$$ \begin{align*}\mathscr{C} \longrightarrow \mathrm{lim}(\cdots \to \mathrm{h}_n \mathscr{C} \to \mathrm{h}_{n-1} \mathscr{C} \to \cdots \to \mathrm{h}_1 \mathscr{C})\end{align*} $$

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

$$ \begin{align*}X \to \cdots \mathrm{h}_n X \to \mathrm{h}_{n-1} X \to \cdots \to \mathrm{h}_1 X \to \pi_0 X\end{align*} $$

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.

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

$$ \begin{align*}\mathrm{h}_n \mathscr{C} \times \Delta^k \cong \mathrm{h}_n \mathscr{C} \times \mathrm{h}_n \Delta^k \cong \mathrm{h}_n (\mathscr{C} \times \Delta^k) \xrightarrow{\mathrm{h}_n F} \mathrm{h}_n \mathscr{D}.\end{align*} $$

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.

3.1 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 $t \in \mathbb {Z}_{\geq 0} \cup \{\infty \}$.

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

$$ \begin{align*}\mathrm{Map}_{\mathscr{C}}(x, y) \simeq \mathrm{Map}_{\mathscr{C}^{K^{\triangleright}}}(G, c_y) \to \mathrm{Map}_{\mathscr{C}^K}(F, c_y)\end{align*} $$

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

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.

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

$$ \begin{align*}G^{\prime}_2{}_{|K_0^{\triangleright}} \colon K_0^{\triangleright} \to \mathscr{C}, \ G^{\prime\prime}_1\colon K_1^{\triangleright} \to \mathscr{C} \text{and} \ G^{\prime}_2 \colon K_2^{\triangleright} \to \mathscr{C}\end{align*} $$

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.

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

$$ \begin{align*}\mathrm{Fun}(K, \mathscr{C}) \to \mathrm{Fun}(K, \mathrm{h}_n \mathscr{C}),\end{align*} $$

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

(3.13)$$ \begin{align} \Phi^K_n: \mathrm{h}_n \mathrm{Fun}(K, \mathscr{C}) \to \mathrm{Fun}(K, \mathrm{h}_n \mathscr{C}). \end{align} $$

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

$$ \begin{align*}\Phi^K_n : \mathrm{h}_n \mathrm{Fun}(K, \mathscr{C}) \to \mathrm{Fun}(K, \mathrm{h}_n \mathscr{C})\end{align*} $$

satisfies the following:

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

2. (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 $\Delta ^{n-d} \times K$.

3. (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

$$ \begin{align*}\mathrm{sk}_n(\Delta^m \times K) \to \mathscr{C}\end{align*} $$

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

$$ \begin{align*}\mathrm{sk}_n(\Delta^m) \times K \to \mathscr{C}\end{align*} $$

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

$$ \begin{align*}\mathrm{sk}_n(\Delta^m \times K) \to \mathrm{sk}_n(\Delta^m) \times K,\end{align*} $$

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

$$ \begin{align*}\mathrm{sk}_{n-1}(\Delta^m \times K) \to \mathrm{sk}_{n-1}(\Delta^m) \times K\end{align*} $$

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

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

$$ \begin{align*}u, \gamma_n \circ \tau^{\prime}_{|\partial \Delta^m} \colon \partial \Delta^m \to \mathrm{h}_n(\mathscr{C}^K)\end{align*} $$

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)

$$ \begin{align*}u, \tau^{\prime}_{| \partial \Delta^m \times K} \colon \partial \Delta^m \times K \to \mathscr{C},\end{align*} $$

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

$$ \begin{align*}H \colon (\partial \Delta^m \times K) \times J \to \mathscr{C}\end{align*} $$

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

$$ \begin{align*}\partial \Delta^m \times K \cup_{\mathrm{sk}_{n-1}(\partial \Delta^m \times K)} \mathrm{sk}_{n-1}(\Delta^m \times K) \subset \Delta^m \times K\end{align*} $$

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

$$ \begin{align*}H' \colon (\Delta^m \times K) \times J \to \mathscr{C},\end{align*} $$

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

$$ \begin{align*}\widetilde{\tau} \colon = H^{\prime}_{| \Delta^m \times K \times \{1\}} \colon \Delta^m \times K \to \mathscr{C}\end{align*} $$

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

$$ \begin{align*}\mathrm{Map}_{\mathrm{h}_n(\mathscr{C}^K)}(F, G) \longrightarrow \mathrm{Map}_{(\mathrm{h}_n \mathscr{C})^K}(\Phi^K_n(F), \Phi^K_n(G))\end{align*} $$

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

(3.18)$$ \begin{align} \mathrm{h}_{n-d}\big(\mathrm{Fun}(K,\mathscr{C})\big) \simeq \mathrm{h}_{n-d}\big(\mathrm{Fun}(K, \mathrm{h}_n \mathscr{C})\big). \end{align} $$

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

$$ \begin{align*}\mathrm{h}_{n-d}(\operatorname{colim}_K) \colon \mathrm{h}_{n-d}\big(\mathrm{Fun}(K, \mathrm{h}_n \mathscr{C})\big) \rightleftarrows \mathrm{h}_{n-d} (\mathscr{C}) \colon \mathrm{h}_{n-d}(c),\end{align*} $$

where c denotes the constant K-diagram functor.

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

$$ \begin{align*}\mathrm{Map}_{(\mathrm{h}_n\mathscr{C})^{K^{\triangleright}}}(G, c_y) \to \mathrm{Map}_{(\mathrm{h}_n\mathscr{C})^K}(F, c_y)\end{align*} $$

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

$$ \begin{align*}\mathrm{Map}_{(\mathrm{h}_n \mathscr{C})^{K^{\triangleright}}}(G, c_y) \simeq \mathrm{Map}_{\mathrm{h}_n \mathscr{C}}(x, y) \simeq \mathrm{Map}_{\mathrm{h}_n(\mathscr{C}^{K^{\triangleright}})}(G, c_y) \to \mathrm{Map}_{\mathrm{h}_n(\mathscr{C}^K)}(F, c_y).\end{align*} $$

Hence it suffices to show that the canonical map

$$ \begin{align*}\mathrm{Map}_{\mathrm{h}_n(\mathscr{C}^K)}(F, c_y) \to \mathrm{Map}_{(\mathrm{h}_n\mathscr{C})^K}(F, c_y)\end{align*} $$

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:

$$ \begin{align*}\mathrm{h}_{n-1} \big(\mathrm{lim}_{K^{\operatorname{op}}} \mathrm{Map}_{\mathscr{C}}(F(-), y)\big) \to \mathrm{lim}_{K^{\operatorname{op}}} \mathrm{h}_{n-1} \big(\mathrm{Map}_{\mathscr{C}}(F(-), y)\big).\end{align*} $$

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

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.

