# Skeleta and categories of algebras

Jonathan Beardsley, Tyler Lawson

###### Abstract

We define a notion of a connectivity structure on an $\infty$-category,
analogous to a $t$-structure but applicable in unstable contexts—such as
spaces, or algebras over an operad. This allows us to generalize notions of
n-skeleta, minimal skeleta, and cellular approximation from the category of
spaces. For modules over an Eilenberg–Mac Lane spectrum, these are closely
related to the notion of projective amplitude.

We apply these to ring spectra, where they can be detected via the cotangent
complex and higher Hochschild homology with coefficients. We show that the
spectra $Y(n)$ of chromatic homotopy theory are minimal skeleta for
$H\mathbb{F}_{2}$ in the category of associative ring spectra. Similarly,
Ravenel’s spectra $T(n)$ are shown to be minimal skeleta for $BP$ in the same
way, which proves that these admit canonical associative algebra structures.

###### Contents

  1. 1 Introduction
    1. 1.1 Main applications
    2. 1.2 Outline
    3. 1.3 Notation and conventions
    4. 1.4 Acknowledgements
  2. 2 Connectivity structures
    1. 2.1 Definitions
    2. 2.2 Examples
    3. 2.3 Basic properties of connectivity
    4. 2.4 Stable connectivity structures
    5. 2.5 Stabilization
    6. 2.6 Cartesian squares
  3. 3 Skeleta
    1. 3.1 Skeletal objects
    2. 3.2 Colimits of skeletal objects
    3. 3.3 Skeleta and minimal skeleta
    4. 3.4 Cells
    5. 3.5 Cellular skeleta
  4. 4 Detecting skeleta
    1. 4.1 Nilpotence
    2. 4.2 Tangent categories
  5. 5 Lifting constructions of skeleta
    1. 5.1 Subductivity
    2. 5.2 Excavation
    3. 5.3 Excavating cells
    4. 5.4 Excavating skeleta
  6. 6 Excision
    1. 6.1 Excisive connectivity structures
    2. 6.2 Excision and augmentations
    3. 6.3 Excision and suspension
  7. 7 Applications
    1. 7.1 Skeleta in derived categories
    2. 7.2 Skeleta for spaces
    3. 7.3 Algebras in module categories
    4. 7.4 Remark on Thom spectra
    5. 7.5 $T(n)$ as associative skeleta
    6. 7.6 $Y(n)$ as associative skeleta
    7. 7.7 Involutions

##  1 Introduction

Within homotopy theory, it is commonly understood (if not always made
explicit) that the homology groups of a space $X$ are closely coupled with how
$X$ can be built as a CW-complex. If $X$ is equivalent to a CW-complex, then
$C_{*}(X)$ is quasi-isomorphic to a chain complex with one free generator for
each cell in the CW-structure. For 1-connected spaces $X$ we can do better,
because a converse holds: if $\widetilde{C}_{*}(X)$ is quasi-isomorphic to a
complex that is levelwise free, then there exists a CW-complex equivalent to
$X$ with one cell for each generator.

The engine that makes this technique possible is that we can understand the
relation between connectivity and cell attachment. For maps of simply-
connected spaces, homology detects connectivity. Further, suppose we have
built the $n$-skeleton for a CW-approximation of $X$: an $n$-connected map
$X^{(n)}\to X$. The set of possible ways to “extend this map an $(n+1)$-cell”
is governed by the relative homotopy group $\pi_{n+1}(X,X^{(n)})$; similarly,
given a map of chain complexes $C\to D$, the set of possible ways to “extend
this map to an $(n+1)$-cell” is governed by the relative homology group
$H_{n+1}(D,C)$. The map

$\pi_{n+1}(X,X^{(n)})\to H_{n+1}(C_{*}X,C_{*}X^{(n)})$

is an isomorphism by the relative Hurewicz theorem, making it possible to lift
the attaching map for each $(n+1)$-dimensional generator uniquely from the
chain level to the space level.

With $1$-connected rational or $p$-complete spaces, the same argument works
but our job is easier: there exists a rational or $p$-complete CW-structure on
$X$ with one cell for each element in a basis of $H_{*}(X;\mathbb{Q})$ or
$H_{*}(X;\mathbb{F}_{p})$ respectively. This argument also works for
connective spectra, and this is a core component of understanding objects in
_stable_ homotopy theory via their homology. This perspective also makes it
clear that, because the Blakers–Massey excision theorem can be used to give an
isomorphism

$\pi_{n}(X,X^{(n)})\to\pi_{n+1}(\Sigma X,\Sigma X^{(n)}),$

it is possible to lift (up to homotopy equivalence) any CW-structure from
$\Sigma X$ to $X$.

This paper studies these types of cell attachments in other categories. Our
main application is to demonstrate that certain spectra in chromatic homotopy
theory have natural multiplicative structures: specifically, that they arise
as “skeleta” in cellular constructions of known ring spectra. The desire to
having an analogous theory of homology for commutative ring spectra, detecting
cell attachment, was one motivation for the development of topological
André–Quillen homology by Kriz, Basterra, and Mandell [Kri95, Bas99, BM05].
However, our first goal will be to understand exactly what it means to say
“$K$ is an $n$-skeleton of $X$” in a rather broad generality.

To say that $K$ is an $n$-skeleton of $X$ in the terms above relies on the
existence of a particular type of _construction_ of $X$ that produces $K$ at
some stage. We hope to demonstrate that many of the useful homotopy-theoretic
properties of a skeleton $K$ can be derived from the definition of
connectivity of a map.

######  Definition.

An object $K$ is _$k$ -skeletal_ if it has the left lifting property with
respect to $k$-connected maps: for any $k$-connected map $X\to Y$, every map
$K\to Y$ lifts to a map $K\to X$.

A map $f\colon\thinspace K\to X$ makes $K$ into a _$k$ -skeleton_ of $X$ if
$K$ is $k$-skeletal and $f$ is $k$-connected.

For example, a space is $k$-skeletal if and only if (in the homotopy category)
it is a retract of a $k$-dimensional CW-complex. However, this definition is
cell-free, interacts well with adjunctions, and has well-behaved
generalizations even if we adopt a flexible notion of what “connectivity”
means. When we _can_ construct objects inductively with cells, the stages will
naturally be skeleta. More, because the definition of skeletal objects is in
terms of lifting properties, we can often detect $k$-skeletal objects by
testing analogues of cohomology.

This will allow us to give criteria under which the construction of a skeleton
can be lifted from one category to another. Our engine for lifting CW-
structures in terms of relative homotopy and homology is a special property of
an _adjunction_ between two categories. The general notion that we introduce
is that of a _subductive adjunction_ : one where a $k$-connected map $X\to Y$
induces a $k$-Cartesian square

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL(Y).}$

Subductive adjunctions capture the situation where attaching maps for cells
can be lifted along the functor $L$. If, instead, $k$-connected maps are taken
to $(k+1)$-Cartesian squares, we have a _strongly subductive adjunction_ ,
where attaching maps lift uniquely.111The term “subductive” is loosely
borrowed from geology; we think of the adjunction as an interaction between
two tectonic plates, with the left adjoint
$L\colon\thinspace\mathcal{C}\to\mathcal{D}$ gradually pushing objects from
$\mathcal{C}$ under the surface. Even though this process can (and typically
does) lose information from the category $\mathcal{C}$, it is still gradual
enough that (with enough effort) we can gradually dig up information about
objects and maps to reassemble them.

###  1.1 Main applications

The proofs of the nilpotence and periodicity theorems by
Devinatz–Hopkins–Smith make heavy use of ring spectra $X(n)$, built as Thom
spectra on the spaces $\Omega SU(n)$. These assemble into a directed system

$\mathbb{S}=X(1)\to X(2)\to X(3)\to\dots\to\operatorname{MU}$

of ring spectra (specifically, $\mathbb{E}_{2}$ ring spectra) and on integral
homology this realizes a filtration of $H_{*}\operatorname{MU}$ by polynomial
subalgebras,

$\mathbb{Z}\to\mathbb{Z}[x_{1}]\to\mathbb{Z}[x_{1},x_{2}]\to\dots\to
H_{*}\operatorname{MU}$

where $|x_{i}|=2i$. When working $p$-locally, however,
$\operatorname{MU}_{(p)}$ has a split summand $\operatorname{BP}$ that is more
tractable for computational applications, and the filtration of
$\operatorname{MU}$ by $X(n)$ can be replaced by a filtration

$\mathbb{S}_{(p)}=T(0)\to T(1)\to T(2)\to\dots\to\operatorname{BP}.$

On homology this realizes a filtration of $H_{*}\operatorname{BP}$,

$\mathbb{Z}_{(p)}\to\mathbb{Z}_{(p)}[t_{1}]\to\mathbb{Z}_{(p)}[t_{1},t_{2}]\to\dots\to
H_{*}\operatorname{BP},$

where $|t_{i}|=2p^{i}-2$. However, the spectrum $T(n)$ is usually constructed
as a split summand of $X(p^{n})_{(p)}$ and this splitting does not grant it
structured multiplication.

We will show that this sequence can be realized in a _canonical_ way as a
sequence of associative ring spectra. We know that $\operatorname{BP}$ exists
and has an associative structure, and we can construct a cellular
approximation by attaching cells inductively—but within the category of
_associative ring spectra_ , rather than just within the category of spectra.
We will show that the $T(n)$ arise quite naturally as skeleta of
$\operatorname{BP}$ as an associative algebra, and that the necessary cells
are detected by topological Hochschild homology with coefficients. (Cellular
constructions, in various categories, of objects related to
$\operatorname{BP}$ are by no means new: see, for example, work of Priddy
[Pri80], Hu–Kriz–May [HKM01], or Baker [Bak14].) By similar methods, it is
also possible to show that the $2$-primary spectra $Y(n)$ of
Mahowald–Ravenel–Shick [MRS01] arise as skeleta of the Eilenberg–Mac Lane
spectrum for $\mathbb{F}_{2}$, although this can be shown more directly using
their construction as Thom spectra.

###  1.2 Outline

In Section 2 we will introduce the notion of a _connectivity structure_ on
$\mathcal{C}$, generalizing the standard notion of “$k$-connectedness” for
spaces and spectra, and prove several properties. In the case where
$\mathcal{C}$ is stable, the most common way to get a connectivity structure
is to have a $t$-structure on $\mathcal{C}$. Several of our categories of
interest, such as categories of algebras, are not stable, but they still
inherit connectivity structures from a forgetful functor to an underlying
stable category.

Section 3 introduces _skeleta_ and _skeletal objects_ in $\mathcal{C}$,
defined in terms of lifting against sufficiently connected maps, and proves
basic properties. We will also discuss the notion of _cells_ in $\mathcal{C}$
and give a version of CW-approximation when there is a sufficient supply of
cells. We will introduce _minimal skeleta_ , which (when they exist) are
unique up to homotopy equivalence.

Section 4 discusses the _detection_ of skeletal objects. The chief application
is to show that tools like stabilization can be used to detect skeletality so
long as $k$-connected maps admit a Postnikov-like decomposition.

Section 5, by contrast, discusses _lifting_ skeleta across an adjunction with
left adjoint $L$. We introduce subductive adjunctions and how they allow
analogues of lifting CW-structures: inductively, we can lift attaching maps of
cells along $L$ from the target category to the source category. This allows
us to obtain results lifting the construction of skeleta, and minimal skeleta,
along $L$.

Section 6 discusses _excision_ theorems, and how their presence often makes
stabilization into a subductive adjunction. This makes it possible to employ
homology theories for ring spectra, such as those introduced by Basterra and
Mandell [BM11].

Finally, Section 7 discusses our applications. For the derived category of a
ring, we find that skeletality is closely related to the notion of _projective
amplitude_ , and for spaces we find that $k$-skeletality is equivalent to a
constraint on the projective amplitude for chains on the universal cover.

We then work with algebra spectra, showing that cellular constructions of
associative algebras are governed by topological Hochschild homology. For
these, we need to make heavy use Ching–Harper’s results on excision for
categories of algebras over an operad.

###  1.3 Notation and conventions

Throughout this paper, we make the assumption that $\mathcal{C}$ is an
$\infty$-category in the sense of [Lur09] that admits pullbacks. We write
$h\mathcal{C}$ for the associated homotopy category.

###  1.4 Acknowledgements

The authors would like to thank Jeremy Hahn for being a close part of
questions and discussions that instigated this project, and Clark Barwick, Tim
Campion, Ian Coley, and Denis Nardin for help with material related to this
paper.

This work is an outgrowth of conversations that took place while the authors
were attending the workshop “Derived algebraic geometry and chromatic homotopy
theory” during the Homotopy Harnessing Higher Structures program at the Isaac
Newton Institute (EPSRC grant numbers EP/K032208/1 and EP/R014604/1). The
first author’s travel to the workshop was funded by NSF grant DMS-1833295. The
first author was also partially supported by NSF grant DMS-1745583 while
working on this paper. The second author was partially supported by NSF grant
DMS-1610408.

##  2 Connectivity structures

###  2.1 Definitions

######  Definition 2.1.

Suppose that $\mathcal{C}$ is an $\infty$-category with homotopy pullbacks. A
_connectivity structure_ $T$ on $\mathcal{C}$ consists of, for each
$k\in\mathbb{Z}$, a collection of _$k$ -connected maps_ in the homotopy
category $h\mathcal{C}$, subject to the following axioms.

  1. 1.

A map that is $k$-connected is also $(k-1)$-connected.

  2. 2.

Equivalences are $k$-connected for all $k$.

  3. 3.

The homotopy pullback of a $k$-connected map is $k$-connected.

  4. 4.

Suppose we have composable maps $f\colon\thinspace X\to Y$ and
$g\colon\thinspace Y\to Z$.

    1. (a)

If $f$ and $g$ are $k$-connected, so is $gf$.

    2. (b)

If $f$ is $(k-1)$-connected and $gf$ is $k$-connected, then $g$ is
$k$-connected.

    3. (c)

If $gf$ is $k$-connected and $g$ is $(k+1)$-connected, then $f$ is
$k$-connected.

When specifying the connectivity structure is necessary, we refer to a map as
$k$-connected in $T$.

A functor $F\colon\thinspace\mathcal{C}\to\mathcal{D}$ _preserves
connectivity_ if, whenever $f$ is $k$-connected, the image $F(f)$ is
$k$-connected.

######  Remark 2.2.

For convenience, we will refer to $f$ as $\infty$-connected if it is
$k$-connected for all $k$. Similarly, we will refer to all maps as
$(-\infty)$-connected. We define a connectivity structure to be _left
complete_ if an $\infty$-connected map is an equivalence.222This is by analogy
with $t$-structures: see §2.4.

The definition of a connectivity structure is invariant under translation.

######  Definition 2.3.

For $d\in\mathbb{Z}$, the connectivity structure $(T+d)$ obtained by _shifting
$T$ by $d$_ is defined as follows. A map $f$ is $k$-connected in $T$ if and
only if $f$ is $(k+d)$-connected in $(T+d)$.

A functor $F\colon\thinspace\mathcal{C}\to\mathcal{D}$ _adds (at least) $d$ to
connectivity_ if, whenever $f$ is $k$-connected, the image $F(f)$ is
$(k+d)$-connected.333In other words, it is a connectivity-preserving functor
$(\mathcal{C},T_{\mathcal{C}}+d)\to(\mathcal{D},T_{\mathcal{D}})$.

######  Definition 2.4.

If $\\{T_{i}\\}$ is a collection of connectivity structures, the _intersection
connectivity structure_ $\cap T_{i}$ is defined as follows. A map $f$ is
$k$-connected in $\cap T_{i}$ if and only if, for all $i$, $f$ is
$k$-connected in $T_{i}$.

######  Definition 2.5.

Suppose that $R\colon\thinspace\mathcal{D}\to\mathcal{C}$ is a functor that
preserves homotopy pullbacks and that $\mathcal{C}$ has a connectivity
structure $T$. Then there is a connectivity structure $R^{-1}(T)$ on
$\mathcal{D}$, defined as follows. A map $f\colon\thinspace X\to Y$ in
$\mathcal{D}$ is $k$-connected in $R^{-1}(T)$ if and only if the image $Rf$ is
$k$-connected in $T$. We refer to this as the connectivity structure _lifted_
from $\mathcal{C}$, and say that the functor $R$ _reflects connectivity_.

This is possible whenever $R$ is a right adjoint, such as a forgetful functor.
The lifted structure is always the maximal structure such that $R$ preserves
connectivity.

######  Definition 2.6.

If $\mathcal{C}$ has homotopy pushouts, a connectivity structure on
$\mathcal{C}$ is _compatible with cobase change_ if the homotopy pushout of
any $k$-connected map is $k$-connected.

Suppose that $\mathcal{C}$ has $K$-indexed homotopy colimits. A connectivity
structure on $\mathcal{C}$ is _compatible with $K$-indexed (homotopy)
colimits_ if, whenever a natural transformation $F\to G$ of $K$-indexed
diagrams is $k$-connected objectwise, the map
$\operatorname*{hocolim}_{K}F\to\operatorname*{hocolim}_{K}G$ is
$k$-connected.

###  2.2 Examples

######  Example 2.7.

Every $\mathcal{C}$ has available the _minimal_ connectivity structure
$T_{min}$, where only equivalences are $k$-connected for $k\neq-\infty$. It
also has the _maximal_ connectivity structure $T_{max}$, where all maps are
$\infty$-connected.

Every functor out of the minimal connectivity structure preserves
connectivity; every functor into the maximal connectivity structure preserves
connectivity.

