# Presentations of algebras in chain complexes

Brice Le Grignou  bricelegrignou "at" gmail.com

###### Abstract.

The goal of this article is to describe several presentations of the infinity
category of algebras over some monad on the infinity category of chain
complexes.

###### Contents

  1. 1 Presentable infinity-categories and algebras
  2. 2 Chain complexes and their algebras
  3. A Chain complexes in non negative degrees

## Introduction

Many categories encountered in Algebra are presentable. This means that for
such a category $\mathsf{C}$, one can find a small category $\mathsf{D}$ and
an equivalence relating $\mathsf{C}$ to the full subcategory of
$\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{Set}\right)$ spanned by
functors $F:\mathsf{D}\to\mathsf{Set}$ that satisfy some locality conditions.

Finite products theories also called Props provides many examples of such
presentable categories in Algebra. For instance, if $R$ is a commutative ring,
the category of $R$-modules may be presented as follows. Let
$\mathsf{Cart}_{R}$ be the full subcategory of $R$-modules spanned by those of
the form $R^{n}$ for $n\in\mathbb{N}$. Then the category of $R$-modules is
equivalent to the full subcategory of
$\mathrm{Fun}\left(\mathsf{Cart}_{R}^{\mathrm{op}},\mathsf{Set}\right)$
spanned by functors that preserves finite products.

Moreover, since $R$-modules are equivalent to such products preserving
functors, then many categories of algebras in $R$-modules have a similar
presentation. Indeed, let us consider a monadic adjunction

${R-\mathsf{Modules}}$${\mathsf{A}}$$\scriptstyle{T_{M}}$$\scriptstyle{U^{M}}$

whose associated monad $M$ preserves reflective coequalisers and filtered
colimits, and let $\mathsf{Cart}_{M}$ be the full subcategory of $\mathsf{A}$
spanned by objects of the form $T_{M}(R^{n})$ for $n\in\mathbb{N}$. Then the
category $\mathsf{A}$ is equivalent to the full subcategory of
$\mathrm{Fun}\left(\mathsf{Cart}_{M}^{\mathrm{op}},\mathsf{Set}\right)$
spanned by functors that preserves finite products.

These results have extensions in the world of $\infty$-categories. Indeed, if
$\mathsf{Ch}_{R}^{\geq 0}$ is the $\infty$-category of chain complexes of
$R$-modules in non negative degrees and if $\mathsf{S}$ is the
$\infty$-category of $\infty$-groupoids, then we have the following result.

######  Proposition.

[Lur18, Corollary 4.7.3.18] Let $U^{M}:\mathsf{A}\to\mathsf{Ch}_{R}$ be a
monadic functor that preserves filtered colimits and geometric realisations
and let $T_{M}$ be its left adjoint. If $\mathsf{Cart}_{M}$ is the full
subcategory of $\mathsf{A}$ spanned by the objects
$T_{M}(R^{n}),n\in\mathbb{N}$, then $\mathsf{A}$ is equivalent to the full
subcategory of
$\mathrm{Fun}\left(\mathsf{Cart}_{M}^{\mathrm{op}},\mathsf{S}\right)$ spanned
by functors that preserve finite products.

The goal of this article is to get similar presentations in the stable
context. More precisely, given a monadic adjunction relating the product
$\mathsf{Ch}_{R}^{K}$ over a set $K$ of the $\infty$-category of chain
complexes $\mathsf{Ch}_{R}$ to an $\infty$-category $\mathsf{A}$

${\mathsf{Ch}_{R}^{K}}$${\mathsf{A}}$$\scriptstyle{T_{M}}$$\scriptstyle{U^{M}}$

whose right adjoint $U^{M}$ preserves filtered colimits and geometric
realisations, we describe several presentations of the $\infty$-category
$\mathsf{A}$.

One kind of presentation is given by finitely cellular objects, that is
objects that are obtained from the initial object by a finite composition of
pushouts of the form

${T_{M}(S^{m}_{k})}$${A}$${T_{M}(0)}$${B}$

for some $k\in K$ and $m\in\mathbb{Z}$, where $S^{m}_{k}$ denotes the object
of $\mathsf{Ch}_{R}^{K}$ whose $k$-component is the shift of $R$ to the
$m^{th}$ degree and whose $k^{\prime}$-component, for $k^{\prime}\neq k$, is
the zero chain complex. For any $n\in\mathbb{Z}\sqcup\\{\infty\\}$, we denote
$\mathsf{Cell}_{M,\leq n}$ the full subcategory of $\mathsf{A}$ spanned by
finitely cellular objects, with the additional restriction that they are built
using a finite composition of pushout as above but so that $m\leq n-1$.

######  Theorem (3).

For any $n\in\mathbb{Z}\sqcup\\{\infty\\}$ the $\infty$-category $\mathsf{A}$
is canonically equivalent to the full subcategory of
$\mathrm{Fun}\left(\mathsf{Cell}_{M,\leq n}^{\mathrm{op}},\mathsf{S}\right)$
spanned by functors $F$ so that

  1. (1)

$F(\emptyset)\simeq\ast$;

  2. (2)

for any pushout square in $\mathsf{Cell}_{M,\leq n}$ as above its image
through $F$ is a pullback.