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 $\mathrm {h}_n \mathscr {C}$:

$$ \begin{align*}\mathrm{h}_{n-d}(\operatorname{colim}_K) \colon \mathrm{h}_{n-d}\big(\mathrm{Fun}(K, \mathrm{h}_n \mathscr{C})\big) \to \mathrm{h}_{n-d}(\mathscr{C}).\end{align*} $$

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

$$ \begin{align*}\epsilon \colon \mathrm{sk}_{n-d+1}\big(\mathrm{Fun}(K, \mathrm{h}_n\mathscr{C})\big) \to \mathrm{sk}_{n-d+1}\big(\mathrm{h}_n(\mathscr{C}^K)\big).\end{align*} $$

(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 $)

$$ \begin{align*}\mathrm{sk}_{n-d+1} \big(\mathrm{Fun}(K, \mathrm{h}_n \mathscr{C}) \big) \to \mathrm{h}_n(\mathscr{C})\end{align*} $$

as the composition

$$ \begin{align*}\mathrm{sk}_{n-d+1}\big(\mathrm{Fun}(K, \mathrm{h}_n\mathscr{C})\big) \xrightarrow{\epsilon} \mathrm{sk}_{n-d+1}\big(\mathrm{h}_n(\mathscr{C}^K)\big) \to \mathrm{h}_n(\mathscr{C}^K) \xrightarrow{\mathrm{h}_n(\operatorname{colim}_K)} \mathrm{h}_n(\mathscr{C}).\end{align*} $$

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

$$ \begin{align*}\Phi^K_1 \colon \mathrm{h}_{1} (\mathscr{C}^K) \to (\mathrm{h}_{1} \mathscr{C})^K\end{align*} $$

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

$$ \begin{align*}\mathrm{h}_0(\operatorname{colim}_K) \colon \mathrm{h}_0 \big(\mathrm{Fun}(K, \mathrm{h}_1 \mathscr{C})\big) \to \mathrm{h}_0(\mathscr{C}),\end{align*} $$

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

(3.26)$$ \begin{align} \mathrm{sk}_0\big(\mathrm{Fun}(K, \mathrm{h}_{ 1} \mathscr{C})\big) \to \mathrm{h}_{ 1}(\mathscr{C}), \end{align} $$

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

$$ \begin{align*}\mathrm{h}_{n-1}\big(\mathrm{Fun}(K, \mathrm{h}_n \mathscr{C})\big) \to \mathrm{h}_{n-1}(\mathscr{C})\end{align*} $$

and partial pushout-functors

$$ \begin{align*}\mathrm{sk}_{n}\big(\mathrm{Fun}(K, \mathrm{h}_n \mathscr{C})\big) \to \mathrm{h}_n(\mathscr{C})\end{align*} $$

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

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:

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

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

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

4. (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:

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

2. (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.)

3. (iii) The full subcategory $\mathsf {Cat_{n}} \subset \mathsf {Cat_{\infty }}$ of n-categories for any $n \geq 1$.

4. (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

$$ \begin{align*}\underline{\mathrm{Hom}}_{\mathsf{Dia}^{\operatorname{op}}}(\mathsf{X}, \mathsf{Y}) = \underline{\mathrm{Hom}}_{\mathsf{Dia}}(\mathsf{Y}, \mathsf{X}) = \mathrm{Fun}(\mathsf{Y}, \mathsf{X}).\end{align*} $$

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

$$ \begin{align*}\mathbb{D} \colon \mathsf{Dia}^{\operatorname{op}} \to \mathsf{Cat_{\infty}}.\end{align*} $$

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

$$ \begin{align*}\mathbb{D} \colon \mathsf{Dia}^{\operatorname{op}} \to \mathsf{Cat_{n}}.\end{align*} $$

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 4.3. 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 $n \geq 1$,

$$ \begin{align*}\mathbb{D}^{(n)}_{\mathscr{C}} \colon \mathsf{Dia}^{\operatorname{op}} \to \mathsf{Cat_{n}}, \ \ \mathsf{X} \mapsto \mathrm{h}_n \big(\mathrm{Fun}(\mathsf{X}, \mathscr{C})\big)\end{align*} $$

defines an n-prederivator.

We define left $\infty $-derivators dually.

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

$$ \begin{align*}\mathbb{D}(-^{\operatorname{op}})^{\operatorname{op}}\colon \mathsf{Dia}^{\operatorname{op}} \to \mathsf{Cat_{\infty}}, \ \mathsf{X} \mapsto \mathbb{D}(\mathsf{X}^{\operatorname{op}})^{\operatorname{op}}\end{align*} $$

is a right $\infty $-derivator.

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

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:

$$ \begin{align*}\mathrm{h}_k \mathbb{D} \colon \mathsf{Dia}^{\operatorname{op}} \to \mathsf{Cat_{n}} \xrightarrow{\mathrm{h}_k} \mathsf{Cat}_k.\end{align*} $$

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. (1) The canonical base change transformation

   $$ \begin{align*}p_! j^* \longrightarrow q^* u_!\end{align*} $$
   is a natural equivalence if u is a cocartesian fibration or if q is a cartesian fibration.

2. (2) The canonical base change transformation

   $$ \begin{align*}q^* u_* \longrightarrow p_* j^*\end{align*} $$
   is a natural equivalence if u is a cartesian fibration or if q is a cocartesian fibration.

Proof. Using the duality $\mathbb {D} \mapsto \mathbb {D}(-^{\operatorname {op}})^{\operatorname {op}}$, it suffices to prove only (1). Indeed, we obtain (2) by applying (1) to the derivator $\mathbb {D}(-^{\operatorname {op}})^{\operatorname {op}}$ in the case of the opposite pullback square in $\mathsf {Dia}$.