######  Example 2.8.

A map $f\colon\thinspace X\to Y$ in the category $\operatorname{\mathcal{S}}$
of spaces is $k$-connected if (for all choices of basepoint) $\pi_{j}(f)$ is
an isomorphism for all $j<k$, and $\pi_{k}(f)$ is a surjection.

######  Example 2.9.

A morphism $f\colon\thinspace X\to Y$ of objects in the category
$\operatorname{Sp}$ of spectra is called $k$-connected if $\pi_{j}(f)$ is an
isomorphism for all $j<k$, and $\pi_{j}(f)$ is a surjection.

More generally, if $\mathcal{C}$ is stable and has a $t$-structure, we can
give $\mathcal{C}$ a connectivity structure by declaring that a map
$f\colon\thinspace X\to Y$ is $k$-connected if its cofiber
$\operatorname*{cofib}(f)$ is $k$-connected: $\tau_{\leq
k}\operatorname*{cofib}(f)$ is trivial (cf. Sections 2.4 and 2.5).

######  Remark 2.10.

By [Lur17, 1.2.1.6], if $\mathcal{C}$ is stable and the connectivity structure
on $\mathcal{C}$ is determined by a $t$-structure, then the connectivity
structure is compatible with cobase change and all homotopy colimits that
exist in $\mathcal{C}$.

######  Example 2.11.

If $\mathcal{C}_{i}$ have connectivity structures, then the product
$\prod_{i}\mathcal{C}_{i}$ has a _product connectivity structure_ : a map
$(X_{i})\to(Y_{i})$ is $k$-connected when each map $X_{i}\to Y_{i}$ is. This
is the intersection of the connectivity structures lifted from each
$\mathcal{C}_{i}$.

######  Example 2.12.

If $\mathfrak{O}$ is an (ordinary) operad, the category of
$\mathfrak{O}$-algebras in $\mathcal{C}$ (if it has pullbacks) has a
connectivity structure lifted from $\mathcal{C}$: the forgetful functor
preserves all homotopy limits that exist in $\mathcal{C}$.

######  Example 2.13.

If $\mathcal{C}$ has a connectivity structure, then the slice and coslice
categories $\mathcal{C}_{/Y}$ and $\mathcal{C}_{X/}$ have a connectivity
structure lifted from $\mathcal{C}$.

######  Example 2.14.

Suppose $\mathcal{X}$ is an $\infty$-topos in the sense of [Lur09]. Then
$\mathcal{X}$ admits a connectivity structure where a morphism is
$k$-connected precisely when it is $(k+1)$-connective in the sense of [Lur09,
6.5.1.10]. Specifically, a morphism $f\colon X\to Y$ in $\mathcal{C}$ is
$k$-connected if it is an effective epimorphism as in [Lur09, 6.2.3.5] and if
$\pi_{i}(f)\simeq\ast$ (for the toposic homotopy groups of [Lur09, 6.5.1]) for
all $0\leq i\leq k$. The first three axioms of Definition 2.1, as well as part
(a) of the fourth axiom, are satisfied as a result of [Lur09, Proposition
6.5.1.16].

For parts (b) and (c) of the fourth axiom, recall that by [Lur09, 6.1.0.6] any
$\infty$-topos $\mathcal{X}$ is a left exact localization of a presheaf
$\infty$-topos
$\mathcal{P}(\mathcal{C})=Fun(\mathcal{C},\operatorname{\mathcal{S}})$. If we
say that $\mathcal{X}\simeq L\mathcal{P}(\mathcal{C})$ for a localization
functor $L$ then a morphism $f\in\mathcal{X}$ is $k$-connected if and only if
it is $k$-connected in $\mathcal{P}(\mathcal{C})$. One direction of this
assertion follows from [Lur09, 6.5.1.15], i.e. if $f\simeq L(f_{0})$ is
$k$-connected then so is $f_{0}$. For the other direction, notice that the
left exact left adjoint $L$ preserves colimits and truncations (cf. [Lur09,
5.5.6.28]) so preserves $k$-connectedness (since $k$-connectedness in an
$\infty$-topos can be determined by truncations [Lur09, 6.5.1.12]). In the
presheaf topos $\mathcal{P}(\mathcal{C})$, $k$-truncation, and therefore
$k$-connectedness, is determined “pointwise” in $\operatorname{\mathcal{S}}$,
the statements follow from their usual proofs in $\operatorname{\mathcal{S}}$.

Note that by [ABFJ20, 3.6.6] the definition of a $k$-connected morphism in an
$\infty$-topos $\mathcal{X}$ described above agrees with the definition of a
$k$-connected morphism given in Example 2.8 when
$\mathcal{X}\simeq\operatorname{\mathcal{S}}$.

######  Remark 2.15.

There is a discrepancy between the definition of $k$-connected in an
$\infty$-topos given in [Lur09] and in our notion of a connectivity structure
in that, in an $\infty$-topos, the concept of being $k$-connected only makes
sense for $k\geq-2$. However we can extend the notion of $k$-connectedness in
an $\infty$-topos described in Example 2.14 by saying that all morphisms are
$k$-connected for any $k\leq-2$.

###  2.3 Basic properties of connectivity

######  Proposition 2.16.

Suppose that we have a retract diagram $A\to X\to A$. Then the map $A\to X$ is
$k$-connected if and only if the map $X\to A$ is $(k+1)$-connected.

###### Proof.

This follows immediately from the second axiom and items (b) and (c) of the
fourth axiom of Definition 2.1. ∎

######  Proposition 2.17.

Suppose that we have a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}}$$\textstyle{Z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the middle vertical map is $(k+1)$-connected and the outer vertical maps
are $k$-connected. Then the induced map of homotopy pullbacks $X\times_{Y}Z\to
X^{\prime}\times_{Y^{\prime}}Z^{\prime}$ is $k$-connected.

###### Proof.

Consider the composite map

$X\times_{Y}Z\to X^{\prime}\times_{Y^{\prime}}Z\to
X^{\prime}\times_{Y^{\prime}}Z^{\prime}.$

We will show that both maps are $k$-connected, which implies the desired
result.

The map $X^{\prime}\times_{Y^{\prime}}Z\to
X^{\prime}\times_{Y^{\prime}}Z^{\prime}$ is the pullback of the $k$-connected
map $Z\to Z^{\prime}$ along the map $X^{\prime}\to Y^{\prime}$, and so it is
$k$-connected.

The map $X^{\prime}\times_{Y^{\prime}}Y\to X^{\prime}$ is the pullback of a
$(k+1)$-connected map $Y\to Y^{\prime}$ along $X^{\prime}\to Y^{\prime}$, and
so it is $(k+1)$-connected. The composite $X\to
X^{\prime}\times_{Y^{\prime}}Y\to X^{\prime}$ is $k$-connected. Therefore, the
map $X\to X^{\prime}\times_{Y^{\prime}}Y$ is $k$-connected. Taking pullback
along the map $Z\to Y$, we find that the map $X\times_{Y}Z\to
X^{\prime}\times_{Y^{\prime}}Z$ is $k$-connected. ∎

By considering the special case where $X=Y=Z=X^{\prime}=Z^{\prime}$, we arrive
at the following.

######  Corollary 2.18.

If $f\colon\thinspace X\to Y$ is $(k+1)$-connected, then the induced map
$\Delta\colon\thinspace X\to X\times_{Y}X$ is $k$-connected.

######  Remark 2.19.

If $\mathcal{C}$ is an $\infty$-topos then [Lur09, 6.5.1.18] gives a partial
converse to the above corollary. Specifically, if the map $\Delta$ is
$k$-connected and also an effective epimorphism then $f$ is $(k+1)$-connected.

As another special case, we can take $Y=Y^{\prime}$ and $Z=Z^{\prime}$.

######  Corollary 2.20.

Let $f\colon\thinspace Z\to Y$ be a morphism in a category $\mathcal{C}$ with
a connectivity structure. Then the pullback functor
$f^{\ast}\colon\thinspace\mathcal{C}_{/Y}\to\mathcal{C}_{/Z}$ preserves
connectivity (for the connectivity structures lifted from $\mathcal{C}$).

The following is a dual argument to Proposition 2.17.

######  Proposition 2.21.

Suppose that the connectivity structure on $\mathcal{C}$ is compatible with
cobase change, and that we have a commutative diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime}}$

where the middle vertical map is $(k-1)$-connected and the outer vertical maps
are $k$-connected. Then the induced map of homotopy pushouts $A\amalg_{B}C\to
A^{\prime}\amalg_{B^{\prime}}C^{\prime}$ is $k$-connected.

###### Proof.

Consider the composite map

$A\amalg_{B}C\to A\amalg_{B}C^{\prime}\to
A^{\prime}\amalg_{B^{\prime}}C^{\prime}.$

We will show that both maps are $k$-connected, which implies the desired
result.

The map $A\amalg_{B}C\to A\amalg_{B}C^{\prime}$ is the homotopy pushout of the
$k$-connected map $C\to C^{\prime}$ along the map $B\to A$, and so it is
$k$-connected.

The map $A\to A\amalg_{B}B^{\prime}$ is the homotopy pushout of a
$(k-1)$-connected map $B\to B^{\prime}$ along $B\to A$, and so it is
$(k-1)$-connected. The composite $A\to A\amalg_{B}B^{\prime}\to A^{\prime}$ is
$k$-connected. Therefore, the map $A\amalg_{B}B^{\prime}\to A^{\prime}$ is
$k$-connected. Taking homotopy pushouts along the map $B^{\prime}\to
C^{\prime}$, we find that the map $A\amalg_{B}C^{\prime}\to
A^{\prime}\amalg_{B^{\prime}}C^{\prime}$ is $k$-connected. ∎

######  Corollary 2.22.

Suppose that $\mathcal{C}$ has homotopy pushouts and that the connectivity
structure on $\mathcal{C}$ is compatible with cobase change. Given any map
$A\to B$, the functor
$B\coprod_{A}(-)\colon\thinspace\mathcal{C}_{A/}\to\mathcal{C}_{B/}$ preserves
connectivity.

######  Corollary 2.23.

Suppose that $\mathcal{C}$ has homotopy pushouts and that the connectivity
structure on $\mathcal{C}$ is compatible with cobase change. If $X\to Y$ is a
$(k-1)$-connected map of objects over $Z$, then the induced map
$Z\amalg_{X}Z\to Z\amalg_{Y}Z$ is $k$-connected.

###  2.4 Stable connectivity structures

Suppose $\mathcal{C}$ is pointed. Proposition 2.17 shows that whenever
$f\colon\thinspace X\to Y$ is $k$-connected, $\Omega f\colon\thinspace\Omega
X\to\Omega Y$ is $(k-1)$-connected: the loop operator $\Omega$ adds at least
$(-1)$ to connnectivity. In the case of a stable category, $\Omega$ is an
autoequivalence, and it is common for connectivity structures to satisfy a
converse.

######  Definition 2.24.

Suppose that $\mathcal{C}$ is stable. A _stable_ connectivity structure on
$\mathcal{C}$ is a connectivity structure such that $f\colon\thinspace X\to Y$
is $k$-connected if and only if $\Omega f\colon\thinspace\Omega X\to\Omega Y$
is $(k-1)$-connected.

In the category of spectra, we can detect connectivity by examining
connectivity of the cofiber. This property is always possible with stable
connectivity structures.

######  Definition 2.25.

Let $\mathcal{C}$ be a pointed category with a connectivity structure. Then we
say that $X\in\mathcal{C}$ is $k$-connected if the map from the zero object
$\ast\to X$ is $k$-connected.

######  Remark 2.26.

Note that the zero object of a pointed category will be $\infty$-connected.

######  Lemma 2.27.

If $\mathcal{C}$ is pointed, then the fiber of any $k$-connected morphism is
$(k-1)$-connected.

###### Proof.

Let $f\colon\thinspace C\to D$ be a $k$-connected morphism in $\mathcal{C}$
and consider the pullback square:

$\textstyle{\operatorname*{fib}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$

By pullback stability the morphism $p$ is $k$-connected, so the zero morphism
$\ast\to\operatorname*{fib}(f)$ is $(k-1)$-connected by Proposition 2.16. ∎

The above definition allows us to give another characterization of the
$k$-connected morphisms in a stable connectivity structure, since stable
categories always have zero objects.

######  Proposition 2.28.

If $\mathcal{C}$ has a stable connectivity structure, then a map
$f\colon\thinspace X\to Y$ is $k$-connected if and only if the cofiber
$\operatorname*{cofib}(f)$ is $k$-connected.

###### Proof.

Because $\mathcal{C}$ is stable, we have the following two homotopy pullback
squares:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega\operatorname*{cofib}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname*{cofib}(f)}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

The left-hand pullback, along with pullback stability of connectivity, shows
that if $\operatorname*{cofib}(f)$ is $k$-connected, then the map $f$ is
$k$-connected.

For the converse, consider the right-hand pullback. It shows that if $f$ is
$k$-connected, then $\Omega\operatorname*{cofib}(f)$ is $(k-1)$-connected by
Lemma 2.27. By stability, the map $\ast\to\operatorname*{cofib}(f)$ is
$k$-connected. ∎

######  Remark 2.29.

Stability of $\mathcal{C}$ and Proposition 2.28 of course immediately also
imply that a map $f\colon\thinspace X\to Y$ is $k$-connected if and only if
$\operatorname*{fib}(f)$ is $(k-1)$-connected, giving a converse to Lemma
2.27.

######  Remark 2.30.

Given a stable connectivity structure on $\mathcal{C}$ we can define the
subcategory of connective objects, $\mathcal{C}_{\geq 0}$, as the full
subcategory of all objects which are $(-1)$-connected. It is not hard to check
that $\mathcal{C}_{\geq 0}$ satisfies the following conditions:

  1. 1.

$\mathcal{C}_{\geq 0}$ contains a zero object;

  2. 2.

$\mathcal{C}_{\geq 0}$ is closed under extensions; and

  3. 3.

$\mathcal{C}_{\geq 0}$ is closed under cofibers.

Conversely, given a stable $\infty$-category $\mathcal{C}$, a full subcategory
$\mathcal{D}\subseteq\mathcal{C}$ determines a stable connectivity structure
on $\mathcal{C}$ so long as it satisfies these three conditions. The third
condition can be replaced by closure under the suspension operator $\Sigma$,
because a cofiber sequence $X\to Y\to\operatorname*{cofib}(f)$ is equivalent
data to an extension $Y\to\operatorname*{cofib}(f)\to\Sigma X$.

More specifically, we can say that a morphism $f\colon\thinspace X\to Y$ is
$k$-connected, for each $k\in\mathbb{Z}$, exactly when
$\operatorname*{cofib}(f)\in\Sigma^{k+1}\mathcal{D}$. Checking that this
determines a connectivity structure on $\mathcal{C}$ is straightforward for
the first three axioms. All three parts of the fourth axiom can be checked by
recalling that if $\mathcal{C}$ is stable then $h\mathcal{C}$ is triangulated
and applying the octahedral axiom.

To see that this connectivity structure is stable, notice that the suspension
functor $\Sigma$ is an equivalence, so preserves cofibers. Thus
$f\colon\thinspace X\to Y$ is $k$-connected, i.e.
$\operatorname*{cofib}(f)\in\Sigma^{k+1}\mathcal{D}$, if and only if
$\operatorname*{cofib}(\Sigma^{-1}f)\simeq\Sigma^{-1}\operatorname*{cofib}(f)\in\Sigma^{k}\mathcal{D}$,
i.e. the desuspended morphism
$\Sigma^{-1}f\colon\thinspace\Sigma^{-1}X\to\Sigma^{-1}Y$ is
$(k-1)$-connected. It also follows immediately that an object
$X\in\mathcal{C}$ is $k$-connected in the sense of Definition 2.25 if and only
if $X\in\Sigma^{k+1}\mathcal{D}$.

######  Example 2.31.

Let $E\in\operatorname{Sp}$ be a spectrum. Then there is a connectivity
structure on $\operatorname{Sp}$ determined by declaring the subcategory of
connective objects to be precisely those spectra $X$ such that the
$E$-homology groups $E_{i}(X)=\pi_{i}(E\otimes X)$ are trivial for all $i<0$.

###  2.5 Stabilization

Recall that a functor is reduced if it preserves terminal objects and excisive
if takes pushout squares to pullback squares. If $\mathcal{C}$ is an
$\infty$-category with finite limits then there is a category
$\operatorname{Sp}(\mathcal{C})$ of _spectrum objects of $\mathcal{C}$_
[Lur17, 1.4.2.8]. Explicitly, $\operatorname{Sp}(\mathcal{C})$ is the full
subcategory
$\operatorname{Exc}_{\ast}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C})\subseteq\operatorname{Fun}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C})$
of reduced, excisive functors from finite pointed spaces to $\mathcal{C}$.

######  Remark 2.32.

For any $\infty$-category $\mathcal{C}$ with finite limits there are functors
$ev_{d}\colon\thinspace\operatorname{Sp}(\mathcal{C})\to\mathcal{C}$ for all
$d\geq 0$ given by evaluation on the $d$-spheres
$S^{d}\in\operatorname{\mathcal{S}_{\ast}^{\text{fin}}}$ as in [Lur17,
1.4.2.20]. For $d>0$ functors can be equivalently defined by composing
$ev_{0}$ with $\Sigma^{d}$. We will often write $\Omega^{\infty}$ instead of
$ev_{0}$.