Another type of presentation, that we call cartesian presentation is closer to
the presentations shown in the unstable case. For any
$n\in\mathbb{Z}\sqcup\\{\infty\\}$, we denote $\mathsf{Cart}_{M,\leq n}$ the
full subcategory of $\mathsf{A}$ spanned by finite coproducts of objects of
the form $T_{M}(S^{m}_{k})$ for $m\leq n$ and $k\in K$.

######  Theorem (4).

For any $n\in\mathbb{Z}\sqcup\\{\infty\\}$ the $\infty$-category $\mathsf{A}$
is canonically equivalent to the full subcategory of
$\mathrm{Fun}\left(\mathsf{Cart}_{M,\leq n}^{\mathrm{op}},\mathsf{S}\right)$
spanned by functors $F$ so that

  1. (1)

$F$ preserves finite products;

  2. (2)

for any pushout square in $\mathsf{Cart}_{M,\leq n}$ of the form

${T_{M}(X)}$${T_{M}(0)}$${T_{M}(0)}$${T_{M}(\Sigma X)}$

its image through $F$ is a pullback.

These two theorems are direct consequences of a single result. Indeed, let us
consider a monadic functor $U^{M}$ from an $\infty$-category $\mathsf{A}$ to
an $\infty$-category $\mathsf{C}$ that preserves sifted colimits. We assume
that $\mathsf{C}$ is $\omega$-presentable in the sense that there exists a
small $\infty$-category $\mathsf{D}$ so that $\mathsf{C}$ is a reflective
subcategory of $\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)$
stable through filtered colimits. Thus we have a sequence of adjunctions

${\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)}$${\mathsf{C}}$${\mathsf{A}.}$$\scriptstyle{i_{!}}$$\scriptstyle{i^{\ast}}$$\scriptstyle{T_{M}}$$\scriptstyle{U^{M}}$

Then, let us consider a full subcategory $j:\mathsf{B}\to\mathsf{A}$ so that
the composite functor from $\mathsf{D}$ to $\mathsf{A}$ factorises through
$j$. Let us also suppose that $\mathsf{B}$ contains only $\omega$-small
objects and that it is stable through finite coproducts. We obtain a square of
right adjoint functors

${\mathrm{Fun}\left(\mathsf{B}^{\mathrm{op}},\mathsf{S}\right)}$${\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)}$${\mathsf{A}}$${\mathsf{C}}$$\scriptstyle{t^{\ast}}$$\scriptstyle{j^{\ast}}$$\scriptstyle{U^{M}}$$\scriptstyle{i^{\ast}}$

Let $\mathsf{E}$ be a reflective subcategory of
$\mathrm{Fun}\left(\mathsf{B}^{\mathrm{op}},\mathsf{S}\right)$. If
$\mathsf{E}$ is stable through filtered colimits, if its functors preserve
finite products and if it contains the image of $\mathsf{A}$ in
$\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)$, then one has
the following result.

######  Theorem (2).

The following assertions are equivalent.

  1. (1)

the functor from $\mathsf{A}$ to $\mathsf{E}$ is an equivalence;

  2. (2)

the composite functor

$\mathsf{E}\xrightarrow{q}\mathrm{Fun}\left(\mathsf{B}^{\mathrm{op}},\mathsf{S}\right)\xrightarrow{t^{\ast}}\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)$

is conservative and sends elements of $\mathsf{E}$ to the essential image of
the functor $i^{\ast}$ from $\mathsf{C}$ to
$\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)$;

  3. (3)

the composite functor

$\mathsf{E}\xrightarrow{q}\mathrm{Fun}\left(\mathsf{B}^{\mathrm{op}},\mathsf{S}\right)\xrightarrow{t^{\ast}}\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)\xrightarrow{i_{!}}\mathsf{C}$

is conservative.

Finally, a recurring context where one can apply these results is provided by
the category of algebras over a $K$-coloured operad $\mathsf{P}$ enriched in
chain complexes of $R$-modules. Indeed, if $\mathsf{P}$ is $\Sigma$-cofibrant
and if the model structure on products of chain complexes may be transferred
to the category of $\mathsf{P}$-algebras along the adjunction

${\mathsf{Chain}_{R}^{K}}$${\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}}$$\scriptstyle{T_{P}}$$\scriptstyle{U^{P}}$

then the resulting adjunction between the $\infty$-categories obtained from
localisation at weak equivalences is monadic and its right adjoint preserves
filtered colimits and geometric realisations.

### Layout

In the first section, we recall definitions and results about presentable
$\infty$-cateogories and monadic functors. Moreover, we prove Theorem 2. The
second section deals with chain complexes. We describe there presentations of
the $\infty$-categories of chain complexes of modules over a ring $R$ and of
some $\infty$-categories of algebras in chain complexes. This latter
presentations correspond to Theorem 3 and Theorem 4. Finally the appendix
gathers some older results about chain complexes in non negative degrees.