Suppose that u is a cocartesian fibration. Applying (Der 2) and using the naturality properties of base change transformations, it suffices to prove the claim only in the case where $W = \Delta ^0$:

Using the following factorization of this square and applying (Der 4), it suffices to prove the claim for the left square in the following diagram:

Note that the horizontal functors in the left square admit left adjoints

$$ \begin{align*}\ell_1 \colon \mathsf{X}_{/ y} \to \mathsf{X}_y\end{align*} $$

$$ \begin{align*}\ell_2 \colon \mathsf{Y}_{/y} \to \Delta^0\end{align*} $$

because u is a cocartesian fibration and $\mathsf {Y}_{/y}$ has a terminal object $(y = y)$. Thus, after applying $\mathbb {D}$, we obtain pairs of adjoint functors $(j^*, \ell _1^*)$ and $(i_{(y=y)}^*, \ell _2^*)$. Therefore, the base change transformation between compositions of left adjoints

(4.12)$$ \begin{align} p_! j^* \longrightarrow i_{(y=y)}^* (p_{u/y})_! \end{align} $$

is conjugate to the natural equivalence

$$ \begin{align*}\mathrm{id} \colon p_{u/y}^* (\ell_2)^* \simeq (\ell_1)^* p^*,\end{align*} $$

which then implies that (4.12) is an equivalence, as required.

Suppose now that q is a cartesian fibration. Then it suffices to show that the conjugate base change transformation

(4.13)$$ \begin{align} u^*q_* \to j_*p^* \end{align} $$

is a natural equivalence. Again, by (Der 2), it suffices to restrict to the case where $X = \Delta ^0$

The proof of the equivalence (4.13) in this case is obtained similarly by dualizing the arguments in the previous proof.

In the context of ordinary derivators, there is a well-known additional axiom (Der 5), which is very useful in practice. Before we state the generalization of this axiom to higher derivators, we define (similarly to the classical case) for every $\infty $-prederivator $\mathbb {D}$ and $\mathsf {X}, \mathsf {Y} \in \mathsf {Dia}$, the underlying $(\mathsf {X}-)$diagram functor

$$ \begin{align*}\mathrm{dia}_{\mathsf{X}, \mathsf{Y}} \colon \mathbb{D}(\mathsf{X} \times \mathsf{Y}) \rightarrow \mathrm{Fun}(\mathsf{X}, \mathbb{D}(\mathsf{Y}))\end{align*} $$

to be the adjoint of the following composition:

$$ \begin{align*}\mathsf{X} \cong \mathrm{Fun}(\Delta^0, \mathsf{X}) \xrightarrow{(- \times \mathsf{Y})} \mathrm{Fun}(\mathsf{Y}, \mathsf{X} \times \mathsf{Y}) \xrightarrow{\mathbb{D}} \mathrm{Fun}(\mathbb{D}(\mathsf{X} \times \mathsf{Y}), \mathbb{D}(\mathsf{Y})).\end{align*} $$

We say that a functor $F\colon \mathscr {C}\to \mathscr {D}$ between $\infty $-categories is n–full if it restricts to $(n-1)$-connected maps between the mapping spaces. For example, a full functor between ordinary categories is $1$-full in this sense.

Remark 4.15. Similarly to the case of ordinary (pre)derivators, we may also consider a stronger form of the last axiom (compare to [Reference Heller19]), which states that the underlying diagram functor

$$ \begin{align*}\mathrm{dia}_{\mathcal{I}, \mathsf{X}} \colon \mathbb{D}(\mathcal{I} \times \mathsf{X}) \rightarrow \mathrm{Fun}(\mathcal{I}, \mathbb{D}(\mathsf{X}))\end{align*} $$

is n-full and essentially surjective for every $\mathcal {I}$ in $\mathsf {Dia}$, which is equivalent in the Joyal model category to a finite 1-dimensional simplicial set. Moreover, the following even stronger form of this axiom fits naturally in the higher categorical context:

1. ($\widetilde {\text {Der 5}_n}$) For every $\mathsf {X} \in \mathsf {Dia}$ and for every $\mathcal {I} \in \mathsf {Dia}$, which is equivalent (in the Joyal model category) to a d-dimensional simplicial set for some $d \leq n + 1$, the underlying diagram functor

   $$ \begin{align*}\mathrm{dia}_{\mathcal{I}, \mathsf{X}} \colon \mathbb{D}(\mathcal{I} \times \mathsf{X}) \rightarrow \mathrm{Fun}(\mathcal{I}, \mathbb{D}(\mathsf{X}))\end{align*} $$
   is $(n+1-d)$-full and essentially surjective.

Note that this axiom for $d = 0$ is also closely related to (Der 1).

Following the definitions of pointed and stable (1-)derivators [Reference Maltsiniotis24, Reference Groth18], we also define pointed and stable $\infty $-(pre)derivators as follows.

4.2 Examples of n-derivators

The examples of n-derivators we are mainly interested in are those where the underlying prederivator arises from an $\infty $-category as in Example 4.3. Let $\mathsf {Dia} \subset \mathsf {Cat_{\infty }}$ be a fixed full subcategory satisfying the conditions (Dia 0)–(Dia 3).

It is often possible to reduce statements about $\infty $-derivators to the corresponding (known) statements about $1$-derivators. This happens, for example, in the case of statements that involve the detection of equivalences. The following proposition shows another instance of this phenomenon and produces many examples of n-derivators from known examples of $1$-derivators.

Proof. (1) $\Rightarrow $ (2) was shown in Proposition 4.17, and the implications (2) $\Rightarrow $ (3) $\Rightarrow $ (4) $\Rightarrow $ (5) are obvious. We prove the implication (5) $\Rightarrow $ (1). We restrict to showing that $\mathscr {C}$ admits $\mathsf {X}$-colimits as the case of $\mathsf {X}$-limits can be treated similarly by duality. Let $F\colon \mathsf {X} \to \mathscr {C}$ be an $\mathsf {X}$-diagram, and let $G \colon \mathsf {X}^{\triangleright } \to \mathscr {C}$ be the diagram that is the image of $F \in \mathbb {D}_{\mathscr {C}}(\mathsf {X})$ under

$$ \begin{align*}u_! \colon \mathbb{D}_{\mathscr{C}}(\mathsf{X}) \longrightarrow \mathbb{D}_{\mathscr{C}}(\mathsf{X}^{\triangleright}),\end{align*} $$

where $u \colon \mathsf {X} \to \mathsf {X}^{\triangleright }$ is the canonical inclusion. As a consequence of (Der 4), the functor $u_!$ is fully faithful because u is so. In particular, there is a canonical isomorphism $F \cong u^* u_!(F),$ and therefore we may assume that G is an extension of the diagram F. We claim that G is a colimit diagram in $\mathscr {C}$ for the functor F. For this, it suffices to prove that for each sieve between (nerves of) finite posets $v \colon \mathsf {Y}' \to \mathsf {Y}$ in $\mathsf {Dia}$, the canonical map

(4.19)

is a $\pi _0$-isomorphism, where $c_y$ denotes the constant functor at $y \in \mathscr {C}$ in the respective functor $\infty $-category. We will do this by first expressing $\pi _0$ of the domain and the target of this map (4.19) in terms of morphism sets in the (1-)category $\mathbb {D}_{\mathscr {C}}(\mathsf {X}^{\triangleright } \times \mathsf {Y})$ and then use the derivator properties of $\mathbb {D}_{\mathscr {C}}$.