######  Definition 2.33.

Let $\mathcal{C}$ be an $\infty$-category with finite limits and a
connectivity structure, and let $\operatorname{Sp}(\mathcal{C})$ be its
associated category of spectra. Say that an object $X$ of
$\operatorname{Sp}(\mathcal{C})$ is connective if for each $d\geq 0$,
$ev_{d}(X)$ is $(d-1)$-connected in $\mathcal{C}_{\ast}$. Let
$\operatorname{Sp}(\mathcal{C})_{\geq 0}$ denote the full subcategory of
$\operatorname{Sp}(\mathcal{C})$ spanned by the connective objects.

######  Proposition 2.34.

If $\mathcal{C}$ has finite limits and a connectivity structure then
$\operatorname{Sp}(\mathcal{C})_{\geq
0}\subseteq\operatorname{Sp}(\mathcal{C})$ determines a stable connectivity
structure on $\operatorname{Sp}(\mathcal{C})$ as in Remark 2.30.

###### Proof.

We only need to check the three conditions of Remark 2.30.

First, the zero object of $\operatorname{Sp}(\mathcal{C})$ is the functor
given by $K\mapsto\ast$ for every
$K\in\operatorname{\mathcal{S}_{\ast}^{\text{fin}}}$. It follows from Remark
2.26 that its image is contained in $\operatorname{Sp}(\mathcal{C})_{\geq 0}$.

Now let $X\to Y\to Z$ be a fiber sequence in $\operatorname{Sp}(\mathcal{C})$
with $X,Z\in\operatorname{Sp}(\mathcal{C})_{\geq 0}$, and fix $d\geq 0$. Then
we have a pullback square

$\textstyle{X(S^{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y(S^{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z(S^{d}),}$

in which the bottom horizontal morphism is $(d-1)$-connected. By pullback
stability we have the top horizontal morphism is also $(d-1)$-connected.
Because $X(S^{d})$ is $(d-1)$-connected, the zero map $\ast\to X(S^{d})$ is
$(d-1)$-connected, so the composite $\ast\to X(S^{d})\to Y(S^{d})$ is
$(d-1)$-connected, and therefore $Y(S^{d})$ is $(d-1)$-connected.

Finally, we show that $\operatorname{Sp}(\mathcal{C})_{\geq 0}$ is closed
under the shift operator $\Sigma$. Recall that for a reduced excisive functor
$X\in\operatorname{Sp}(\mathcal{C})$ we have $\Sigma X(K)\simeq X(K\wedge
S^{1})$ (this follows for instance from [Lur17, 1.4.2.13 (2)]). If $X(S^{d})$
is $(d-1)$-connected for all $d$, then $(\Sigma X)(S^{d})\simeq X(S^{d+1})$ is
$d$-connected (and therefore $(d-1)$-connected) for all $d$. ∎

Recall the following definition from [Lur17, 6.1.1.6]:

######  Definition 2.35.

An $\infty$-category $\mathcal{C}$ is differentiable if it satisfies the
following conditions:

  1. 1.

$\mathcal{C}$ admits finite limits;

  2. 2.

$\mathcal{C}$ admits sequential colimits; and

  3. 3.

sequential colimits commute with finite limits.

######  Lemma 2.36.

Suppose that $\mathcal{C}$ is differentiable and has finite colimits. Then the
functor
$\Omega^{\infty}\colon\thinspace\operatorname{Sp}(\mathcal{C})\to\mathcal{C}$
has a left adjoint $\Sigma^{\infty}$.

###### Proof.

This is [Lur17, 6.2.3.16], but we give an explicit description of the functor
$\Sigma^{\infty}$ for future use. The functor described in [Lur17] is a
composite. An object $C\in\mathcal{C}_{\ast}$ determines a functor
$f_{C}\in\operatorname{Fun}(\operatorname{\mathcal{S}^{\text{fin}}},\mathcal{C}_{\ast})$
by declaring $f_{C}(\ast)\simeq C$ and extending by homotopy colimits. This
may be extended to a functor
$f_{C}^{+}\in\operatorname{Fun}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C})$
by setting

$f_{C}^{+}(K)\simeq f_{C}(K)\amalg_{C}\ast$

where the morphism $C\simeq f_{C}(\ast)\to f_{C}(K)$ is determined by the base
point of $K$. This gives a functor
$\mathcal{C}_{\ast}\to\operatorname{Fun}_{\ast}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C})$,
left adjoint to evaluation on $S^{0}$. There is an inclusion
$\operatorname{Exc}_{\ast}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C})\hookrightarrow\operatorname{Fun}_{\ast}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C})$
whose left adjoint is given by the excisive approximation $F\mapsto P_{1}F$
(cf. the proof of [Lur17, 6.2.1.12]). Thus the functor $\Sigma^{\infty}$ takes
$C$ to the functor $P_{1}f_{C}^{+}$. ∎

######  Remark 2.37.

When $\mathcal{C}$ is also pointed, [Lur17, 6.1.1.28] gives an equivalence
$P_{1}F\simeq\operatorname*{colim}_{n}\Omega^{n}\circ F\circ\Sigma^{n}$, where
$\Omega^{n}$ is formed in $\mathcal{C}$ and $\Sigma^{n}$ is formed in
$\operatorname{\mathcal{S}_{\ast}^{\text{fin}}}$. Thus $\Sigma^{\infty}$ is
the composite functor

$C\mapsto\operatorname*{colim}_{n}\left(\Omega^{n}\circ
f_{C}^{+}\circ\Sigma^{n}\right).$

The fact that $\mathcal{C}$ is pointed also implies an equivalence
$f_{C}^{+}(S^{d})\simeq\Sigma^{d}C$ giving the familiar formula

$(\Sigma^{\infty}C)(S^{d})\simeq\operatorname*{colim}_{n}\Omega^{n}\Sigma^{n+d}C.$

In particular,
$\Omega^{\infty}\Sigma^{\infty}C\simeq\operatorname*{colim}_{n}\Omega^{n}\Sigma^{n}C$.

######  Proposition 2.38.

Suppose that $\mathcal{C}$ is pointed, differentiable, and has finite
colimits, and that the connectivity structure on $\mathcal{C}$ is compatible
with cobase change and sequential colimits. Then the stabilization functor
$\Sigma^{\infty}\colon\thinspace\mathcal{C}\to\operatorname{Sp}(\mathcal{C})$
preserves connectivity.

###### Proof.

By Proposition 2.28 it suffices to show that if $g\colon\thinspace C\to D$ is
$k$-connected in $\mathcal{C}$ then $\operatorname*{cofib}(\Sigma^{\infty}g)$
is $k$-connected in $\operatorname{Sp}(\mathcal{C})$. By stability, we may
equivalently show that
$\operatorname*{fib}(\Sigma^{\infty}g)\simeq\Omega\operatorname*{cofib}(\Sigma^{\infty}g)$
is $(k-1)$-connected. To say that $\operatorname*{fib}(\Sigma^{\infty}g)$ is
$(k-1)$-connected is equivalent to saying that
$\Omega^{k}\operatorname*{fib}(\Sigma^{\infty}g)\in\operatorname{Sp}(\mathcal{C})_{\geq
0}$, i.e. $ev_{d}\Omega^{k}\operatorname*{fib}(\Sigma^{\infty}g)$ is
$(d-1)$-connected in $\mathcal{C}$ for all $d\geq 0$. But from [Lur17,
1.4.2.20] we have that

$ev_{d}\Omega^{k}\operatorname*{fib}(\Sigma^{\infty}g)\simeq\Omega^{\infty}\Sigma^{d}\Omega^{k}\operatorname*{fib}(\Sigma^{\infty}g)\simeq
ev_{d-k}\operatorname*{fib}(\Sigma^{\infty}g)$

because $\Sigma$ and $\Omega$ are homotopy inverses in
$\operatorname{Sp}(\mathcal{C})$. Now notice that by the description of the
functor $\Sigma^{\infty}$ given in Lemma 2.36 and Remark 2.37, we can write
$ev_{d-k}\Sigma^{\infty}g$ as the colimit of the morphisms

$\Omega^{n}f_{C}^{+}(S^{d-k+n})\simeq\Omega^{n}\Sigma^{d-k+n}C\to\Omega^{n}\Sigma^{d-k+n}D\simeq\Omega^{n}f_{D}^{+}(S^{d-k+n})$

where $\Omega$ is being applied in $\mathcal{C}$ rather than
$\operatorname{Sp}(\mathcal{C})$. We have the following string of
equivalences:

$\displaystyle\operatorname*{fib}\left(\operatorname*{colim}_{n}\left(\Omega^{n}\Sigma^{d-k+n}C\to\Omega^{n}\Sigma^{d-k+n}D\right)\right)$
$\displaystyle\simeq\operatorname*{colim}_{n}\left(\operatorname*{fib}\left(\Omega^{n}\Sigma^{d-k+n}C\to\Omega^{n}\Sigma^{d-k+n}D\right)\right)$
$\displaystyle\simeq\operatorname*{colim}_{n}\left(\Omega^{n}\operatorname*{fib}\left(\Sigma^{d-k+n}C\to\Sigma^{d-k+n}D\right)\right)$

in which the first equivalence uses the fact that finite limits commute with
sequential colimits in a differentiable category. By Corollary 2.23 we have
that $\Sigma^{d-k+n}C\to\Sigma^{d-k+n}D$ is $(d+n)$-connected, so by Lemma
2.27 we have that
$\operatorname*{fib}\left(\Sigma^{d-k+n}C\to\Sigma^{d-k+n}D\right)$ is
$(d+n-1)$-connected. The result follows from Proposition 2.17 and the fact
that the connectivity structure of $\mathcal{C}$ is compatible with sequential
colimits. ∎

In the unpointed case, Corollary 2.22 allows us to compose with a disjoint
basepoint functor $X\mapsto X\amalg\ast$.

######  Corollary 2.39.

Suppose that $\mathcal{C}$ is differentiable, admits pushouts, and has an
initial object. Suppose that the connectivity structure on $\mathcal{C}$ is
compatible with cobase change and sequential colimits. Then the stabilization
functor
$\Sigma^{\infty}_{+}\colon\thinspace\mathcal{C}\to\operatorname{Sp}(\mathcal{C})$
preserves connectivity.

Finally, in the relative case, we have the following assembly of the above
results.

######  Theorem 2.40.

Suppose that $\mathcal{C}$ is differentiable, admits pushouts, and has an
initial object. Suppose that the connectivity structure on $\mathcal{C}$ is
compatible with cobase change and sequential colimits. Then, for any object
$Z\in\mathcal{C}$, the stabilization functor

$\Sigma^{\infty}_{Z,+}\colon\thinspace\mathcal{C}_{/Z}\to\operatorname{Sp}(\mathcal{C}_{/Z})$

and its pointed variant $\Sigma^{\infty}_{Z}$ preserve connectivity.

###  2.6 Cartesian squares

######  Definition 2.41.

A commutative square

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W}$

in $\mathcal{C}$ is $k$-Cartesian if the induced map $X\to Y\times_{W}Z$ to
the homotopy pullback is $k$-connected (cf. [Goo92, 1.3]).444More generally,
just as in the classical case we have notions of $k$-Cartesian cubes, and
$k$-coCartesian cubes, in terms of connectivity of maps to total homotopy
pullbacks and from total homotopy pushouts

As expected, there is a “stacking lemma” for $k$-Cartesian squares.

######  Proposition 2.42.

Given a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{\prime}}$

with both squares $k$-Cartesian, the outside rectangular diagram is also
$k$-Cartesian.

###### Proof.

The map $X\to Z\times_{Z^{\prime}}X^{\prime}$ is the composite

$X\to
Y\times_{Y^{\prime}}X^{\prime}\to(Z\times_{Z^{\prime}}Y^{\prime})\times_{Y^{\prime}}X^{\prime}\simeq
Z\times_{Z^{\prime}}X^{\prime}.$

The first map is $k$-connected, and the second is the pullback of a
$k$-connected map. ∎

######  Remark 2.43.

We say that _pullbacks reflect connectivity_ in $\mathcal{C}$ if the
connectivity of any pullback of $f$ is the same as the connectivity of $f$.
For example, this is true of stable connectivity structures by Proposition
2.28.

When pullbacks reflect connectivity, the proof of Proposition 2.42 can be
upgraded to show that Cartesianness determines a connectivity structure on the
arrow category of $\mathcal{C}$.

##  3 Skeleta

###  3.1 Skeletal objects

######  Definition 3.1.

An object $A$ is _$k$ -skeletal_ if, for any $k$-connected map $X\to Y$, every
map $A\to Y$ in $h\mathcal{C}$ lifts to a map $A\to X$.

A map $A\to B$ is _relatively $k$-skeletal_ if it is $k$-skeletal when viewed
as a map in the undercategory $\mathcal{C}_{A/}$.

######  Remark 3.2.

If $\mathcal{C}$ has an initial object $\emptyset$, then an object $A$ is
$k$-skeletal if and only if the map $\emptyset\to A$ is relatively
$k$-skeletal.

######  Remark 3.3.

Relative $k$-skeletality can be phrased as a lifting property: given a
commutative diagram in $\mathcal{C}$

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{Y}$

where $\gamma$ is $k$-connected, we ask for the existence of a lift $B\to X$
together with coherences. This asks that the map to the homotopy pullback

$\operatorname{Map}(B,X)\to\operatorname{Map}(B,Y)\times_{\operatorname{Map}(A,Y)}\operatorname{Map}(A,X)$

is surjective on $\pi_{0}$.

More explicitly, we can consider $\delta$ to be a point of the mapping space
$\mathcal{C}_{A/}(\beta,\gamma\alpha)$. Note that, given some
$\phi\in\mathcal{C}_{A/}(\beta,\alpha)$, we can compose with $\gamma$ to get a
point $\gamma\phi\in\mathcal{C}_{A/}(\beta,\gamma\alpha)$, i.e. there is a map
of spaces
$\gamma_{\ast}\colon\thinspace\mathcal{C}_{/A}(\beta,\alpha)\to\mathcal{C}_{A/}(\beta,\gamma\alpha)$.
By [Lur09, 5.5.5.12] the fiber over $\delta$ of this map is precisely the
space of maps from $\alpha$ to $\beta$ in $(\mathcal{C}_{A/})_{/\delta}$.
Therefore relative $k$-skeletality can be rephrased as asking for
$\operatorname{Lift}(\beta,\alpha)$, the mapping space in
$(\mathcal{C}_{A/})_{/\delta}$, to be non-empty. Or, equivalently, for the
morphism $\gamma_{\ast}$ to be be surjective on $\pi_{0}$. This should be
compared to the discussion regarding unique lifts in [Lur09, 5.2.8.3].

######  Example 3.4.

The sphere $S^{k}$ is $k$-skeletal in the category
$\operatorname{\mathcal{S}}$: a $k$-connected map $X\to Y$ is a surjection on
$\pi_{0}$ and on $\pi_{k}$ at any basepoint.

######  Proposition 3.5.

If $A$ is $k$-skeletal and $X\to Y$ is $(k+d)$-connected for $d\geq 0$, then
the map of function spaces $\operatorname{Map}(A,X)\to\operatorname{Map}(A,Y)$
is $d$-connected.

###### Proof.

Recall the following inductive characterization of connectedness: a map $U\to
V$ of topological spaces is $0$-connected if it is surjective on $\pi_{0}$,
and it is $d$-connected for $d>0$ if it is $0$-connected and the map
$U\to\operatorname*{holim}(U\to V\leftarrow U)$ to the homotopy fiber product
is $(d-1)$-connected (cf. for instance [Lur09, 6.5.1.18]).

We now prove the proposition by induction on $d$. Asking that the map
$\operatorname{Map}(A,X)\to\operatorname{Map}(A,Y)$ be $0$-connected is the
same as asking that every map $A\to Y$ in $h\mathcal{C}$ lifts to $X$, which
is implied by the definition of $k$-skeletality.

Therefore, to complete the induction we need to show that if $X\to Y$ is
$(k+d)$-connected, the diagonal map

$\operatorname{Map}(A,X)\to\operatorname*{holim}(\operatorname{Map}(A,X)\to\operatorname{Map}(A,Y)\leftarrow\operatorname{Map}(A,X))$

is $(d-1)$-connected. However, the functor $\operatorname{Map}(A,-)$ preserves
homotopy limits; therefore, this is equivalent to asking that the map

$\operatorname{Map}(A,X)\to\operatorname{Map}(A,\operatorname*{holim}(X\to
Y\leftarrow X)),$

induced by the diagonal $\Delta\colon\thinspace X\to\operatorname*{holim}(X\to
Y\leftarrow X)$, is $(d-1)$-connected. However, by Corollary 2.18 the map
$\Delta$ is $k+(d-1)$-connected, and so the inductive hypothesis completes the
proof. ∎

######  Corollary 3.6.

If $A$ is $k$-skeletal and $f\colon\thinspace X\to Y$ is $(k+1)$-connected,
then every map $A\to Y$ in $h\mathcal{C}$ lifts uniquely to a map $A\to X$ in
$h\mathcal{C}$.

###### Proof.

The map of spaces $\operatorname{Map}(A,X)\to\operatorname{Map}(A,Y)$ is an
isomorphism on $\pi_{0}$, which equivalently says that
$\operatorname{Hom}_{h\mathcal{C}}(A,X)\to\operatorname{Hom}_{h\mathcal{C}}(A,Y)$
is an isomorphism. ∎

######  Definition 3.7.

Let $\phi\colon\thinspace X\to Y$ and $f\colon\thinspace A\to Y$ be maps in
$\mathcal{C}$. Then we define $\operatorname{Lift}(f,\phi)$ to be the
following homotopy pullback:

$\textstyle{\operatorname{Lift}(f,\phi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Map}(A,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\ast}}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\operatorname{Map}(A,Y).}$

######  Remark 3.8.

By [Lur09, 5.5.5.12] the space $\operatorname{Lift}(f,\phi)$ is precisely the
space of morphisms from $f$ to $\phi$ in $\mathcal{C}_{/Y}$.

######  Proposition 3.9.

Let $L\colon\thinspace\mathcal{C}\to\mathcal{D}$ be a functor with a right
adjoint $R$. If $R$ preserves $k$-connected morphisms then $L$ preserves
$k$-skeletal objects.

###### Proof.

Let $A$ be $k$-skeletal in $\mathcal{C}$ and suppose we have a diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{L(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Y}$

in $\mathcal{D}$. Then by adjunction we have a diagram

$\textstyle{R(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R(f)}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{adj(\phi)}$$\textstyle{R(Y)}$

in $\mathcal{C}$ and by the hypothesis, $R(f)$ is $k$-connected. Therefore
there is a lift

$\textstyle{R(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R(f)}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{adj(\phi)}$$\scriptstyle{\widetilde{adj(\phi)}}$$\textstyle{R(Y)}$

by the $k$-skeletality of $A$. Applying $L$ to above diagram and using the
counit natural transformation of the adjunction, we obtain the following
commutative diagram in $h\mathcal{D}$:

$\textstyle{LR(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{LR(f)}$$\scriptstyle{\epsilon_{X}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{L(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L(adj(\phi))}$$\scriptstyle{L(\widetilde{adj(\phi)})}$$\textstyle{LR(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{Y}}$$\textstyle{Y}$

However, the lower composite is $\phi$. ∎

######  Corollary 3.10.

The left adjoint $L$ preserves relatively $k$-skeletal morphisms.

###### Proof.

The adjunction between $L$ and $R$ induces an adjunction between the slice
categories $\mathcal{C}_{A/}$ and $\mathcal{D}_{L(A)/}$ by [Lur09, 5.2.5.1],
and we can apply the proposition to the slice categories. ∎

###  3.2 Colimits of skeletal objects

A standard result about left-lifting properties implies the following.

######  Proposition 3.11.

Relatively $k$-skeletal maps are closed under equivalences, coproducts,
composition, pushout, and retracts.

######  Proposition 3.12.

A coproduct of objects is $k$-skeletal if all the summands are $k$-skeletal.

###### Proof.

Let $\\{A_{j}\\}_{j\in J}$ be a collection of $k$-skeletal objects, and
suppose $\phi\colon\thinspace X\to Y$ is a $k$-connective morphism. Let
$i_{j}\colon\thinspace A_{j}\to\coprod A_{j}$ be the standard inclusions and
let $f\colon\thinspace\coprod A_{j}\to Y$ be any map so that $\coprod f\circ
i_{j}\simeq f$ in $\mathcal{C}_{/Y}$. By Remark 3.8 we have that

$\operatorname{Lift}(f,\phi)\simeq\operatorname{Lift}\left(\amalg_{j}f\circ
i_{j},\phi\right)\simeq\prod_{j}\operatorname{Lift}(f\circ i_{j},\phi).$

Because all $A_{j}$ are $k$-skeletal, the right-hand side is nonempty;
therefore, the left-hand side is nonempty. ∎

######  Remark 3.13.

This result can be upgraded to an if-and-only-if statement if _enough maps
exist_ : if there exist maps $A_{j}\to Y$ for every object $Y$. For example,
this is true if $\mathcal{C}$ is pointed.

######  Proposition 3.14.

Suppose that we have a homotopy pushout diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{B}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{C}}$$\textstyle{P.}$

If $B$ and $C$ are $k$-skeletal and $A$ is $(k-1)$-skeletal, then the homotopy
pushout $P$ is $k$-skeletal.

###### Proof.

Let $f\colon\thinspace X\to Y$ be a $k$-connected map, $\phi\colon\thinspace
P\to Y$ any map, and consider the following diagram:

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{B}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Y}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{C}}$

Because $B$ and $C$ are $k$-skeletal, there are lifts $\widetilde{\phi
i_{B}}\colon\thinspace B\to X$ and $\widetilde{\phi i_{C}}\colon\thinspace
C\to X$. We can obtain lifts of $\phi i_{B}p$ and $\phi i_{C}q$ to $X$ by
simply composing $p$ and $q$ with $\widetilde{\phi i_{B}}$ and
$\widetilde{\phi i_{C}}$ respectively. Using the fact that $f$ is not just
$(k-1)$-connected, but $k$-connected, and Proposition 3.6, both of these lifts
must agree in $h\mathcal{C}$ because they all lift $\phi i_{B}p\simeq\phi
i_{C}q$. Call that unique lift $\psi\colon\thinspace A\to X$. Since
$\psi\simeq\widetilde{\phi i_{B}}p\simeq\widetilde{\phi i_{C}}q$, the
universal property of $P$ induces a lift $\widetilde{\phi}\colon\thinspace
P\to X$. ∎

###  3.3 Skeleta and minimal skeleta

######  Definition 3.15.

A map $f\colon\thinspace A\to X$ is a _$k$ -skeleton_ if $A$ is $k$-skeletal
and $f$ is $k$-connected. If $f$ is $(k+1)$-connected, we refer to this as a
_minimal $k$-skeleton_.

Similarly, a factorization $A\to B\to X$ is a _relative $k$-skeleton_ if it is
a $k$-skeleton in the undercategory $\mathcal{C}_{A/}$.

######  Remark 3.16.

Notice that saying a factorization $A\to B\to X$ of a map $A\to X$ is a
relative $k$-skeleton is equivalent to saying that $A\to B$ is relatively
$k$-skeletal and that $B\to X$ is $k$-connected. We will show in Section 3.5
that in good cases, every map of $\mathcal{C}$ admits such a factorization. In
other words, the classes of relatively $k$-skeletal maps and $k$-connected
maps are the left and right classes of a weak factorization system,
respectively.

######  Example 3.17.

Every $k$-skeletal object is a minimal $k$-skeleton of itself.

######  Example 3.18.

Suppose that $A$ is a CW-complex with $k$-skeleton $A^{(k)}$. Then these
spaces are built, inductively, by a sequence of (homotopy) pushout diagrams:

$\textstyle{\coprod
S^{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{(k-1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod
D^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{(k)}}$

By Proposition 3.12, the space $\coprod S^{k-1}$ is $(k-1)$-skeletal and the
space $\coprod D^{k}$ is $0$-skeletal. Inductively assuming that $A^{(k-1)}$
is $(k-1)$-skeletal, then by Proposition 3.14 we find that $A^{(k)}$ is
$k$-skeletal.

The maps $A^{(k)}\to A$ are then $k$-skeleta. More generally, if $X$ is a
space and $A\to X$ is a weak equivalence from a CW-complex, then the maps
$A^{(k)}\to X$ are $k$-skeleta.

The following proposition shows that a minimal skeleton is a retract of any
other skeleton.

######  Proposition 3.19.

Suppose that $A\to X$ is a minimal $k$-skeleton and $B\to X$ is another
$k$-skeleton. Then there exists a map $r\colon\thinspace B\to A$ of objects
over $X$, unique in the homotopy category, and it admits a section.

###### Proof.

Because the map $A\to X$ is $(k+1)$-connected by assumption, Corollary 3.6
implies that in the homotopy category $h\mathcal{C}$ there is a unique lift
$r\colon\thinspace B\to A$ over $X$. To show the existence of a section, the
definitions imply that there is a lift $i\colon\thinspace A\to B$ over $X$.
The maps $ri$ and $\operatorname{id}_{A}$ are then two lifts $A\to A$ along a
$(k+1)$-connected map $A\to X$, and another application of Corollary 3.6
implies that $ri=\operatorname{id}_{A}$ in $h\mathcal{C}$. ∎

If we have two minimal $k$-skeleta of $X$, we can then arrive at the following
conclusion.

######  Proposition 3.20.

Suppose that $A_{1}\to X$ and $A_{2}\to X$ are minimal $k$-skeleta of $X$.
Then in the homotopy category $h\mathcal{C}$ there exists a unique isomorphism
$A_{1}\simeq A_{2}$ over $X$.

######  Corollary 3.21.

Any $(k+1)$-connected map between $k$-skeletal objects is an equivalence.

######  Example 3.22.

Retracts of $k$-dimensional CW-complexes are $k$-skeletal. For the converse,
suppose $X$ is a $k$-skeletal space, and construct a CW-approximation $A\to
X$. Then $A^{(k)}\to X$ is a $k$-skeleton and $X\to X$ is a minimal
$k$-skeleton, so $X$ is a retract of a $k$-dimensional CW-complex $A^{(k)}$.

In particular, a connected $1$-skeletal space $X$ is a retract of a wedge of
circles. This means $X$ is a $K(G,1)$ for $G$ a retract of a free group; such
a group $G$ is free, and choosing generators gives an equivalence from a wedge
of circles to $X$.

###  3.4 Cells

######  Definition 3.23.

A _$k$ -cell_ is a $(k-1)$-connected map $A\to B$ such that $A$ is
$(k-1)$-skeletal and the map is relatively $k$-skeletal.

We say that a set $S_{k}$ of $k$-cells is a set of _generating $k$-cells_ for
the connectivity structure if it satisfies the following property: a map is
$k$-connected if and only if it is $(k-1)$-connected and satisfies the right
lifting property with respect to the maps in $S_{k}$.

A set of _generating cells_ is a choice of sets $S_{k}$ of generating
$k$-cells for all $k$; if one exists, we say that the connectivity structure
is _determined by cells_.

######  Definition 3.24.

A cell $A\to B$ is _$j$ -bounded_ if $B$ is $j$-skeletal.

######  Example 3.25.

For $k\geq 0$ the inclusion $S^{k-1}\to D^{k}$ is a ($0$-bounded) $k$-cell in
$\operatorname{\mathcal{S}}$, and the connectivity structure on spaces is
determined by these cells.

######  Example 3.26.

Similarly, for $k\in\mathbb{Z}$ the maps $\Sigma^{k-1}\mathbb{S}\to\ast$ and
$\ast\to\Sigma^{k}\mathbb{S}$ are both $k$-cells ($(-\infty)$-bounded and
$k$-bounded respectively) for the standard connectivity structure on the
category $\operatorname{Sp}$ of spectra. The connectivity structure on spectra
is determined by the first type of cells: a map $f\colon\thinspace X\to Y$ has
the right lifting property with respect to $\Sigma^{k-1}\mathbb{S}\to\ast$ if
and only if $\pi_{k}\operatorname*{cofib}(f)$ is trivial.

By contrast, the connectivity structure is not determined by the latter cells.
A $(k-1)$-connected map $f\colon\thinspace X\to Y$ has the right lifting
property with respect to $\ast\to\Sigma^{k}\mathbb{S}$ if and only if
$\pi_{k}(X)$ surjects onto $\pi_{k}(Y)$, which does not ensure that the map
$\pi_{k-1}(X)\to\pi_{k-1}(Y)$ is an isomorphism.

######  Example 3.27.

Consider the category $\operatorname{Sp}_{\geq 0}$ of connective spectra. The
connectivity structure on $\operatorname{Sp}$ restricts to one on
$\operatorname{Sp}_{\geq 0}$.555The inclusion $\operatorname{Sp}_{\geq
0}\to\operatorname{Sp}$ does not preserve homotopy pullbacks, and so the
connectivity structure on $\operatorname{Sp}_{\geq 0}$ is not lifted from
$\operatorname{Sp}$. Instead, it is lifted from $\operatorname{\mathcal{S}}$
along the functor $\Omega^{\infty}$. The maps $\Sigma^{k-1}\mathbb{S}\to\ast$
for $k\geq 1$ are also $k$-cells in $\operatorname{Sp}_{\geq 0}$, but they do
not determine the connectivity structure: they do not detect surjectivity on
$\pi_{0}$. To repair this, we must also include the $0$-cell
$\ast\to\mathbb{S}$.

Alternatively, the maps
$\Sigma^{\infty}_{+}S^{k-1}\to\Sigma^{\infty}_{+}D^{k}$ and
$\ast\to\Sigma^{\infty}_{+}D^{0}$ are also generating cells for
$\operatorname{Sp}_{\geq 0}$.

######  Proposition 3.28.

Suppose that $\mathcal{D}$ has a connectivity structure lifted from
$\mathcal{C}$ along a functor $R\colon\thinspace\mathcal{D}\to\mathcal{C}$
with left adjoint $L$.

If $L$ preserves connectivity, then $L$ takes $k$-cells to $k$-cells. If,
additionally, there are sets $S_{k}$ of generating $k$-cells that determine
the connectivity structure on $\mathcal{C}$, then the sets $L(S_{k})$ are
generating $k$-cells that determine the connectivity structure on
$\mathcal{D}$.

###### Proof.

Suppose $A\to B$ is a $k$-cell in $\mathcal{C}$. Then $L(A)$ is
$(k-1)$-skeletal by Proposition 3.9 and $L(A)\to L(B)$ is relatively
$k$-skeletal by Corollary 3.10. Therefore, if $L$ preserves connectivity then
$L(A)\to L(B)$ is a $k$-cell.

Suppose now that the connectivity structure on $\mathcal{C}$ is determined by
cells. A map $f$ in $\mathcal{D}$ is $n$-connected if and only if $Rf$ is
$n$-connected, and this is true if and only if $Rf$ has the right lifting
property with respect to the generating cells $A\to B$ in $S_{k}$ for $k\leq
n$; by adjunction, this is true if and only if $f$ has the right lifting
property with respect to the cells $LA\to LB$. ∎

######  Example 3.29.

If $S$ is an associative ring spectrum, the category $\operatorname{LMod}_{S}$
of left $S$-modules has a connectivity structure lifted from the forgetful
functor $R\colon\thinspace\operatorname{LMod}_{S}\to\operatorname{Sp}$. There
is a left adjoint $L(X)=S\otimes X$, and it preserves connectivity if and only
if $S$ is connective. In this case, the maps $\Sigma^{k-1}S\to\ast$ are
$k$-cells that determine the connectivity structure on
$\operatorname{LMod}_{S}$. Similarly, $\Sigma^{k-1}S\to\ast$ for $k\geq 1$ and
$\ast\to S$ are cells that determine the connectivity structure on
$(\operatorname{LMod}_{S})_{\geq 0}$.

######  Example 3.30.

Consider the category $\operatorname{CAlg}(\operatorname{Sp})$ of commutative
algebras in $\operatorname{Sp}$, with the connectivity structure lifted from
$\operatorname{Sp}$. The left adjoint to the forgetful functor is the free
algebra functor $\mathbb{P}$:

$\mathbb{P}(X)\simeq\coprod_{k\geq 0}X^{\otimes k}_{h\Sigma_{k}}.$

The functor $\mathbb{P}$ does not preserve connectivity in general, because
the symmetric power functors don’t: the map $S^{-1}\to\ast$ is
$(-1)$-connected, but the map $(S^{-1}\otimes S^{-1})_{h\Sigma_{2}}\to\ast$ is
only $(-2)$-connected. However, if we restrict attention to the category
$\operatorname{Sp}_{\geq 0}$ of connective spectra, the free functor
$\mathbb{P}$ does preserve connectivity. Therefore, the maps
$\mathbb{P}(\Sigma^{k-1}\mathbb{S})\to\mathbb{S}$ and
$\mathbb{S}\to\mathbb{P}(S)$ are generating cells for the connectivity
structure on $\operatorname{CAlg}(\operatorname{Sp}_{\geq 0}$) lifted from
$\operatorname{Sp}_{\geq 0}$.

(A similar result applies to an arbitrary operad acting on the category
$(\operatorname{LMod}_{S})_{\geq 0}$ of connective modules over a connective
commutative ring spectrum.)

######  Example 3.31.

Similarly, if $R$ is a connective $\mathbb{E}_{2}$-ring spectrum then there is
a connectivity structure on $\operatorname{Alg}(\operatorname{LMod}_{R})_{\geq
0}$ lifted along the forgetful functor
$\operatorname{Alg}(\operatorname{LMod}_{R})_{\geq
0}\to(\operatorname{LMod}_{R})_{\geq 0}$. This connectivity structure is
determined by the cells $\mathbb{T}^{R}(\Sigma^{k-1}R)\to R$ for $k\geq 1$ and
$R\to\mathbb{T}^{R}(R)$, where
$\mathbb{T}^{R}\colon\thinspace\operatorname{LMod}_{R}\to\operatorname{Alg}(\operatorname{LMod}_{R})$
is the free $R$-algebra functor.

(A similar result applies to $\mathbb{E}_{k}$-algebras in the category of left
modules over a connective $\mathbb{E}_{k+1}$-ring spectrum.)

###  3.5 Cellular skeleta

######  Definition 3.32.

We say that a set $S_{k}$ of $k$-cells _is sufficient for $k$-skeleta_ if the
following properties hold.

  1. 1.

The set $S_{k}$ is a set of generating $k$-cells for $\mathcal{C}$.

  2. 2.

$\mathcal{C}$ has homotopy colimits.

  3. 3.

Given any cells $A_{i}\to B_{i}$ in $S_{k}$, any cobase change of the map
$\coprod A_{i}\to\coprod B_{i}$ is $(k-1)$-connected.

######  Remark 3.33.

In particular, the third item is automatic if the connectivity structure is
compatible with cobase change.