First, note that we have a canonical isomorphism of morphism sets

(4.20)$$ \begin{align} \begin{aligned} \mathbb{D}_{\mathscr{C}}(\mathsf{X}^{\triangleright} \times \mathsf{Y})(\pi_{\mathsf{X}^{\triangleright}, \mathsf{Y}}^*(G), c_y) &= \pi_0 \big(\mathrm{Map}_{\mathscr{C}^{\mathsf{X}^{\triangleright} \times \mathsf{Y}}}(\pi_{\mathsf{X}^{\triangleright}, \mathsf{Y}}^*(G), c_y)\big) \\ &\cong \pi_0 \big(\mathrm{Map}(\mathsf{Y}, \mathrm{Map}_{\mathscr{C}^{\mathsf{X}^{\triangleright}}}(G, c_y)) \big) \end{aligned} \end{align} $$

where $\pi _{\mathsf {X}^{\triangleright }, \mathsf {Y}} \colon \mathsf {X}^{\triangleright } \times \mathsf {Y} \to \mathsf {X}$ denotes the projection functor. Similarly, we have canonical isomorphisms

(4.21)$$ \begin{align} \begin{aligned} \mathbb{D}_{\mathscr{C}}(\mathsf{X} \times \mathsf{Y})(\pi_{\mathsf{X}, \mathsf{Y}}^*(F), c_y) \cong \pi_0 \big(\mathrm{Map}(\mathsf{Y}, \mathrm{Map}_{\mathscr{C}^{\mathsf{X}}}(F, c_y)) \big) \\ \mathbb{D}_{\mathscr{C}}(\mathsf{X} \times \mathsf{Y}^{'})(\pi_{\mathsf{X}, \mathsf{Y}^{'}}^*(F), c_y) \cong \pi_0 \big(\mathrm{Map}(\mathsf{Y}^{'}, \mathrm{Map}_{\mathscr{C}^{\mathsf{X}}}(F, c_y)) \big) \end{aligned} \end{align} $$

where $\pi _{\mathsf {X}, \mathsf {Y}}$ and $\pi _{\mathsf {X}, \mathsf {Y}'}$ denote again the projection functors.

Then consider the following pullback diagram in $\mathsf {Dia}$:

Since the horizontal functors are cartesian fibrations, it follows from (Der 4) and Proposition 4.11(1) that the canonical base change morphism in $\mathbb {D}_{\mathscr {C}}(\mathsf {X}^{\triangleright } \times \mathsf {Y})$

(4.22)$$ \begin{align} \pi_{\mathsf{X}^{\triangleright}, \mathsf{Y}}^*(G) = \pi_{\mathsf{X}^{\triangleright}, \mathsf{Y}}^* u_!(F) \stackrel{\cong}{\leftarrow} (u \times 1)_! \pi_{\mathsf{X}, \mathsf{Y}}^*(F) \end{align} $$

is an isomorphism.

Let $i \colon \mathsf {A} = \mathsf {X}^{\triangleright } \times \mathsf {Y}' \cup _{\mathsf {X} \times \mathsf {Y}'} \mathsf {X} \times \mathsf {Y} \subset \mathsf {X}^{\triangleright } \times \mathsf {Y}$ denote the full subcategory. This is again in $\mathsf {Dia}$ by (Dia 1) and (Dia 3) because it can be described as the pullback of a functor $\mathsf {X}^{\triangleright } \times \mathsf {Y} \to \Delta ^1 \times \Delta ^1$ along the ‘upper corner’ inclusion $\Delta ^1 \cup _{\Delta ^0} \Delta ^1 \to \Delta ^1 \times \Delta ^1$. Note that using (4.21), we can identify the set of components of the target of (4.19) canonically with the morphism set

(4.23)$$ \begin{align} \begin{aligned} \mathbb{D}_{\mathscr{C}}(\mathsf{A})(i^* \pi^*_{\mathsf{X}^{\triangleright}, \mathsf{Y}}(G), c_y) = \mathbb{D}_{\mathscr{C}}(\mathsf{A})(i^*\pi_{\mathsf{X}^{\triangleright}, \mathsf{Y}}^*u_!(F), c_y) \\ \cong \pi_0(\textrm{target of } (4.19)) \end{aligned} \end{align} $$

and the map (4.19) on $\pi _0$ agrees using the identifications (4.20) and (4.23) with the map defined by the restriction functor

$$ \begin{align*}i^* \colon \mathbb{D}_{\mathscr{C}}(\mathsf{X}^{\triangleright} \times \mathsf{Y}) \to \mathbb{D}_{\mathscr{C}}(\mathsf{A}).\end{align*} $$

Since i is full, it follows from (Der 4) that the unit transformation $1 \to i^* i_!$ of the adjunction $(i_!, i^*)$ is a natural isomorphism. Therefore it suffices to show that the counit morphism

(4.24)$$ \begin{align} i_! i^* \big( \pi^*_{\mathsf{X}^{\triangleright}, \mathsf{Y}}(G)\big) \to \pi^*_{\mathsf{X}^{\triangleright}, \mathsf{Y}} (G) \end{align} $$

is an isomorphism.