######  Proposition 3.34.

Suppose that, for all $k>N$, we have a set of $k$-cells $S_{k}$ that is
sufficient for $k$-skeleta. Then an $N$-connected map $f\colon\thinspace A\to
X$ admits relative $n$-skeleta for any $n$.

###### Proof.

The factorization $A\to A\xrightarrow{f}X$ is a relative $n$-skeleton for any
$n\leq N$: the map $A\to A$ is relatively $n$-skeletal because it is initial
in the undercategory $\mathcal{C}_{/A}$, and the map $f$ is $N$-connected by
assumption.

We then apply a standard cellular approximation method. Suppose inductively
that we have found a relative $(n-1)$-skeleton $A\to X^{(n-1)}\to X$. Let
$S_{n}=\\{A_{j}\to B_{j}\\}_{j\in J}$ be a set of generating $n$-cells for the
connectivity structure, and consider the collection $T$ of all (up to
equivalance) commutative diagrams

$\textstyle{A_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{(n-1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X.}$

Factor the map $X^{(n-1)}\to X$ through the homotopy pushout

$\textstyle{\coprod_{t\in
T}A_{j_{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{(n-1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod_{t\in
T}B_{j_{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{(n)}.}$

By assumption, this cobase change $X^{(n-1)}\to X^{(n)}$ is $(n-1)$-connected.
Therefore, the factorization $X^{(n-1)}\to X^{(n)}\to X$ shows that the map
$X^{(n)}\to X$ is at least $(n-1)$-connected using Definition 2.1. Moreover,
since $\coprod A_{j}\to\coprod B_{j}$ is relatively $n$-skeletal, so is the
map $X^{(n-1)}\to X^{(n)}$ and hence the composite $A\to X^{(n-1)}\to X^{(n)}$
by Proposition 3.11.

To prove that $X^{(n)}\to X$ is $n$-connected, it then suffices to show that
any diagram

$\textstyle{A_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{(n)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X.}$

has a lift when the left-hand vertical arrow is a cell in the generating set
$S_{n}$. However, in that case $A_{j}$ is $(n-1)$-skeletal, and so the topmost
map factors through a map $A_{j}\to X^{(n-1)}$; the resulting commutative
diagram is then equivalent to a map in $T$, and so there is a lift $B_{j}\to
X^{(n)}$ by construction of the homotopy pushout. ∎

######  Remark 3.35.

We refer to this type of construction of a factorization $A\to B\to X$, via
iterated pushout, as a _cellular_ construction of a map using the cells in
$S_{k}$—regardless of whether or not $B$ is a relative skeleton of $X$.

######  Remark 3.36.

Suppose that the connectivity structure of $\mathcal{C}$ is determined by sets
of $k$-cells $S_{k}$ and all of the $S_{k}$ are sufficient for $k$-skeleta.
Then the above proposition implies that for every $k$ there is a weak
factorization system on $h\mathcal{C}$ whose left class is the class of
relative $k$-skeleta and whose right class is the class of $k$-connected
morphisms.

######  Corollary 3.37.

If $\emptyset$ is an initial object and $\emptyset\to X$ is $N$-connected,
then $X$ admits $n$-skeleta for any $n$.

######  Example 3.38.

All objects in the category $\operatorname{\mathcal{S}}$ of spaces have
skeleta, and the proof above is a standard construction of a CW-approximation
to any space using the cellular approximation theorem.

######  Example 3.39.

Suppose $S$ is a connective ring spectrum. The forgetful functor
$\operatorname{LMod}_{S}\to\operatorname{Sp}$ detects homotopy colimits and
connectivity, and so $\operatorname{LMod}_{S}$ satisfies the assumptions of
the proposition. Therefore, an object in $\operatorname{LMod}_{S}$ has an
$n$-skeleton so long as it is $N$-connected for some $N>-\infty$.

##  4 Detecting skeleta

###  4.1 Nilpotence

######  Definition 4.1.

Suppose $E$ is a collection of maps in $\mathcal{C}$. The class of _$E$
-nilpotent maps_ is the smallest collection of maps in $h\mathcal{C}$ which
contains $E$ and is closed under base change, transfinite composition,
products, filtered homotopy limits, and retracts.

######  Example 4.2.

Let $E$ be the collection of maps $\ast\to K(A,n)$ in
$\operatorname{\mathcal{S}}$ for $n\geq 2$. Recall that if a space $X$ is
nilpotent in the classical sense (i.e. $X$ has nilpotent fundamental group,
which acts nilpotently on the higher homotopy groups) then it can be written
as a limit of a tower $X\simeq\lim(\cdots\to Y_{n}\to Y_{n-1}\to\cdots
Y_{2}\to Y_{1}\to Y_{0}\simeq\ast)$ in which $Y_{n}\to Y_{n-1}$ is the fiber
of a map $Y_{n-1}\to K(A,m)$ for some $A$ and some $m\geq 2$ (cf. [MP12,
3.2.2]). Then it follows from Definition 4.1 that the terminal morphism
$X\to\ast$ is $E$-nilpotent.

######  Proposition 4.3.

An object $A$, or a map $A\to B$, has the left lifting property with respect
to $E$-nilpotent maps if and only it has the left lifting property with
respect to maps in $E$.

###### Proof.

This is the assertion that “right lifting properties” are closed under the
operations that define $E$-nilpotent maps. ∎

######  Corollary 4.4.

Suppose $R\colon\thinspace\mathcal{D}\to\mathcal{C}$ is a functor with a left
adjoint $L$ and $E$ is a collection of maps in $\mathcal{D}$. An object $A$ of
$\mathcal{C}$ has the lifting property with respect to $R(E)$-nilpotent maps
if and only if $LA$ has the lifting property with respect to maps in $E$.

######  Example 4.5.

Suppose that $S$ is a connective ring spectrum, and let $E$ be the class of
maps $\ast\to\Sigma^{n}HN$ for $N$ a $\pi_{0}(S)$-module and $n\geq k+1$. Then
any $k$-connected map $X\to Y$ in $\operatorname{LMod}_{S}$ is $E$-nilpotent:
one can apply the Postnikov tower to its cofiber. Therefore, if a left
$S$-module $M$ has a $k$-skeletal image

$H\pi_{0}(S)\otimes_{S}M$

in the category $\operatorname{LMod}_{H\pi_{0}(S)}$, then $M$ is $k$-skeletal.
The converse holds by Proposition 3.9.

###  4.2 Tangent categories

We begin with a reminder about tangent $\infty$-categories.

Suppose that $\mathcal{C}$ is a presentable $\infty$-category. Let
$T_{\mathcal{C}}\to\operatorname{Fun}(\Delta^{1},\mathcal{C})$ be a _tangent
bundle_ to $\mathcal{C}$ in the sense of [Lur17, 7.3.1.9, 7.3.1.10], i.e. the
category of excisive functors
$\operatorname{Exc}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C})$.
There are functors

$ev_{d}\colon\thinspace T_{\mathcal{C}}\to\mathcal{C}$

for $d\geq 0$ given by evaluation at $S^{d}$,

$U\colon\thinspace T_{\mathcal{C}}\to\mathcal{C}$

given by evaluation at $\ast$, and a map

$T_{\mathcal{C}}\to\operatorname{Fun}(\Delta^{1},\mathcal{C})$

that takes $X$ to $X(S^{0})\to X(\ast)$. By restricting to functors which take
$\ast$ to a fixed object $Z\in C$ we obtain _pointed_ excisive functors
$\operatorname{Exc}_{\ast}(\operatorname{\mathcal{S}_{\ast}^{\text{fin}}},\mathcal{C}_{/Z})$.
In other words, the fiber over an object $Z\in\mathcal{C}$, along the map
$U\colon\thinspace T_{\mathcal{C}}\to\mathcal{C}$, is equivalent to the
category $\operatorname{Sp}(\mathcal{C}_{/Z})$ discussed in Section 2.5.

The results of [Lur17, 7.3.1] imply that for any morphism $f\colon\thinspace
W\to Z$ in $\mathcal{C}$ there is an pair of functors
$f^{\ast}\colon\thinspace\operatorname{Sp}(\mathcal{C}_{/Z})\to\operatorname{Sp}(\mathcal{C}_{/W})$
and
$f_{!}\colon\thinspace\operatorname{Sp}(\mathcal{C}_{/W})\to\operatorname{Sp}(\mathcal{C}_{/Z})$
that form an adjunction $f_{!}\dashv f^{\ast}$. The functor $f_{*}$ is
described as follows: given an excisive functor
$Y\colon\thinspace\operatorname{\mathcal{S}_{\ast}^{\text{fin}}}\to\mathcal{C}$
with $Y(\ast)=Z$,

$(f^{\ast}Y)(K)=Y(K)\times_{Z}W.$

Its left adjoint $f_{!}$ is described as follows: given an excisive functor
$X\colon\thinspace\operatorname{\mathcal{S}_{\ast}^{\text{fin}}}\to\mathcal{C}$
with $X(\ast)=W$, the image $f_{!}X$ is the excisive approximation to the
functor sending $K$ to the pushout $X(K)\amalg_{W}Z$.

Up to equivalence of categories, the objects of $T_{\mathcal{C}}$ are pairs
$(Z,M)$ of an object $Z$ of $\mathcal{C}$ and an object $M$ of the stable
category $\operatorname{Sp}(\mathcal{C}_{/Z})$. A morphism from $(W,M)$ to
$(Z,N)$ in $T_{\mathcal{C}}$ is equivalent to a pair of morphisms $(f,\phi)$
where $f\colon\thinspace W\to Z$ is a morphism in $\mathcal{C}$ and
$\phi\colon\thinspace M\to f^{\ast}(N)$ is a morphism in
$\operatorname{Sp}(\mathcal{C}_{/W})$ (or, equivalently, a morphism
$\phi\colon\thinspace f_{!}(M)\to N$ in
$\operatorname{Sp}(\mathcal{C}_{/Z})$). Under this identification, the functor
$T_{\mathcal{C}}\to\operatorname{Fun}(\Delta^{1},\mathcal{C})$ sends the
object $(Z,M)$ to the object $\Omega^{\infty}M\to Z$ in
$\mathcal{C}_{/Z}\subseteq\operatorname{Fun}(\Delta^{1},\mathcal{C})$.

######  Proposition 4.6.

Suppose $\mathcal{C}$ has a connectivity structure. Then there is a _fiberwise
connectivity structure_ on $T_{\mathcal{C}}$, defined as follows. A map $X\to
Y$ in $T_{\mathcal{C}}$ is $k$-connected if the map $UX\to UY$ is an
equivalence and each map $ev_{n}X\to ev_{n}Y$ is $(k-n)$-connected for $n\geq
0$.

###### Proof.

It is clear that $k$-connected maps are $(k-1)$-connected and that
equivalences are $\infty$-connected. Moreover, connectivity is preserved by
pullback because pullbacks in the category $T_{\mathcal{C}}$ are pullbacks of
functors to $\mathcal{C}$.

Suppose that we have composable maps $f\colon\thinspace X\to Y$ and
$g\colon\thinspace Y\to Z$ in $T_{\mathcal{C}}$, and two out of three of $f$,
$g$, and $gf$ are $k$-connected for some $k$. Then the maps $UX\to UY\to UZ$
are all equivalences. Therefore, this diagram is equivalent to a diagram in
the full subcategory $\operatorname{Sp}(\mathcal{C}_{/UX})$. The restriction
of connectivity to this subcategory is the stable connectivity structure of
Definition 2.24, where the 2-out-of-3 axioms for connectivity are satisfied. ∎

######  Remark 4.7.

Note that the functors $ev_{0}$ and $U$ both preserve connectivity; the former
by definition and the latter because equivalences are $k$-connected for all
$k$.

######  Remark 4.8.

When restricted to the fiber over $Z\in\mathcal{C}$ the connectivity structure
of Proposition 4.6 recovers the stable connectivity structure on
$\operatorname{Sp}(\mathcal{C}_{/Z})$ induced by stabilizing the connectivity
structure on $\mathcal{C}_{/Z}$ (which is in turn induced by the connectivity
structure on $\mathcal{C}$). It is not hard to check that, for a morphism
$f\colon\thinspace Z\to W$ in $\mathcal{C}$, the pullback functor $f^{\ast}$
preserves connectivity. The pushforward $f_{!}$ preserves connectivity if the
connectivity structure is compatible with cobase change and $\mathcal{C}$ is
differentiable.

######  Remark 4.9.

The base-change functors $f^{*}$ induced by maps $f\colon\thinspace W\to Z$
give the following interpretation of the fiberwise connectivity structure. The
map $T_{\mathcal{C}}\to\mathcal{C}$ is a Cartesian fibration, whose fiber over
$Z$ is the stable category $\operatorname{Sp}(\mathcal{C}_{/Z})$. Each fiber
has the associated stable connectivity structure from Definition 2.5, and a
map $f$ in $\mathcal{C}$ induces a connectivity-preserving functor $f^{*}$.
This fibration determines, by straightening, a contravariant functor from
$\mathcal{C}$ to a category of stable categories and connectivity-preserving
functors. If the connectivity structure is compatible with cobase change and
$\mathcal{C}$ is differentiable, a dual statement holds: the coCartesian
fibration $T_{\mathcal{C}}\to\mathcal{C}$ classifies a covariant functor from
$\mathcal{C}$ to stable categories and connectivity-preserving functors.

######  Remark 4.10.

If $\mathcal{C}$ is an $\infty$-topos it can be shown that $T_{\mathcal{C}}$
is also an $\infty$-topos. In this case $T_{\mathcal{C}}$ has the connectivity
structure described in Example 2.14. Note that this connectivity structure is
very different from the connectivity structure on $T_{\mathcal{C}}$ described
in Proposition 4.6. More specifically, a morphism $f\colon\thinspace X\to Y$
in $T_{\mathcal{C}}$ is $k$-connected in the toposic connectivity structure on
$T_{\mathcal{C}}$ exactly when $UX\to UY$ is $k$-connected in the toposic
connectivity structure on $\mathcal{C}$.

######  Proposition 4.11.

An object $A\in T_{\mathcal{C}}$ is $k$-skeletal if and only if it is
$k$-skeletal in the subcategory $\operatorname{Sp}(\mathcal{C}_{/UA})$.

###### Proof.

One direction is clear: objects which are skeletal in $T_{\mathcal{C}}$ are
still skeletal in the full subcategory.

For the converse, suppose that $A$ is $k$-skeletal in
$\operatorname{Sp}(\mathcal{C}_{/UA})$, and that we have a $k$-connected map
$g\colon\thinspace X\to Y$ in $T_{\mathcal{C}}$. Let $f$ be a map
$f\colon\thinspace A\to Y$ which we would like to lift, with associated base-
change functor
$f^{*}\colon\thinspace\operatorname{Sp}(\mathcal{C}_{/UY})\to\operatorname{Sp}(\mathcal{C}_{/UA})$.

The $k$-connected map $g$ determines an equivalence $UY\simeq UX$ and a
$k$-connected map $X\to g^{*}Y\in\operatorname{Sp}(\mathcal{C}_{/UX})$. The
map $UA\to UY$ automatically lifts to a map $h\colon\thinspace A\to UX$, and a
lift of $f$ is equivalent to asking for a lift of the map $A\to
f^{*}Y=h^{*}g^{*}Y$ along $h^{*}X\to h^{*}g^{*}Y$ in
$\operatorname{Sp}(\mathcal{C}_{/UA})$. However, $h^{*}X\to h^{*}g^{*}Y$ is
still $k$-connected, and so this lift exists. ∎

The forgetful functor $U\colon\thinspace T_{\mathcal{C}}\to\mathcal{C}$ has a
left adjoint: the absolute cotangent functor
$\mathbb{L}\colon\thinspace\mathcal{C}\to T_{\mathcal{C}}$, sending
$A\in\mathcal{C}$ to
$\mathbb{L}(A)\in\operatorname{Sp}(\mathcal{C}_{/A})\subset T_{\mathcal{C}}$
[Lur17, 7.3.2.14].

######  Definition 4.12.

A map $f\colon\thinspace X\to Y$ in $\mathcal{C}$ is _nilpotent_ if it is
$\Omega^{\infty}(E)$-nilpotent, where $E$ is the class of morphisms $f$ in
$T_{\mathcal{C}}$ whose underlying morphism $Uf$ is an equivalence.

######  Remark 4.13.

We may think of such a morphism $X\to Y$ as being a map of abelian group
objects in some slice category $\mathcal{C}_{/Z}$.

The following lemma gives a good supply of nilpotent morphisms in a
presentable $\infty$-category:

######  Lemma 4.14.

If $\mathcal{C}$ is presentable and a map $f\colon\thinspace A\to B$ is a
square-zero morphism in the sense of [Lur17, 7.4.1.6] then it is nilpotent.

###### Proof.

Note that by [Lur17, 7.4.1.7], any square-zero extension $f\colon\thinspace
A\to B$ by an object $M\in\operatorname{Sp}(\mathcal{C}_{/B})$ is a pullback
of a map $B\to B\oplus\Sigma M$. This latter map is
$\Omega^{\infty}(0_{B}\to\Sigma M)$ where $0_{B}\to\Sigma M$ is the image of
the zero morphism in $\operatorname{Sp}(\mathcal{C}_{/B})$ under its inclusion
(as the fiber over $B$) into $T_{\mathcal{C}}$. Therefore $A\to B$ is the
pullback of a nilpotent morphism. ∎

######  Proposition 4.15.

Suppose that all $k$-connected maps in $\mathcal{C}$ are nilpotent. Then an
object $A$ of $\mathcal{C}$ is $k$-skeletal if and only if the absolute
cotangent complex $\mathbb{L}(A)$ is $k$-skeletal in
$\operatorname{Sp}(\mathcal{C}_{/A})$.

###### Proof.

The object $A$ is $k$-skeletal in $\mathcal{C}$ if and only if $\mathbb{L}(A)$
is $k$-skeletal in $T_{\mathcal{C}}$ by Proposition 4.4. We can then deduce
the result from Proposition 4.11. ∎

######  Remark 4.16.

Recall that $T_{\mathcal{C}}\to\operatorname{Fun}(\Delta^{1},\mathcal{C})$ is
the stable envelope, in the sense of [Lur17, 7.3.1.1], of the presentable
fibration $\operatorname{Fun}(\Delta^{1},\mathcal{C})\to\mathcal{C}$ given by
evaluating a morphism at its target. One can generalize Definition 4.12 and
say that for any presentable fibration
$p\colon\thinspace\mathcal{C}\to\mathcal{D}$ with stable envelope
$q\colon\thinspace\mathcal{C}^{\prime}\to\mathcal{C}$, an object of
$\mathcal{C}$ is nilpotent if it is in the essential image of $q$. In the case
that we take $p$ to be the presentable fibration
$\operatorname{\mathcal{S}}\to\Delta^{0}$, the classically nilpotent spaces
are nilpotent objects of $\operatorname{\mathcal{S}}$ (cf. Example 4.2).

##  5 Lifting constructions of skeleta

###  5.1 Subductivity

######  Definition 5.1.

Suppose that we have an adjunction

$\mathcal{C}\mathop{\rightleftarrows}^{L}_{R}\mathcal{D}.$

We say that the adjunction is $k$-_subductive_ on $\mathcal{C}$ if, whenever a
map $X\to Y$ in $\mathcal{C}$ is $k$-connected, the square

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL(Y)}$

is always $k$-Cartesian, and _strongly $k$-subductive_ if the square is always
$(k+1)$-Cartesian.

The _subduction zone_ is the set of $k$ such that the adjunction is
$k$-subductive. If it is subductive for all $k$, we simply refer to it as
_subductive_. (Similar definitions apply for terms such as the _strong
subduction zone_.)

######  Example 5.2.

Consider the $\Sigma-\Omega$ adjunction on pointed spaces. The condition of
$k$-subductivity asks whether, for a $k$-connected map $X\to Y$, the diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega\Sigma
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega\Sigma
Y}$

is $k$-Cartesian. If the spaces are allowed to be disconnected, this is
generically false. If we restrict attention to pointed connected spaces, this
adjunction is subductive: subductivity is equivalent to asking for
surjectivity of the map

$\pi_{k+1}(Y,X)\to\pi_{k+2}(\Sigma Y,\Sigma X),$

which is a consequence of the Blakers–Massey excision theorem for any $k\geq
1$. If we further restrict to pointed 1-connected paces, this adjunction is
strongly subductive.

######  Example 5.3.

Suppose that $f\colon\thinspace R\to S$ is a map of connective ring spectra,
determining an adjunction between $\operatorname{LMod}_{R}$ and
$\operatorname{LMod}_{S}$. Then this adjunction is subductive if and only if,
for any $k$-connected map $M\to N$, the square

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\otimes_{R}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\otimes_{R}N}$

is $k$-Cartesian. Taking fibers, we find that subductivity is equivalent to
showing that for any $(k-1)$-connected object $F$, the map $F\to
S\otimes_{R}F$ is $k$-connected. On $\pi_{k}$, this is the map

$\pi_{k}(F)\to\pi_{0}(S)\otimes_{\pi_{0}(R)}\pi_{k}(F),$

Therefore, this adjunction is subductive if and only if the map $R\to S$ is
$0$-connected. Similarly, the adjunction is strongly subductive if and only if
the map $R\to S$ is $1$-connected.

######  Proposition 5.4.

Suppose that we have a strongly $k$-subductive adjunction

$\mathcal{C}\mathop{\rightleftarrows}^{L}_{R}\mathcal{D}$

such that $R$ preserves $(k+1)$-connectivity, and that $f\colon\thinspace X\to
Y$ is $k$-connected in $\mathcal{C}$. If $Lf\colon\thinspace LX\to LY$ is
$(k+1)$-connected, then $f$ is $(k+1)$-connected.

###### Proof.

The map $f$ factors as a composite

$X\to RL(X)\times_{RL(Y)}Y\to Y.$

The first map is $(k+1)$-connected by strong $k$-subductivity, and the second
is a pullback of the $(k+1)$-connected map $RL(X)\to RL(Y)$. ∎

Applying the previous proposition inductively gives the following result.

######  Corollary 5.5.

Suppose that we have an adjunction

$\mathcal{C}\mathop{\rightleftarrows}^{L}_{R}\mathcal{D}$

such that $\\{k,k+1,\dots,N\\}$ are in the strong subduction zone, that
$f\colon\thinspace X\to Y$ in $\mathcal{C}$ is $k$-connected, and that $L$ and
$R$ preserve connectivity. Then $f$ is $N$-connected if and only if $Lf$ is
$N$-connected.

######  Proposition 5.6.

Let $k$ and $j$ be integers with $j\geq 0$. Suppose that we have a (strongly)
$k$-subductive adjunction

$\mathcal{C}\mathop{\rightleftarrows}^{L}_{R}\mathcal{D}$

and a (strongly) $(k+j)$-subductive adjunction

$\mathcal{D}\mathop{\rightleftarrows}^{M}_{S}\mathcal{E},$

that $L$ increases connectivity by at least $j$, and that $R$ decreases
connectivity by no more than $j$. Then the composite adjunction is also
(strongly) $k$-subductive.

###### Proof.

Suppose that $f\colon\thinspace X\to Y$ is $k$-connected. Let $\epsilon$ be
$0$ in the case of subductivity, and $1$ in the case of strong subductivity;
we wish to show $X\to Y\times_{RSML(Y)}RSML(X)$ is $(k+\epsilon)$-connected.

From [RV21, Proposition 2.1.9], we know that the unit of the composite
adjunction is given the composite transformation $id_{\mathcal{C}}\to RL\to
RSML$. Therefore we have a composite of cospans:

${X}$${X}$${Y}$${X}$${{RL(X)}}$${Y}$${{RL(Y)}}$${{RSML(X)}}$${{RSML(Y)}}$$\scriptstyle{id_{X}}$$\scriptstyle{id_{X}}$

which induces a factorization of the universal map $X\to
Y\times_{RSML(Y)}RSML(X)$. Thus we need to prove that the composite map

$X\to Y\times_{RL(Y)}RL(X)\to
Y\times_{RL(Y)}\left(RL(Y)\operatorname*{\times}_{RSML(X)}RSML(X)\right)\simeq
Y\times_{RSML(Y)}RSML(X)$

is $(k+\epsilon)$-connected. The first map is $(k+\epsilon)$-connected because
the first adjunction is (strongly) $k$-subductive, and so it suffices to prove
that the second map is $(k+\epsilon)$-connected.

Because $L$ increases connectivity by at least $j$, the map
$Lf\colon\thinspace LX\to LY$ is $(k+j)$-connected, and therefore the map

$LX\to LY\times_{SML(X)}SML(Y)$

is $(k+j+\epsilon)$-connected because the second adjunction is (strongly)
$(k+j)$-subductive. Applying $R$, the map

$RL(X)\to RL(Y)\times_{RSML(X)}RSML(X)$

is $(k+\epsilon)$-connected. Taking the fiber product with $Y$ over $RL(Y)$,
we find that the map

$Y\times_{RL(Y)}RL(X)\to Y\times_{RSML(X)}RSML(X)$

is $(k+\epsilon)$-connected, as desired. ∎

If the $\Sigma-\Omega$ adjunction is (strongly) subductive, we get the
following result.

######  Corollary 5.7.

Suppose $\mathcal{C}$ is pointed and has pushouts, that connectivity is
compatible with cobase change, and that $\\{k,k+1,\dots,k+n-1\\}$ are in the
(strong) subduction zone of the $\Sigma-\Omega$ adjunction. Then the
$\Sigma^{n}-\Omega^{n}$ adjunction is (strongly) $k$-subductive.

###### Proof.

The functors $\Omega$ and $\Sigma$ both shift connectivity by $1$ in opposing
directions, the latter by Corollary 2.23. Therefore, iterated application of
Proposition 5.6 implies that the composite $\Sigma^{n}-\Omega^{n}$ adjunction
is subductive. ∎

######  Proposition 5.8.

Suppose that $\mathcal{C}$ is differentiable, pointed, and admits pushouts.
Suppose that the connectivity structure on $\mathcal{C}$ is compatible with
cobase change and sequential colimits.

If the $\Sigma-\Omega$ adjunction is (strongly) subductive, then the
$\Sigma^{\infty}-\Omega^{\infty}$ adjunction between $\mathcal{C}$ and
$\operatorname{Sp}(\mathcal{C})$ is (strongly) subductive.

###### Proof.

The proof of Proposition 2.38 shows that
$\Omega^{\infty}\Sigma^{\infty}X\simeq\operatorname*{colim}_{m}\Omega^{m}\Sigma^{m}X$.
Therefore we need to prove that, for any $k$-connected map $X\to Y$, the
square

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname*{colim}_{m}\Omega^{m}\Sigma^{m}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname*{colim}_{m}\Omega^{m}\Sigma^{m}Y}$

is $(k+\epsilon)$-Cartesian. For any individual $m$, this diagram is
$(k+\epsilon)$-Cartesian by Corollary 5.7. The fiber product, and the
connectivity of the map from $X$ to it, are preserved by sequential colimits
because $\mathcal{C}$ is differentiable. ∎

###  5.2 Excavation

######  Definition 5.9.

Let $L\colon\thinspace\mathcal{C}\to\mathcal{D}$ be a colimit-preserving
functor. Suppose that we have a set $S$ of maps in $\mathcal{C}$, a map $X\to
Y$ in $\mathcal{C}$, and a commutative diagram

$\textstyle{\coprod_{t\in
T}LA_{j_{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod_{t\in
T}LB_{j_{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LY}$

where the left-hand map is a coproduct of maps in $LS$. We say that this can
be excavated from $\mathcal{D}$ if the square diagram from each term in the
coproduct is equivalent to the image of a diagram in $L$. In particular, the
diagram lifts to a commutative diagram

$\textstyle{\coprod_{t\in
T}A_{j_{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\coprod_{t\in
T}B_{j_{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

in $\mathcal{C}$, and the constructed factorization $LX\to Z\to LY$ through
the pushout lifts to a constructed factorization $X\to W\to Y$.

More generally, if we have sets of cells $S_{k}$ and a map $X\to Y$ with a
factorization

$LX=Z^{(n-1)}\to Z^{(n)}\to Z^{(n+1)}\to\dots\to Z^{(N)}\to LY,$

where each map $Z^{(k-1)}\to Z^{(k)}$ is cellularly constructed from maps in
$LS_{k}$, then we say that this can be excavated if each stage can be
excavated: there exists a lift to a sequence

$X=W^{(n-1)}\to W^{(n)}\to W^{(n+1)}\to\dots\to W^{(N)}\to Y,$

in $\mathcal{C}$, where each map $W^{(k-1)}\to W^{(k)}$ is cellularly
constructed from the maps in $S_{k}$, lifting the construction of $Z^{(k)}$.

###  5.3 Excavating cells

The following lemma assures us that, when we have a subductive adjunction,
“cells can be excavated so long as their ranges can.”

######  Lemma 5.10.

Suppose that

$\mathcal{C}\mathop{\rightleftarrows}^{L}_{R}\mathcal{D}$

is a $k$-subductive adjunction and that $f\colon\thinspace X\to Y$ is a
$k$-connected map in $\mathcal{C}$. Then, for any $(k+1)$-cell $A\to B$ in
$\mathcal{C}$, a commutative diagram

$\textstyle{LA\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LX\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LB\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LY}$

can be excavated from $\mathcal{D}$ if and only if the map $LB\to LY$ lifts to
$\mathcal{C}$.

###### Proof.

Showing that this diagram is the image of one in $L$ is the same as asking
that we can complete the commutative diagram

---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{RL(Y)}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By adjunction, lifting $LB\to LY$ to a map $B\to Y$ is the same as
constructing the bottom commutative triangle.

Suppose that a lift $B\to Y$ is chosen. Then we have a map $A\to
B\times_{RL(Y)}RL(X)\to Y\times_{RL(Y)}RL(X)$, and what remains is to lift it
along the map $X\to Y\times_{RL(Y)}RL(X)$. However, this map is $k$-connected
by assumption and $A$ is $k$-skeletal, so such a lift exists. ∎

######  Remark 5.11.

The lifting requirement holds automatically in several important cases, such
as if the object $B$ is a zero object of $\mathcal{C}$. Our principal case
will use boundedness: lifting holds if the cell is $j$-bounded and the unit
$Y\to RL(Y)$ is $j$-connected.

######  Corollary 5.12.

Fix a $k$-subductive adjunction
$\mathcal{C}\mathop{\rightleftarrows}^{L}_{R}\mathcal{D}$. Let $S_{k+1}$ be a
set of $j$-bounded $(k+1)$-cells in $\mathcal{C}$.

Suppose that $Y\to RL(Y)$ is $j$-connected, that we have a $k$-connected map
$f\colon\thinspace X\to Y$ in $\mathcal{C}$ and a choice of factorization
$LX\to Z\to LY$ where $LX\to Z$ is cellularly constructed from the maps in
$LS_{k+1}$. Then $LX\to Z\to LY$ can be excavated from $\mathcal{D}$.

###  5.4 Excavating skeleta

Our goal in this section is to expand the previous section inductively, and
show that by using subductive adjunctions
$\mathcal{C}\rightleftarrows\mathcal{D}$, we can lift cellular constructions
from $\mathcal{D}$. In particular, this will allow us to construct minimal
skeleta.

######  Theorem 5.13.

Suppose that we have an adjunction

$\mathcal{C}\mathop{\rightleftarrows}^{L}_{R}\mathcal{D}$

such that $\\{n,n+1,\dots,N-1\\}$ are in the subduction zone. Fix sets $S_{k}$
of $j$-bounded $k$-cells in $\mathcal{C}$ for $n<k\leq N$.

Suppose $Y\to RL(Y)$ is $j$-connected, that we have an $n$-connected map
$f\colon\thinspace X\to Y$ in $\mathcal{C}$, and a cellular construction
$LX=Z^{(n)}\to Z^{(n+1)}\to\dots\to Z^{(N)}\to LY$ using the cells in
$LS_{k}$. Then the cellular construction of $Z^{(N)}$ can be excavated from
$\mathcal{D}$ to a sequence $X=W^{(n)}\to W^{(n+1)}\to\dots\to W^{(N)}\to Y$.

If $L$ reflects $N$-connectivity, then the map $W^{(N)}\to Y$ is an
$N$-skeleton if the original map $Z^{(N)}\to Y$ was. If $L$ also reflects
$(N+1)$-connectivity, this skeleton is minimal if the original map was.

###### Proof.

Starting with $W^{(n)}=X$, inductively apply Corollary 5.12 to excavate the
cellular constructions of $(LW)^{(k-1)}\simeq Z^{(k-1)}\to Z^{(k)}\to LY$ to
cellular constructions of $W^{(k-1)}\to W^{(k)}\to Y$.

Because the $S_{k}$ are $k$-cells, each map $W^{(k-1)}\to W^{(k)}$ is
relatively $k$-cellular.

If $L$ reflects connectivity appropriately, and the map $(LW)^{(N)}\simeq
Z^{(N)}\to LY$ is $N$-connected, then the map $W^{(N)}\to Y$ is $N$-connected
and hence an $N$-skeleton. If the original was a minimal $N$-skeleton, then
this connectivity estimate improves, making $W^{(N)}$ into a minimal
$N$-skeleton of $Y$. ∎

######  Example 5.14.

The $\Sigma-\Omega$ adjunction between $0$-connected pointed spaces and
$1$-connected pointed spaces has subduction zone $[1,\infty)$ as in Example
5.2. The connectivity structure in both cases is determined by $0$-bounded
cells $S^{k}\to D^{k+1}$, which are preserved by suspension.

For a path-connected space $Y$, the map $Y\to\Omega\Sigma Y$ is $0$-connected.
Given a 1-connected map $X\to Y$ between 0-connected spaces, we can therefore
excavate any relative CW-factorization $\Sigma X\to Z^{(2)}\to\dots\to
Z^{(N+1)}\to\Sigma Y$ to a relative CW-factorization $X\to W^{(1)}\to\dots\to
W^{(N)}\to Y$.

On $1$-connected spaces, $\Sigma$ reflects connectivity. Therefore, if $Y$ is
$1$-connected, any cellular construction of an $(N+1)$-skeleton for $\Sigma Y$
can be excavated to a cellular construction of an $N$-skeleton for $Y$.

##  6 Excision

In this section we will show that many of these important properties hold for
categories of algebras and modules over an operad in spectra. The fundamental
tools to prove this are excision theorems, and excision theorems are hard
work. For our applications, we will be appealing to the work of Ching–Harper
on excision for algebras in a category of module spectra [CH16]. However, our
use of their main result is simple enough that we can axiomatize it in this
section.