Consider the following pullback diagram in $\mathsf {Dia}$:

The bottom functor q is a cartesian fibration because it is the composition of cartesian fibrations. Therefore, it follows from (Der 4) and Proposition 4.11(1) that the canonical base change morphism in $\mathbb {D}_{\mathscr {C}}(\mathsf {A})$:

(4.25)$$ \begin{align} q^*(G) = q^*u_!(F) \stackrel{\cong}{\leftarrow} j_! \pi_{\mathsf{X}, \mathsf{Y}}^*(F) \end{align} $$

is an isomorphism. As a consequence of (4.22) and (4.25), we obtain canonical isomorphisms as follows:

$$ \begin{align*} \begin{aligned} \pi_{\mathsf{X}^{\triangleright}, \mathsf{Y}}^*(G) = \pi_{\mathsf{X}^{\triangleright}, \mathsf{Y}}^* u_!(F) \cong (u \times 1)_! \pi_{\mathsf{X}, \mathsf{Y}}^*(F) \\ \cong i_! j_! \pi_{\mathsf{X}, \mathsf{Y}}^*(F) \cong i_! q^* u_!(F) \\ = i_! i^* \pi^*_{\mathsf{X}^{\triangleright}, \mathsf{Y}}(G). \end{aligned} \end{align*} $$

This implies that (4.24) is an isomorphism; therefore, using the adjunction $(i_!, i^*)$ as explained above, it follows that the map in (4.19) is a $\pi _0$-isomorphism.

Remark 4.26. We point out a significant simplification of the proof of Theorem 4.18 in the case where $\mathsf {Dia}$ is large enough that it can detect equivalences of $\infty $-groupoids – that is, in the case where the following holds: a map of $\infty $-groupoids $X \to X'$ is an equivalence if and only if

$$ \begin{align*}\pi_0(\mathrm{Map}(\mathsf{Y}, X)) \longrightarrow \pi_0(\mathrm{Map}(\mathsf{Y}, X'))\end{align*} $$

is an isomorphism for every $\mathsf {Y} \in \mathsf {Dia}$. This happens, for example, when $\mathsf {Dia}$ contains all posets – not just the finite ones. Assuming that $\mathsf {Dia}$ is large enough in this sense, then we may restrict to the case $\mathsf {Y}' = \varnothing $ in the proof above, in which case the proof becomes more immediate.

5.1 Preliminaries

We first recall the $\infty $-categorical version of Waldhausen’s $\mathrm {S}_{\bullet }$-construction [Reference Waldhausen39, Reference Blumberg, Gepner and Tabuada3]. Let $\mathscr {C}$ be a pointed $\infty $-category that admits finite colimits. For every $n \geq 0$, let $\mathrm {Ar}[n]$ denote the (nerve of the) category of morphisms of the poset $[n]$. The $\infty $-category $\mathrm {S}_n \mathscr {C}$ is the full subcategory of $\mathrm {Fun}(\mathrm {Ar}[n], \mathscr {C})$ spanned by the objects $F \colon \mathrm {Ar}[n] \to \mathscr {C}$ such that:

1. (i) $F(i \to i)$ is a zero object for all $i \in [n]$.

2. (ii) For every $i \leq j \leq k$, the following diagram in $\mathscr {C}$

   
   is a pushout.

The construction is clearly functorial in $[n]$, $n \geq 0$, and $\mathrm {S}_{\bullet } \mathscr {C}$ defines a simplicial object of pointed $\infty $-categories, which is functorial in $\mathscr {C}$ with respect to functors that preserve zero objects and pushouts.

We denote by $\mathrm {S}_{\bullet }^{\simeq } \mathscr {C}$ the associated simplicial object of pointed $\infty $-groupoids, which is obtained by passing pointwise to the maximal $\infty $-subgroupoids of $\mathrm {S}_{\bullet } \mathscr {C}$. For $n \geq 1$, the $\infty $-groupoid $\mathrm {S}_{n}^{\simeq } \mathscr {C}$ is equivalent to $\mathrm {Map}(\Delta ^{n-1}, \mathscr {C})$. Moreover, we have $\mathrm {S}^{\simeq }_{0} \mathscr {C} \simeq \Delta ^0$, so we may regard the geometric realization $|\mathrm {S}^{\simeq }_{\bullet } \mathscr {C}|$ as canonically pointed by a zero object in $\mathscr {C}$. The Waldhausen K-theory of $\mathscr {C}$ is defined to be the space

$$ \begin{align*}K(\mathscr{C}) : = \Omega | S_{\bullet}^{\simeq} \mathscr{C} |.\end{align*} $$

If $\mathscr {C}$ arises from a nice Waldhausen category, this definition of K-theory agrees up to homotopy equivalence with the Waldhausen K-theory of the corresponding Waldhausen category (see [Reference Blumberg, Gepner and Tabuada3]). The definition of K-theory is functorial with respect to functors $F \colon \mathscr {C} \to \mathscr {C}'$, which preserve the zero object and finite colimits.

Following [Reference Waldhausen39, Lemma 1.4.1] and [Reference Muro and Raptis26, Proposition 4.2.1], we consider also the following simpler model for Waldhausen K-theory. Restricting pointwise to the objects of $\mathrm {S}_{\bullet } \mathscr {C}$, we obtain a simplicial set

$$ \begin{align*}\textsf{s}_{\bullet} \mathscr{C} \colon \Delta^{\operatorname{op}} \to \mathsf{Set}, \ \ [n] \mapsto \textsf{s}_n \mathscr{C} : = (\mathrm{S}_n \mathscr{C})_0.\end{align*} $$

There is a canonical comparison map, given by the inclusion of objects,

$$ \begin{align*}\iota \colon \Omega |\textsf{s}_{\bullet} \mathscr{C}| \longrightarrow \Omega | \mathrm{S}_{\bullet}^{\simeq} \mathscr{C}| = K(\mathscr{C}).\end{align*} $$

5.2 Derivator K-theory

We extend the definition of derivator K-theory of Maltsiniotis [Reference Maltsiniotis24] and Garkusha [Reference Garkusha14, Reference Garkusha15] to general pointed right $\infty $-derivators. As in the case of ordinary derivators, this definition is based on the following intrinsic notion of cocartesian square.

Let $i \colon \ulcorner = \Delta ^1 \cup _{\Delta ^0} \Delta ^1 \to \square = \Delta ^1 \times \Delta ^1$ denote the ‘upper corner’ inclusion. For any right $\infty $-derivator $\mathbb {D}$ (with domain $\mathsf {Dia}$), we have an adjunction

$$ \begin{align*}i_! \colon \mathbb{D}(\ulcorner) \rightleftarrows \mathbb{D}(\square) \colon i^*.\end{align*} $$

Definition 5.2. Let $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ be a right $\infty $-derivator with domain $\mathsf {Dia}$. An object $F \in \mathbb {D}(\square )$ is called cocartesian if the canonical morphism

$$ \begin{align*}i_! i^*(F) \to F\end{align*} $$

is an equivalence in $\mathbb {D}(\square )$.