###  6.1 Excisive connectivity structures

######  Definition 6.1.

Fix $\mathcal{C}$ with homotopy pushouts and an initial object, with a
connectivity structure that is compatible with cobase change. We say that the
connectivity structure on $\mathcal{C}$ _satisfies algebraic excision_ if the
following conditions hold.

  1. 1.

All maps $X\to Y$ in $\mathcal{C}$ are $(-1)$-connected.

  2. 2.

Let $W$ be a nonempty finite set with power set $\mathcal{P}(W)$, viewed as a
poset, and $X\colon\thinspace\mathcal{P}(W)\to\mathcal{C}$ a “$W$-cube” in
$\mathcal{C}$. Suppose that:

    1. (a)

for each nonempty subset $V\subset W$, the $V$-cube $X|_{\mathcal{P}(V)}$ is
$k_{V}$-coCartesian, and

    2. (b)

$k_{U}\leq k_{V}$ for all $U\subset V\subset W$.

Then $X$ is $k$-Cartesian, where $k$ is the minimum of
$-|W|+\sum_{V\in\lambda}(k_{V}+1)$ over all partitions $\lambda$ of $W$ by
nonempty finite sets.

######  Remark 6.2.

The category $\operatorname{\mathcal{S}}$ of spaces does not satisfy this type
of excision: excision estimates for spaces differ by a shift.666In principle,
pointed connected spaces are equivalent to topological groups via the loop
space/classifying space relationship, and topological groups _do_ satisfy
algebraic excision; this can help make sense of the shift in connectivity.

######  Example 6.3.

Suppose $\mathfrak{O}$ is a connective operad in the category of modules over
a connective commutative symmetric ring spectrum $S$, with category
$\operatorname{Alg}_{\mathfrak{O}}$ of _connective_ algebras. Then [CH16,
Theorem 1.7] is precisely that $\operatorname{Alg}_{\mathfrak{O}}$ satisfies
algebraic excision. This includes models for the category of $\mathbb{E}_{n}$
$S$-algebras, where $S$ is a commutative ring spectrum.777The paper [CH16] is
written in terms of operads $\mathfrak{O}$ in symmetric spectra, using the
Quillen adjunction between positive stable model structures on
$\operatorname{Mod}_{S}$ and $\operatorname{Alg}_{\mathfrak{O}}$. However,
passage to the associated $\infty$-category preserves structures of rings,
modules, and algebras, and the definitions of Cartesian and coCartesian cubes
are defined using homotopy limits and colimits, which are equally well
computed in $\operatorname{Alg}_{\mathfrak{O}}$ or the associated
$\infty$-category. Moreover, in the case where $\mathfrak{O}$ is the
suspension spectrum of an ordinary operad $\mathcal{O}$, algebras can be
rectified: the $\infty$-category associated to
$\operatorname{Alg}_{\mathfrak{O}}$ is a model for the category of algebras
for the associated $\infty$-operad.

######  Remark 6.4.

The property of having algebraic excision is inherited by slice and coslice
categories, because the colimits and limits of deleted cubical diagrams are
calculated in the underlying $\infty$-category.

###  6.2 Excision and augmentations

######  Proposition 6.5.

Suppose that $\mathcal{C}$ satisfies algebraic excision and has a final
object, and that the map $\emptyset\to\ast$ from the initial object to the
final object is $j$-connected. Then there is an adjunction

$\mathcal{C}\rightleftarrows\mathcal{C}_{\ast}$

that is subductive if $j\geq 0$, and strongly subductive if $j>0$.

###### Proof.

The left adjoint is the functor $(-)_{+}=(-)\amalg\ast$. By pushing out along
the map $\emptyset\to\ast$, we find that $X\to X_{+}$ is always $j$-connected.
As a result, consider the coCartesian square

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{+},}$

viewed as a $2$-cube. If the map $X\to Y$ is $k$-connected, then algebraic
excision shows that the square is $n$-Cartesian, where
$n=\min\\{-2+(k+1)+(j+1),-2+\infty\\}=k+j$, as desired. ∎

######  Corollary 6.6.

Suppose that $\mathcal{C}$ satisfies algebraic excision and that $\emptyset\to
Z$ is $j$-connected. Then the adjunction
$\mathcal{C}_{/Z}\rightleftarrows(\mathcal{C}_{/Z})_{\ast}$, between objects
over $Z$ and objects augmented over $Z$, is subductive if $j\geq 0$ and
strongly subductive if $j>0$. The unit $X\to X\amalg Z$ is always
$j$-connected.

###  6.3 Excision and suspension

######  Proposition 6.7.

Suppose that $\mathcal{C}$ satisfies algebraic excision and that $X\to Z$ is a
$k$-connected map in $\mathcal{C}$, where $k\geq 0$. Then the map
$X\to\Omega_{Z}\Sigma_{Z}X$ is $2k$-connected.

###### Proof.

Consider a homotopy pushout diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{Z}X,}$

viewed as a $2$-cube. The two arrows $X\to Z$ are $k$-connected by assumption,
and the full cube is a pushout cube and hence is $\infty$-coCartesian.
Algebraic excision then shows that it is $n$-Cartesian where
$n=\min\\{-2+(k+1)+(k+1),-2+\infty\\}=2k$. By definition, this means that the
map $X\to\Omega_{Z}\Sigma_{Z}X$ is $2k$-connected. ∎

######  Remark 6.8.

In particular, this implies that the map $\Omega_{Z}\Sigma_{Z}X\to Z$ is still
$k$-connected.

######  Proposition 6.9.

Suppose that $\mathcal{C}$ satisfies algebraic excision. Let $X\to Y\to Z$ be
maps, where $X\to Y$ is $k$-connected and the maps $X\to Z$ and $Y\to Z$ are
$\ell$-connected, for $k+1\geq\ell\geq 0$. Then the diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{Z}\Sigma_{Z}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega_{Z}\Sigma_{Z}Y}$

is $(k+\ell)$-Cartesian.

###### Proof.

Consider the cubical diagram:

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{Z}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{Z}Y}$

The natural map from $X$ to the total homotopy pullback of the cube with the
initial vertex deleted is equivalent to the map from $X$ to the homotopy
pullback in the proposition statement, and hence it suffices for us to show
that this cube is $(k+\ell)$-Cartesian.

The maps $X\to Z$ are $\ell$-connected, and the map $X\to Y$ is $k$-connected.
The top and bottom faces are homotopy pushout diagrams, and hence
$\infty$-coCartesian; in particular, the whole cube is $\infty$-coCartesian by
[CH16, 3.8(b)]. Because the map $X\to Y$ is $k$-connected and the map $Z\to Z$
is $\infty$-connected, the left-hand and back faces are both
$(k+1)$-coCartesian, again by [CH16, 3.8(b)].

The inclusion of each initial face into a larger face improves connectivity,
and so the excision estimate applies. The whole cube is $n$-Cartesian, where

$n=\min\\{-3+(k+1)+(\ell+1)+(\ell+1),-3+(k+1)+\infty,-3+(\ell+1)+(k+2),-3+\infty\\}=k+\ell,$

as desired. ∎

######  Corollary 6.10.

If $\mathcal{C}$ satisfies algebraic excision, the $\Sigma_{Z}-\Omega_{Z}$
adjunction is subductive when restricted to objects $X$ such that $X\to Z$ is
$0$-connected, and strongly subductive when restricted to objects such that
$X\to Z$ is $1$-connected.

######  Proposition 6.11.

Suppose that $\mathcal{C}$ satisfies algebraic excision, and that $X\to Y$ is
a map of objects over $Z$ such that $X\to Z$ and $Y\to Z$ are $1$-connected.
Then, for $k\geq 0$, the map $X\to Y$ is $k$-connected if and only if the map
$\Omega_{Z}\Sigma_{Z}X\to\Omega_{Z}\Sigma_{Z}Y$ is $k$-connected.

###### Proof.

Because the maps $X\to Z$ and $Y\to Z$ are $1$-connected, the map $X\to Y$ is
$0$-connected. Therefore, we can apply Proposition 5.4. ∎

##  7 Applications

###  7.1 Skeleta in derived categories

In this section we will examine what skeletality means for an object in the
classical derived category of a ring $R$, with the connectivity structure
determined by the $t$-structure as in Example 2.9.

######  Definition 7.1.

Let $R$ be an ordinary ring. We say that a complex of $R$-modules has
_projective amplitude in $[a,b]$_ if it is equivalent to a complex of
projectives concentrated in degrees $a$ through $b$.

######  Proposition 7.2.

A complex of $R$-modules has projective amplitude in $[a,b]$ if and only if it
is $(a-1)$-connected and $b$-skeletal.

###### Proof.

Suppose $A$ is a complex with the given projective amplitude; since the
properties we need to show are invariant under equivalence, we may assume that
$A$ is a complex of projectives concentrated in degrees $a$ through $b$. Then
its homology is clearly concentrated in degrees $a$ through $b$, so it is
$(a-1)$-connected. Moreover, if $X\to Y$ is a $b$-connected map of complexes,
its cofiber $Y/X$ is $(b+1)$-connected, and hence $[A,Y/X]=0$ via a
hypercohomology spectral sequence with $E_{1}$-term

$\operatorname{Hom}_{R}(A_{s},H_{t}(Y/X))\Rightarrow
H_{t-s}\operatorname{Hom}(A,Y/X).$

This implies the desired lifting property by the long exact sequence for
$[A,-]$.

To prove the converse, we proceed by induction on $b-a$. By first applying a
shift $\Sigma^{-a}$, we may assume without loss of generality that $a=0$.

For the base case, suppose that $A$ is $(-1)$-connected and $0$-skeletal; we
wish to show that $A$ is equivalent to a projective complex concentrated in
degree $0$. Without loss of generality we may assume $A$ is a complex of
projectives in nonnegative degrees. Any surjective map of (discrete)
$R$-modules $M\to N$ can be viewed as a $0$-connected map of complexes
concentrated in degree $0$. The map $[A,M]\to[A,N]$ is therefore surjective;
however, this is isomorphic to the map

$\operatorname{Hom}_{R}(H_{0}A,M)\to\operatorname{Hom}_{R}(H_{0}A,N).$

Since $M\to N$ was an arbitrary surjection, $H_{0}(A)$ is a projective module
$P$. Both $A$ and $P$ are $0$-skeletal, and the maps $A\to P$ and $P\to P$ are
both $1$-connected. This makes both $A$ and $P$ minimal $0$-skeleta of $P$,
and hence equivalent by Proposition 3.20.

Now suppose by induction that we have shown the result for $b-1$. Given a
complex $A$ which is $(-1)$-connected and $b$-skeletal, let $P\to H_{0}(A)$ be
a surjection from a projective module, with a lift to a map $P\to A$. Then the
cofiber $A/P$ is $0$-connected, and it is $b$-skeletal by Proposition 3.14. By
induction it then has projective amplitude in $[1,b]$. The complex $A$ is
equivalent the cofiber of the map $\Sigma^{-1}A/P\to P$, and thus has
projective amplitude in $[0,b]$. ∎

######  Proposition 7.3.

Suppose that $R$ is a principal ideal domain. Then a complex $M$ of
$R$-modules is $k$-skeletal if and only if $H_{k}(M)$ is free and $H_{*}M=0$
for $*>k$.

###### Proof.

Since principal ideal domains have projective dimension $1$, any complex of
$R$-modules splits: there is an equivalence

$\bigoplus_{d}\Sigma^{d}H_{d}(M)\simeq M.$

Therefore, $M$ is $k$-skeletal if and only if each $\Sigma^{d}H_{d}(M)$ is
$k$-skeletal by Proposition 3.12, which is true if and only if $H_{d}(M)$ is
$(k-d)$-skeletal.

By the previous proposition, this is equivalent to $H_{d}(M)$ having
projective amplitude in $[0,k-d]$. Since $R$ has projective dimension $1$,
every discrete module has projective amplitude in $[0,1]$, and so this is
automatically satisfied for $d\leq k-1$. When $d=k$, we must have that
$H_{k}(M)$ has projective amplitude in $[0,0]$, which is equivalent to asking
that $H_{k}(M)$ is projective, and hence free. When $d>k$, we must have that
$H_{k}(M)$ is trivial. ∎

######  Corollary 7.4.

Suppose that $R$ is a field. Then a complex $M$ of $R$-modules is $k$-skeletal
if and only if $H_{*}(M)=0$ for $*>k$.

We now consider the construction of minimal skeleta.

######  Proposition 7.5.

Suppose that $A\to M$ is a map of chain complexes over $R$ that is a
$(k-1)$-skeleton, with cofiber $M/A$. Then there exists a minimal $k$-skeleton
$B\to M$ if and only if $H_{k}(M/A)$ is a projective $R$-module and
$H_{k+1}(M/A)=0$.

###### Proof.

Suppose that $B\to M$ is a minimal $k$-skeleton. Then there exists a unique
lift $A\to B$ over $M$ by Corollary 3.6. The object $B/A$ is $(k-1)$-connected
and $k$-skeletal by Proposition 3.14, and hence equivalent to a shift
$\Sigma^{k}P$ of a projective $R$-module.

Because of the identification

$(M/A)/(B/A)\simeq M/B,$

the map $B\to M$ is $(k+1)$-connected if and only if the map $B/A\to M/A$ is
$(k+1)$-connected. However, for the map $\Sigma^{k}P\to M/A$ to be
$(k+1)$-connected we must have that $H_{k}(M/A)=P$ and $H_{k+1}(M/A)=0$.

Conversely, suppose that $H_{k}(M/A)$ is a projective module $P$ and that
$H_{k+1}(M/A)=0$. Then, since $M/A$ is $(k-1)$-connected, there is a
$(k+2)$-connected map $M/A\to\Sigma^{k}P$, which has a section because
$\Sigma^{k}P$ is $k$-skeletal; the section is $(k+1)$-connected. Let $C$ be
the cofiber of the map $\Sigma^{k}P\to M/A$; it is $(k+1)$-connected. Let $B$
be the fiber of the map $M\to C$; the map $B\to M$ is $(k+1)$-connected. The
octahedral axiom implies that there is a cofiber sequence

$\Sigma^{k-1}P\to A\to B$

and hence $B$ is $k$-skeletal. Therefore, $B$ is a minimal $k$-skeleton of
$M$. ∎

######  Corollary 7.6.

Suppose $R$ is a principal ideal domain. Then a chain complex of $R$-modules
$M$ admits a minimal $k$-skeleton if and only if $H_{k}(M)$ is free and
$H_{k+1}(M)=0$.

###### Proof.

Let $A$ be any $(k-1)$-skeleton of $M$. Then $H_{k+1}(M/A)=H_{k+1}(M)$, and so
these groups are either both zero or both nonzero. There is also an exact
sequence

$0\to H_{k}(M)\to H_{k}(M/A)\to H_{k-1}(A).$

The group $H_{k-1}(A)$ is free by Proposition 7.3, and since $R$ is a
principal ideal domain the image of the map $H_{k}(M/A)\to H_{k-1}(A)$ is also
free. Therefore, the group $H_{k}(M/A)$ splits as a direct sum of $H_{k}(M)$
and a free module, and so $H_{k}(M)$ is projective if and only if $H_{k}(M/A)$
is. ∎

Because there is an equivalence between complexes of $R$-modules and modules
over the Eilenberg–Mac Lane spectrum $HR$ [SS03, 5.1.6], taking homology
groups to homotopy groups, we arrive at the following.

######  Proposition 7.7.

Suppose $R$ is a principal ideal domain. An $HR$-module $M$ is $k$-skeletal if
and only if $\pi_{k}(M)$ is projective and $\pi_{*}(M)=0$ for $*>k$. An
$HR$-module $M$ admits a minimal $k$-skeleton if and only of $\pi_{k}(M)$ is
free and $\pi_{k+1}(M)=0$.

In particular, suppose $R$ is a field. An $HR$-module $M$ is $k$-skeletal if
and only if $\pi_{*}(M)=0$ for $\ast>k$. An $HR$-module $M$ admits a minimal
$k$-skeleton if and only if $\pi_{k+1}(M)=0$.

######  Corollary 7.8.

If $R$ is a principal ideal domain, then any $N$-skeleton of an
$(n-1)$-connected $HR$-module $M$ admits a cellular construction using the
cells $(\\{\Sigma^{k-1}HR\to\ast\\})_{n\leq k\leq N}$.

###  7.2 Skeleta for spaces

For $k\geq 2$, any $k$-connected map of spaces $X\to Y$ is nilpotent: the
relative Postnikov tower expresses each path component as a limit of pullbacks

$\textstyle{P_{n}(Y,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\pi_{1}Y,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{n-1}(Y,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(\pi_{1}Y,1)\ltimes
K(\pi_{n}(Y,X),n+1).}$

As a result, $X$ is $k$-skeletal for $k\geq 2$ if and only if its image in the
stable category $\operatorname{Sp}(\operatorname{\mathcal{S}}_{/X})$ is
$k$-skeletal.

Following Waldhausen, this stable category is equivalent to the category of
functors $X\to\operatorname{Sp}$, and the image of $X$ is the constant functor
with value $\mathbb{S}$. In the case where $X$ is connected, this is
equivalent to the category of modules over the spherical group algebra
$\mathbb{S}[\Omega X]$, and the image of $X$ is the trivial module
$\mathbb{S}$.

The Postnikov truncation map $\mathbb{S}[\Omega X]\to H\mathbb{Z}[\pi_{1}X]$
is $1$-connected. As in Example 4.5 we find that $X$ is $k$-skeletal if and
only if the left $H\mathbb{Z}[\pi_{1}X]$-module

$\mathbb{S}\otimes_{\mathbb{S}[\Omega X]}H\mathbb{Z}[\pi_{1}X]\simeq
H\mathbb{Z}\otimes\widetilde{X}$

has projective amplitude in $[0,k]$, where $\widetilde{X}$ is the universal
cover. This is equivalent to asking that the complex $C_{*}(\widetilde{X})$ of
$\mathbb{Z}[\pi_{1}X]$ has projective amplitude in $[0,k]$. This is related,
but somewhat orthogonal, to Wall’s finiteness obstruction.

###  7.3 Algebras in module categories

Throughout this section we assume that $S$ is a commutative ring spectrum and
$\operatorname{Alg}_{\mathbb{E}_{n}}(S)$ is the $\infty$-category of
$\mathbb{E}_{n}$-algebras in left $S$-modules. Recall that there is a
connectivity structure on $\operatorname{Alg}_{\mathbb{E}_{n}}(S)_{\geq 0}$,
lifted from $\operatorname{LMod}_{S}$, that is described in Examples 3.30 and
3.31. In particular, this connectivity structure is determined by the
($0$-bounded) cells $\mathbb{T}^{S}_{\mathbb{E}_{n}}(\Sigma^{k-1}S)\to S$ for
$k\geq 1$ and $S\to\mathbb{T}_{\mathbb{E}_{n}}^{S}(S)$, where
$\mathbb{T}_{\mathbb{E}_{n}}^{S}\colon\thinspace\operatorname{LMod}_{S}\to\operatorname{Alg}_{\mathbb{E}_{n}}(\operatorname{LMod}_{S})$
is the free $\mathbb{E}_{n}$ $S$-algebra functor. Moreover, Proposition 3.28
and the fact that the connectivity structure on
$\operatorname{Alg}_{\mathbb{E}_{n}}(S)$ is compatible with cobase change
imply that these cells are sufficient for $k$-skeleta for all $k\geq 0$, in
the sense of Section 3.5.

As discussed in Example 6.3, the category
$\operatorname{Alg}_{\mathbb{E}_{n}}(S)$ satisfies algebraic excision by
[CH16, Theorem 1.7], and hence all the results of Section 6 apply.

######  Proposition 7.9.

Any connective $\mathbb{E}_{n}$ $S$-algebra $A$ has a (convergent) Postnikov
tower

$\dots\to P_{2}A\to P_{1}A\to P_{0}A$

in $\mathbb{E}_{n}$-algebras, where each stage is given by a pullback diagram

$\textstyle{P_{m+1}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{m}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{m}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{m}A\oplus\Sigma^{m+2}H\pi_{m+1}A}$

in which the right-hand map is $\Omega^{\infty}$ of a map in
$\operatorname{Sp}(\operatorname{Alg}_{\mathbb{E}_{n}}(R)_{/P_{m}A})$. In
particular, the maps $P_{m+1}A\to P_{m}A$ are nilpotent for $m>0$.

###### Proof.

It follows from [Lur17, 7.1.3.19] that every connective $\mathbb{E}_{n}$
$S$-algebra has a Postnikov tower. Moreover, by Lemma 4.14 and [Lur17,
7.4.1.28] we have that each of the morphisms $P_{m+1}A\to P_{m}A$ is
nilpotent. It remains to show that each level of this tower can be obtained by
the given pullback. By [Lur17, 7.4.1.7] we have that $P_{n+1}A\to P_{m}A$ is
some pullback of a nilpotent morphism. To check that it is a pullback
specifically of a morphism $P_{m}A\to P_{m}A\oplus\Sigma^{m+2}H\pi_{m+1}A$ it
suffices to identify the fiber of $P_{m+1}A\to P_{m}A$, which is
$\Sigma^{m+1}H\pi_{m+1}A$ by [Lur17, 7.1.3.14] parts (3) and (5). ∎

######  Proposition 7.10.

Suppose $f\colon\thinspace A\to B$ is a morphism of $\mathbb{E}_{n}$
$S$-algebras spectra such that $\pi_{0}(f)$ is surjective and has nilpotent
kernel. Then $f$ is nilpotent.

###### Proof.

Recall that a surjective map of discrete rings $R\to R^{\prime}$ with
nilpotent kernel $J$ can always be written as a limit of square-zero
extensions: the tower of quotients $\dots\to R/J^{3}\to R/J^{2}\to R/J\cong
R^{\prime}$. By [Lur17, 7.4.1.21], if $P_{0}A\to P_{0}B$ is a square-zero
extension of discrete rings then it is square-zero as a map of
$\mathbb{E}_{n}$ $S$-algebras and so, by Lemma 4.14, is nilpotent. Therefore
the map $\pi_{0}(f)\colon\thinspace P_{0}A\to P_{0}B$ is nilpotent.

Now, using the Postnikov towers of $A$ and $B$, factor the map $A\to B$ as a
transfinite composite

$A\to\dots\to P_{2}A\times_{P_{2}B}B\to P_{1}A\times_{P_{1}B}B\to
P_{0}A\times_{P_{0}B}B\to B.$

Notice that $\operatorname*{holim}_{m}(P_{m}A\times_{P_{m}B}B)\simeq
A\times_{B}B$, so $A$ is the homotopy limit of the above tower. Therefore, it
suffices to show that all of the maps in the composition are nilpotent.

The last map is the base-change of the map $P_{0}A\to P_{0}B$ along the map
$B\to P_{0}B$, and so it is nilpotent by the discrete case proven above.

Each of the remaining maps factors as

$P_{m+1}A\times_{P_{m+1}B}B\to P_{m+1}A\times_{P_{m}B}B\to
P_{m}A\times_{P_{m}B}B.$

The second map is the base-change of the composite $P_{m+1}A\to P_{m}A\to
P_{m}B$ along the map $B\to P_{m}B$, and so it is nilpotent. The first map is
the base-change of the map

$P_{m+1}B\to P_{m+1}B\times_{P_{m}B}P_{m+1}B$

along $P_{m+1}A\times_{P_{m}B}B\to P_{m+1}B\times_{P_{m}B}P_{m+1}B$, and so it
suffices to show that $P_{m+1}B\to P_{m+1}A\times_{P_{m}B}P_{m+1}B$ is
nilpotent.

However, $P_{m+1}B$ is a square-zero extension of $P_{m}B$ via a map that
becomes trivial when restricted to $P_{m+1}B$. Therefore, the pullback is the
trivial square-zero extension

$P_{m+1}B\oplus\Sigma^{m+1}H\pi_{m+1}B\to P_{m+1}B,$

and the map $P_{m+1}B\to P_{m+1}B\oplus\Sigma^{m+1}H\pi_{m+1}B$ is the image
of the map $0\to\Sigma^{m+1}H\pi_{m}B$ in the stable category
$\operatorname{Sp}((\operatorname{Alg}_{\mathbb{E}_{n}})_{/P_{m+1}B})$. ∎

######  Corollary 7.11.

A $k$-connected map of $\mathbb{E}_{n}$ $R$-algebras $f\colon\thinspace A\to
B$ is nilpotent whenever $k>0$.

Recall that the stable category
$\operatorname{Sp}(\operatorname{Alg}^{\mathbb{E}_{n}}(S)_{/A})$ is identified
with the category of ($S$-linear) $\mathbb{E}_{n}$ $A$-modules [Lur17,
7.3.4.18].

######  Definition 7.12.

For a commutative ring spectrum $S$ and an $\mathbb{E}_{n}$ $S$-algebra $A$,
we define $\mathcal{L}^{S}(A)$ to be the relative Loday construction, or
relative factorization homology object:

$\mathcal{L}^{S}(A)=S\operatorname*{\otimes}_{\int_{S^{n-1}}S}\int_{S^{n-1}}A.$

The category of $S$-linear $\mathbb{E}_{n}$ $A$-modules is equivalent to the
category of modules over $\mathcal{L}^{S}(A)$ [Lur17, 7.3.5.3].

By combining Corollary 7.11 with Proposition 4.15 we immediately obtain the
following result:

######  Proposition 7.13.

A connective $\mathbb{E}_{n}$ $S$-algebra $A$ is $k$-skeletal for $k>0$ if and
only if its absolute cotangent complex $\mathbb{L}^{\mathbb{E}_{n}}(A)$ is
$k$-skeletal in the stable category
$\operatorname{LMod}_{\mathcal{L}^{S}(A)}$.

We recall the following formula for this absolute cotangent complex.

######  Theorem 7.14 ([Fra13, 2.26], [Lur17, 7.3.5.1]).

There is a cofiber sequence

$\mathcal{L}^{S}(A)\to A\to\Sigma^{n}\mathbb{L}^{\mathbb{E}_{n}}(A).$

of $\mathcal{L}^{S}(A)$-modules.

######  Corollary 7.15.

For $k\geq 1-n$, the absolute cotangent complex
$\mathbb{L}^{\mathbb{E}_{n}}(A)$ is $k$-skeletal if $A$ is a $(k+n)$-skeletal
$\mathcal{L}^{S}(A)$-module.

###### Proof.

The object $\Sigma^{-n}\mathcal{L}^{S}(A)$ is always a $(-n)$-skeletal module
and by assumption $\Sigma^{-n}A$ is a $k$-skeletal module, and so this follows
by Proposition 3.14. ∎

######  Remark 7.16.

For $k\geq 2-n$, the converse is also true.

If $A$ is connective, the Loday construction $\mathcal{L}^{S}(A)$ is
connective. Under these circumstances, Example 4.5 now gives us the following
result.

######  Theorem 7.17.

A connective $\mathbb{E}_{n}$ $S$-algebra $A$ is $k$-skeletal for $k\geq 1$ if
the $H\pi_{0}\left(\mathcal{L}^{S}(A)\right)$-module

$H\pi_{0}\left(\mathcal{L}^{S}(A)\right)\operatorname*{\otimes}_{\mathcal{L}^{S}(A)}A$

is $(n+k)$-skeletal.

When $n=1$, $\mathcal{L}^{S}(A)=A\otimes_{S}A^{op}$ as an algebra, and
$\pi_{0}\left(\mathcal{L}^{S}(A)\right)=\pi_{0}(A)\otimes_{\pi_{0}(S)}\pi_{0}(A)^{op}$.
The above theorem specializes to a statement about relative topological
Hochschild homology.

######  Proposition 7.18.

Suppose $A$ is a connective $\mathbb{E}_{1}$ $S$-algebra and that $\pi_{0}(A)$
is a localized quotient of $\pi_{0}S$ (in particular, a commutative ring).
Then $A$ is $k$-skeletal for $k\geq 1$ if the topological Hochschild homology
object

$\operatorname{THH}^{S}(A;H\pi_{0}A)\simeq
H\pi_{0}(A)\otimes_{A\otimes_{S}A^{op}}A$

is a $(k+1)$-skeletal $H\pi_{0}(A)$-module.

This allows the detection of skeleta. We next turn our attention to the
construction of skeleta and minimal skeleta.

######  Lemma 7.19.

Let $S$ be a commutative ring spectrum and $A$ an $\mathbb{E}_{n}$ $S$-algebra
for $n\geq 1$ with $\pi_{0}(A)$ commutative. If the unit map $S\to A$ is
$0$-connected then there is an isomorphism
$\pi_{0}(A)\simeq\pi_{0}(\mathcal{L}^{S}(A))$.

###### Proof.

When $n>1$, the result follows immediately from the long exact sequence in
homotopy groups applied to the cofiber sequence of Theorem 7.14 and does not
require the $0$-connectivity assumption on the unit nor the commutativity
condition on $\pi_{0}(A)$. When $n=1$ however we use both conditions along
with the isomorphism
$\pi_{0}\left(\mathcal{L}^{S}(A)\right)=\pi_{0}(A)\otimes_{\pi_{0}(S)}\pi_{0}(A)^{op}$
to deduce the result. ∎

######  Lemma 7.20.

If $A$ is an $\mathbb{E}_{n}$ $S$-algebra for a commutative ring spectrum $S$
and the unit $S\to A$ is $0$-connected then the unit
$A\to\Omega^{\infty}\Sigma_{+}^{\infty}A\simeq A\oplus\mathbb{L}_{A}$ is
$0$-connected.

###### Proof.

First note that, because $S\to A$ is $0$-connected, Corollary 6.6 implies that
$A\to A_{+}\simeq A\oplus A$ is also $0$-connected. Note that, by virtue of
$A_{+}$ being an object of pointed $\mathbb{E}_{n}$ $S$ algebras over $A$,
that there is a retract $A\to A_{+}\to A$ and therefore $A_{+}\to A$ is
$1$-connected by Proposition 2.16. By Proposition 6.7 we then have that
$A_{+}\to\Omega_{A}\Sigma_{A}A_{+}$ is $2$-connected.

Now let $W$ be a set with $m$-elements and consider the cocartesian $m$-cube
$F\colon\thinspace\mathcal{P}(W)\to\operatorname{Alg}_{\mathbb{E}_{n}}(S)_{/A}$
with $A_{+}$ as the initial vertex, $\Sigma^{m-1}A_{+}$ as the final vertex,
and $A$ for every other vertex. Given a subset $V\subseteq W$ with $|V|=k$,
the homotopy colimit over the diagram $\mathcal{P}(V)-V$ is
$\Sigma_{A}^{k-1}A_{+}$. By iterating Proposition 2.21 we see that
$\Sigma^{k-1}_{A}A_{+}\to\Sigma^{k-1}_{A}A\simeq A$ is $k$-connected, so the
subcube of $F$ associated to $V$ is $k$-cocartesian (except when $V=W$ in
which case it is $\infty$-coCartesian). In other words, in the notation of
Definition 6.1, $k_{V}=|V|$ whenever $|V|<|W|$ and $k_{V}=\infty$ when $V=W$.
Thus the entire cube $F$ must be $K$-cartesian where $K$ is the minimum
$\\{-|W|+\Sigma_{V\in\lambda}\left(|V|+1\right)=|\lambda|:|\lambda|>1\\}\cup\\{\infty\\}$.
Therefore $K=2$. Thus $A_{+}\to\Omega^{m}_{A}\Sigma_{A}^{m}A_{+}$ is at least
$2$-connected for all $m\geq 0$.

Note that because the connectivity structure on
$\operatorname{Alg}_{\mathbb{E}_{n}}(S)_{/A}$ is lifted along a forgetful
functor
$\operatorname{Alg}_{\mathbb{E}_{n}}(S)_{/A}\to\operatorname{LMod}_{S}$ which
preserves sifted colimits, it is compatible with, in particular, sequential
colimits. Thus
$A_{+}\to\operatorname*{colim}_{m}\Omega^{m}_{A}\Sigma^{m}_{A}A_{+}\simeq
A\oplus\mathbb{L}_{A}$ is $2$-connected, and so $A\to A\oplus\mathbb{L}_{A}$
is $0$-connected. ∎

The following is well known to experts, but the authors could not readily find
a reference using the $\infty$-categorical language of this paper.

######  Lemma 7.21.

Let $f\colon\thinspace B\to A$ be a morphism of $\mathbb{E}_{n}$ $S$-algebra
spectra for $S$ a commutative ring spectrum. Then the value of the functor
$(\Sigma^{\infty}_{+})_{A}\colon\thinspace
Alg^{\mathbb{E}_{n}}_{/A}\to\operatorname{Sp}(Alg^{\mathbb{E}_{n}}_{/A})$
applied to $f$ is equivalent to
$\mathcal{L}^{S}(A)\otimes_{\mathcal{L}^{S}(B)}\mathbb{L}^{\mathbb{E}_{n}}_{B}$.

###### Proof.

We check that
$\mathcal{L}^{S}(A)\otimes_{\mathcal{L}^{S}(-)}\mathbb{L}^{\mathbb{E}_{n}}_{(-)}$
is left adjoint to $\Omega^{\infty}_{A}$. Let $f^{\ast}\vdash f_{\ast}$ be the
push-pull adjunction between $Alg^{\mathbb{E}_{n}}_{/A}$ and
$Alg^{\mathbb{E}_{n}}_{/B}$ and $F^{\ast}\vdash F_{\ast}$ its associated
stabilization (cf. [Lur18, Proposition 5.4.1]). We check that
$F_{\ast}\mathbb{L}_{B}^{\mathbb{E}_{n}}\simeq\mathcal{L}^{S}(A)\otimes_{\mathcal{L}^{S}(B)}\mathbb{L}^{\mathbb{E}_{n}}_{B}$
and $\Sigma^{\infty}_{A,+}B$ corepresent the same functor in
$\operatorname{Sp}(Alg_{A}^{\mathbb{E}_{n}})$. For any
$M\in\operatorname{Sp}(Alg_{/A}^{\mathbb{E}_{n}})$ we have the following
string of (natural in $M$) equivalences of mapping spaces:

$\displaystyle Alg_{/A}^{\mathbb{E}_{n}}(f_{\ast}B,\Omega^{\infty}_{A}M)$
$\displaystyle\simeq
Alg_{/B}^{\mathbb{E}_{n}}(B,f^{\ast}\Omega^{\infty}_{A}M)$
$\displaystyle\simeq
Alg^{\mathbb{E}_{n}}_{/B}(B,\Omega^{\infty}_{B}F^{\ast}M)$
$\displaystyle\simeq\operatorname{Sp}(Alg_{/B})(\mathbb{L}_{B}^{\mathbb{E}_{n}},F^{\ast}M)$
$\displaystyle\simeq\operatorname{Sp}(Alg_{/B})(F_{\ast}\mathbb{L}_{B}^{\mathbb{E}_{n}},M).$