Let $\mathbb {D}$ be a pointed right $\infty $-derivator (with domain $\mathsf {Dia}$). Before we define the K-theory of $\mathbb {D}$, we first need to introduce some more notation. For every $0 \leq i \leq j \leq k \leq n$, we denote by $i_{i,j,k} \colon \square \to \mathrm {Ar}[n]$ the inclusion of the following square in $\mathrm {Ar}[n]$:

We define $\mathrm {S}_n \mathbb {D}$ to be the full subcategory of $\mathbb {D}(\mathrm {Ar}[n])$, which is spanned by the objects $F \in \mathbb {D}(\mathrm {Ar}[n])$ such that

1. (i) $F_{(i \to i)}$ is a zero object for all $i \in [n]$.

2. (ii) For every $i \leq j \leq k$, the object $i_{i,j,k}^*(F) \in \mathbb {D}(\square )$, which may be depicted as follows

   
   is cocartesian in $\mathbb {D}(\square )$.

The assignment $[n] \mapsto \mathrm {S}_n \mathbb {D}$ defines a simplicial object of pointed $\infty $-categories. Moreover, it is natural with respect to strict morphisms between pointed right $\infty $-derivators that preserve zero objects and cocartesian squares.

Let $\mathrm {S}_{\bullet }^{\simeq } \mathbb {D}$ denote the simplicial object of pointed $\infty $-groupoids, which is obtained by passing pointwise to the maximal $\infty $-subgroupoids of $\mathrm {S}_{\bullet } \mathbb {D}$. We have $\mathrm {S}_0^{\simeq } \mathbb {D} \simeq \Delta ^0$, and we regard the geometric realization $|\mathrm {S}_{\bullet }^{\simeq } \mathbb {D}|$ as based at a zero object of $\mathbb {D}(\Delta ^0)$.

Definition 5.3. Let $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ be a pointed right $\infty $-derivator with domain $\mathsf {Dia}$. The derivator K-theory of $\mathbb {D}$ is defined to be the space

$$ \begin{align*}K(\mathbb{D}) : = \Omega | \mathrm{S}_{\bullet}^{\simeq} \mathbb{D}|.\end{align*} $$

We note that the definition of derivator K-theory is functorial with respect to strict morphisms $F \colon \mathbb {D} \to \mathbb {D}'$, which preserve the zero object and cocartesian squares. Moreover, derivator K-theory is invariant under those strict morphisms that are pointwise equivalences of $\infty $-categories.

This definition of derivator K-theory clearly agrees with the usual derivator K-theory for ordinary derivators [Reference Garkusha14, Reference Muro and Raptis26]. Moreover, this definition of derivator K-theory is also an extension of the K-theory of pointed $\infty $-categories with finite colimits to pointed right $\infty $-derivators; by definition, for any pointed $\infty $-category $\mathscr {C}$ with finite colimits, the K-theory of $\mathscr {C}$ agrees with the derivator K-theory of $\mathbb {D}^{(\infty )}_{\mathscr {C}}$.

5.3 Comparison with Waldhausen K-theory

Let $\mathscr {C}$ be a pointed $\infty $-category with finite colimits. Applying pointwise the homotopy n-category functor to the simplicial object $[k] \mapsto \mathrm {S}^{\simeq }_k \mathscr {C}$, we obtain a new simplicial object of (pointed) $\infty $-groupoids,

$$ \begin{align*}\mathrm{h}_{n} \mathrm{S}_{\bullet}^{\simeq} \mathscr{C}: \Delta^{\operatorname{op}} \to \mathsf{Grpd}_{\infty}, \ [k] \mapsto \mathrm{h}_{n} \big(\mathrm{S}^{\simeq}_k \mathscr{C}\big),\end{align*} $$

and there is a canonical comparison map:

$$ \begin{align*}\mathrm{S}_{\bullet}^{\simeq} \mathscr{C} \longrightarrow \mathrm{h}_n \big(\mathrm{S}_{\bullet}^{\simeq} \mathscr{C}\big).\end{align*} $$

(We note that the n-groupoid $\mathrm {h}_n(\mathscr {D}^{\simeq })$ is the maximal $\infty $-subgroupoid of $\mathrm {h}_n(\mathscr {D})$ for every $\infty $-category $\mathscr {D}$; this can be seen directly from the definitions of $\mathrm {h}_n(-)$ and $(-)^{\simeq }$. As a result, the order of the operations $\mathrm {h}_n(-)$ and $(-)^{\simeq }$ does not play an essential role in the definition of $\mathrm {h}_n \mathrm {S}_{\bullet }^{\simeq } \mathscr {C}$.)

Let $\mathbb {D}^{(n)}_{\mathscr {C}}$ be the pointed right n-derivator associated to $\mathscr {C}$ with domain $\mathcal {D}ir_{f}$ (see Proposition 4.17). As a consequence of the natural identification between $\mathrm {h}_n \big (\mathrm {S}_{\bullet }^{\simeq } \mathscr {C} \big )$ and $\mathrm {S}_{\bullet }^{\simeq } \mathbb {D}^{(n)}_{\mathscr {C}},$ we obtain a canonical comparison map from the Waldhausen K-theory of $\mathscr {C}$ to derivator K-theory:

$$ \begin{align*}\mu_n: K(\mathscr{C}) \to K(\mathbb{D}^{(n)}_{\mathscr{C}}).\end{align*} $$

In addition, the natural morphisms of simplicial objects $\mathrm {h}_n\big (\mathrm {S}_{\bullet }^{\simeq } \mathscr {C}\big ) \to \mathrm {h}_{n-1}\big (\mathrm {S}_{\bullet }^{\simeq } \mathscr {C}\big )$, for $n> 1$, define a tower of derivator K-theories for $\mathscr {C}$ that is compatible with the comparison maps $\mu _n$:

In the case of ordinary derivators, Maltsiniotis conjectured in [Reference Maltsiniotis24] that the comparison map $\mu _1$ is a weak equivalence for exact categories. The comparison map $\mu _1$ was subsequently studied in [Reference Garkusha15, Reference Muro25, Reference Muro and Raptis26, Reference Muro and Raptis27]. It is known (see [Reference Muro and Raptis27]) that $\mu _1$ is not a weak equivalence for general $\mathscr {C}$, and moreover that $\mu _1$ will fail to be a weak equivalence even for exact categories if derivator K-theory satisfies localization – a property that was also conjectured by Maltsiniotis [Reference Maltsiniotis24].

We prove a general result about the connectivity of the comparison map $\mu _n$ for general $\mathscr {C}$ and $n \geq 1$. This connectivity estimate is also a small improvement of the known estimate for $n=1$ that was shown by Muro [Reference Muro25].

We will need the following useful elementary fact about simplicial spaces.

Proof. (of Theorem 5.5) By Lemma 5.6, it suffices to show that the map of $\infty $-groupoids

$$ \begin{align*}\mathrm{S}_k^{\simeq} \mathscr{C} \to \mathrm{h}_{n} \big(\mathrm{S}_k^{\simeq} \mathscr{C}\big)\end{align*} $$

is $(n+2-k)$-connected for all $k \geq 0$. This holds since the map is an equivalence for $k = 0$ and $(n+1)$-connected for $k> 0$ (Example 2.9).

5.4 Waldhausen K-theory of derivators

Waldhausen K-theory for pointed right 1-derivators was defined in [Reference Muro and Raptis26]. It was shown in [Reference Muro and Raptis26] that this K-theory of derivators agrees with the usual Waldhausen K-theory for all well-behaved Waldhausen categories [Reference Muro and Raptis26, Theorem 4.3.1]. We consider an analogous definition of K-theory for general pointed right $\infty $-derivators.

Let $\mathbb {D}$ be a pointed right $\infty $-derivator with domain $\mathsf {Dia}$. Let $\mathrm {S}_{\bullet \bullet } \mathbb {D}$ be the bisimplicial set whose set of $(n,m)$-simplices $\mathrm {S}_{n,m} \mathbb {D}$ is the set of objects

$$ \begin{align*}F \in \operatorname{Ob} \big(\mathbb{D}(\Delta^m \times \mathrm{Ar}[n])\big)\end{align*} $$

such that

1. (1) For every $j\colon [0] \to [m]$, the object $(j \times \mathrm {id})^*(F) \in \operatorname {Ob} \big (\mathbb {D}(\mathrm {Ar}[n])\big )$ is in $\mathrm {S}_n \mathbb {D}$.

2. (2) The underlying diagram functor associated to F,

   $$ \begin{align*}\mathrm{dia}_{\Delta^m, \mathrm{Ar}[n]}(F) \colon \Delta^m \to \mathbb{D}(\mathrm{Ar}[n]),\end{align*} $$
   takes values in equivalences.

The bisimplicial operators of $\mathrm {S}_{\bullet \bullet } \mathbb {D}$ are again defined using the structure of the underlying $\infty $-prederivator. Moreover, it is easy to see that the construction is functorial in $\mathbb {D}$ with respect to strict morphisms that preserve the zero objects and cocartesian squares. We regard the geometric realization $| \mathrm {S}_{\bullet \bullet } \mathbb {D}|$ as based at a zero object of $\mathbb {D}(\Delta ^0)$.

Definition 5.9. Let $\mathbb {D} \colon \mathsf {Dia}^{\operatorname {op}} \to \mathsf {Cat_{\infty }}$ be a pointed right $\infty $-derivator with domain $\mathsf {Dia}$. The Waldhausen K-theory of $\mathbb {D}$ is defined to be the space

$$ \begin{align*}K^W(\mathbb{D}) := \Omega | \mathrm{S}_{\bullet \bullet} \mathbb{D}|.\end{align*} $$

Following [Reference Muro and Raptis26], we also consider the analogue of the $\textsf {s}_{\bullet }$-construction in this context. Restricting pointwise to the objects of $\mathrm {S}_{\bullet } \mathbb {D}$, we obtain a simplicial set

$$ \begin{align*}\textsf{s}_{\bullet} \mathbb{D} \colon \Delta^{\operatorname{op}} \to \mathsf{Set}, \ \ [n] \mapsto \textsf{s}_n \mathbb{D} := \mathrm{S}_{n, 0} \mathbb{D} = (\mathrm{S}_n \mathbb{D})_0.\end{align*} $$

There is a canonical comparison map, given by the inclusion of $0$-simplices,

$$ \begin{align*}\iota \colon \Omega |\textsf{s}_{\bullet} \mathbb{D}| \longrightarrow \Omega | \mathrm{S}_{\bullet \bullet} \mathbb{D}| = K^W(\mathbb{D}).\end{align*} $$

This is the analogue of the comparison map in Proposition 5.1 for pointed right $\infty $-derivators.

Proposition 5.10. (a) The comparison map $\iota $ is a weak equivalence. (b) Let $\mathscr {C}$ be a pointed $\infty $-category with finite colimits, and let $n \in \mathbb {Z}_{\geq 1} \cup \{\infty \}$. There is a commutative diagram of weak equivalences:

5.5 Universal property of derivator K-theory

The comparison maps $\{\mu _n\}$ from Waldhausen K-theory to derivator K-theory can be defined more generally for pointed right $\infty $-derivators. Given a pointed right $\infty $-derivator $\mathbb {D}$ (with domain $\mathsf {Dia}$), the underlying diagram functors define a bisimplicial map as follows,

$$ \begin{align*}\mathrm{dia}_{\Delta^m, \mathrm{Ar}[n]} \colon \mathrm{S}_{n,m} \mathbb{D} \to (\mathrm{S}_{n}^{\simeq} \mathbb{D})_m,\end{align*} $$

which after passing to the geometric realization and taking loop spaces defines a comparison map:

$$ \begin{align*}\mu \colon \Omega |\textsf{s}_{\bullet} \mathbb{D}| \stackrel{(5.10)}{\simeq} K^W(\mathbb{D}) \to K(\mathbb{D}).\end{align*} $$

A universal property of this comparison map in the case of $1$-derivators was shown in [Reference Muro and Raptis26, Theorem 5.2.2]. More specifically, it was shown that $\mu $ is homotopically initial among all natural transformations from Waldhausen K-theory to a functor that is invariant under (pointwise) equivalences of pointed right derivators. The proof of this universal property extends similarly to our present $\infty $-categorical context.

Let $\mathsf {Der}$ denote the (ordinary) category of pointed right $\infty $-derivators and strict morphisms that preserve the zero object and cocartesian squares. It will be convenient to work with the simpler model for the Waldhausen K-theory of derivators given by the $\textsf {s}_{\bullet }$-construction. This will be denoted by

$$ \begin{align*}K^{W, \operatorname{Ob}} \colon \mathsf{Der} \to \mathrm{Top}, \ \ \mathbb{D} \mapsto \Omega | \textsf{s}_{\bullet} \mathbb{D}|,\end{align*} $$

where $\mathrm {Top}$ denotes the ordinary category of topological spaces. Then we may regard the comparison map $\mu $ as a natural transformation $K^{W, \operatorname {Ob}} \to K$ between functors defined on $\mathsf {Der}$.

Definition 5.11. The category $\mathscr {E}$ of invariant approximations to Waldhausen K-theory is the full subcategory of the comma category $K^{W, \operatorname {Ob}} \downarrow \mathrm {Top}^{\mathsf {Der}}$ spanned by the objects $(\eta \colon K^{W, \operatorname {Ob}} \to F)$ such that $F \colon \mathsf {Der} \to \mathrm {Top}$ sends pointwise equivalences in $\mathsf {Der}$ to weak equivalences. A morphism in $\mathscr {E}$

is a weak equivalence if the components of u are weak equivalences.

We recall from [Reference Dwyer, Hirschhorn, Kan and Smith10] that an object $x \in \mathcal {C}$ in an (ordinary) category with weak equivalences $(\mathcal {C}, \mathcal {W})$ (satisfying in addition the ‘2-out-of-6’ property) is homotopically initial if there are functors $F_0, F_1 \colon \mathcal {C} \to \mathcal {C}$ that preserve the weak equivalences and a natural transformation $\phi \colon F_0 \Rightarrow F_1$ such that

1. (i) $F_0$ is naturally weakly equivalent to the constant functor at $x \in \mathcal {C}$.

2. (ii) $F_1$ is naturally weakly equivalent to the identity functor on $\mathcal {C}$.

3. (iii) $\phi _x \colon F_0(x) \to F_1(x)$ is a weak equivalence.

A homotopically initial object in $(\mathcal {C}, \mathcal {W})$ defines an initial object in the associated $\infty $-category.

Proof. (Sketch) The proof is similar to [Reference Muro and Raptis26, Theorem 5.2.2], so we only give a sketch of the proof. Given $\mathbb {D} \in \mathsf {Der}$ and $m \geq 0$, let $\mathbb {D}^{\simeq }_{m}$ denote the $\infty $-prederivator whose value at $\mathsf {X} \in \mathsf {Dia}$ is the full subcategory $\mathrm {Fun}_{\simeq }(\Delta ^m, \mathbb {D}(\mathsf {X})) \subset \mathrm {Fun}(\Delta ^m, \mathbb {D}(\mathsf {X}))$ spanned by the functors $\Delta ^m \to \mathbb {D}(\mathsf {X})$, which take values in equivalences. This $\infty $-prederivator is pointwise equivalent to $\mathbb {D}$ and therefore also a pointed right $\infty $-derivator. Varying $m \geq 0$, we obtain a simplicial object $(\mathbb {D}^{\simeq }_{m})_{m \geq 0}$ in $\mathsf {Der}$ with $\mathbb {D}^{\simeq }_0 = \mathbb {D}$.

For the proof of the theorem, it suffices to note that every object in $\mathscr {E}$,

$$ \begin{align*}(\eta_{\mathbb{D}} \colon K^{W, \operatorname{Ob}}(\mathbb{D}) \to F(\mathbb{D}))_{\mathbb{D} \in \mathsf{Der}}\end{align*} $$

is naturally weakly equivalent (as object in $\mathscr {E}$) to the composite functor

$$ \begin{align*}\big(K^{W, \operatorname{Ob}}(\mathbb{D}) \to ||K^{W, \operatorname{Ob}}(\mathbb{D}^{\simeq}_{\bullet})|| \xrightarrow{||\eta||} ||F(\mathbb{D}^{\simeq}_{\bullet})||\big)_{\mathbb{D} \in \mathsf{Der}}.\end{align*} $$

This uses the fact that F respects pointwise equivalences in $\mathsf {Der}$. Moreover, the first map above defines a natural transformation that is canonically identified with $\mu $. As a result, we have constructed a zigzag of natural transformations from the constant endofunctor at $\mu $ to $\mathrm {id}_{\mathscr {E}}$ satisfying (i)–(iii).

6.1 Revisiting the properties of homotopy n-categories

Let $\mathscr {C}$ be a pointed $\infty $-category with finite colimits. Then the associated homotopy n-category $\mathrm {h}_n \mathscr {C}$ satisfies the following:

1. (a) $\mathrm {h}_n \mathscr {C}$ is a pointed n-category.

2. (b) The suspension functor $\Sigma _{\mathscr {C}} \colon \mathscr {C} \to \mathscr {C}$ induces a functor $\Sigma \colon \mathrm {h}_n \mathscr {C} \to \mathrm {h}_n \mathscr {C}$. This is an equivalence if and only if $\mathscr {C}$ is stable (see [Reference Lurie21, Corollary 1.4.2.27]).

3. (c) $\mathrm {h}_n \mathscr {C}$ admits finite coproducts and weak pushouts of order $n-1$. Moreover, these are preserved by the functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ (Proposition 3.20).

4. (d) For every $x \in \mathrm {h}_n \mathscr {C}$, there is a natural weak pushout of order $n-1$,

   

Assuming that $\mathscr {C}$ is a stable $\infty $-category, then the adjoint equivalence $(\Sigma _{\mathscr {C}}, \Omega _{\mathscr {C}})$ induces an adjoint equivalence $\Sigma \colon \mathrm {h}_n \mathscr {C} \rightleftarrows \mathrm {h}_n \mathscr {C} \colon \Omega $. Moreover, by duality, $\mathrm {h}_n \mathscr {C}$ also satisfies in this case the following dual versions of (c)–(d):

1. (c)^′ $\mathrm {h}_n \mathscr {C}$ admits finite products and weak pullbacks of order $n-1$. Moreover, these are preserved by the functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$.

2. (d)^′ For every $x \in \mathrm {h}_n \mathscr {C}$, there is a natural weak pullback of order $n-1$

   

In addition, if $\mathscr {C}$ is a stable $\infty $-category, $\mathrm {h}_n \mathscr {C}$ satisfies the following property:

1. (e) A square in $\mathrm {h}_n \mathscr {C}$ is a weak pushout of order $n-1$ if and only if it is a weak pullback of order $n-1$.