### Acknowledgment

I would like to thank Edouard Balzin for his comments and Gabriele Vezzosi for
presenting me the article [Lur11] which has inspired this paper.

### Notations and conventions

  * $\triangleright$

For us, $\infty$-category will mean locally small $(\infty,1)$-category. Since
we mainly use references from the books Higher Topos Theory [Lur17] and Higher
Algebra [Lur18] by Lurie, we will assume that $\infty$-categories are encoded
using quasi-categories.

  * $\triangleright$

We consider two universes $\mathcal{U}<\mathcal{V}$. Small will mean
$\mathcal{U}$-small. Sets and $\infty$-groupoids will be small while many
categories and $\infty$-categories considered will be $\mathcal{U}$-large, but
still $\mathcal{V}$-small.

  * $\triangleright$

$\mathsf{S}$ will denote the $\infty$-category of groupoids (given for
instance by the simplicial nerve of the simplicial category of Kan complexes).

  * $\triangleright$

The first infinite cardinal will be denoted $\omega$. We will use the
denomination $\omega$-small and $\omega$-presentable instead of compact and
compactly generated. Moreover, a functor that preserves filtered colimits will
be called $\omega$-accessible.

  * $\triangleright$

As in [Lur17], geometric realisation means colimit of a simplicial diagram.

  * $\triangleright$

In a stable $\infty$-category, the loop functor is denoted $\Omega$ and the
suspension functor is denoted $\Sigma$.

##  1\. Presentable infinity-categories and algebras

In this section, we recall some definitions and results about presentable
$\infty$-categories and monadic adjunctions. Then, we prove our main result
that gives conditions for an $\infty$-category to present an $\infty$-category
of algebras in another $\infty$-category equipped with a presentation.

###  1.1. Presheaves infinity categories and their subcategories

######  Definition 1.

Given a small $\infty$-category $\mathsf{D}$, its presheaves $\infty$-category
$\mathsf{Psh}(\mathsf{D})$ is the mapping $\infty$-category from
$\mathsf{D}^{\mathrm{op}}$ to $\mathsf{S}$

$\mathsf{Psh}(\mathsf{D})=\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right).$

######  Proposition 1.

[Lur17, Theorem 5.1.5.6, Corollary 5.5.2.9, Remark 5.5.2.10] The presheaves
$\infty$-category $\mathsf{Psh}(\mathsf{D})$ is the free cocomplete
$\infty$-category built from $\mathsf{D}$ in the sense that it is cocomplete
and for any cocomplete $\infty$-category $\mathsf{C}$, the two functors

$\mathrm{Fun}^{\mathrm{L}}\left({\mathsf{Psh}(\mathsf{D})},{\mathsf{C}}\right)\hookrightarrow\mathrm{Fun}^{\mathrm{cc}}\left({\mathsf{Psh}(\mathsf{D})},{\mathsf{C}}\right)\to\mathrm{Fun}\left(\mathsf{D},\mathsf{C}\right)$

are both equivalences where
$\mathrm{Fun}^{\mathrm{L}}\left({\mathsf{Psh}(\mathsf{D})},{\mathsf{C}}\right)$
and
$\mathrm{Fun}^{\mathrm{cc}}\left({\mathsf{Psh}(\mathsf{D})},{\mathsf{C}}\right)$
denote respectively the full subcategories of
$\mathrm{Fun}\left(\mathsf{Psh}(\mathsf{D}),\mathsf{C}\right)$ spanned
respectively by left adjoint functors and functors that preserve small
colimits.

######  Proposition 2.

[Lur17, Corollary 5.1.2.3] The presheaves $\infty$-category
$\mathsf{Psh}(\mathsf{D})$ is also complete. Moreover, the limits and colimits
are computed objectwise in the sense that for any object $d\in\mathsf{D}$, the
evaluation at $d$ functor

$\mathsf{Psh}(\mathsf{D})=\mathrm{Fun}\left(\mathsf{D}^{\mathrm{op}},\mathsf{S}\right)\xrightarrow{F\mapsto
F(d)}\mathsf{S}$

preserves limits and colimits.

######  Definition 2.

Let $\kappa$ be a regular cardinal. The $\kappa$ ind completion
$\mathsf{Ind}_{\kappa}(\mathsf{D})$ of a small $\infty$-category $\mathsf{D}$
is the full subcategory of $\mathsf{Psh}(\mathsf{D})$ spanned by functors
$F:\mathsf{D}^{\mathrm{op}}\to\mathsf{S}$ whose Grothendieck construction
(that is the source of the right fibration that classifies this functor) is a
filtered $\infty$-category.

######  Proposition 3.

[Lur17, Corollary 5.3.5.4] A functor $F\in\mathsf{Psh}(\mathsf{D})$ belongs to
$\mathsf{Ind}_{\kappa}(\mathsf{D})$ if and only if there exists a
$\kappa$-filtered diagram $\mathsf{I}\to\mathsf{D}$ so that $F$ is the colimit
in $\mathsf{Psh}(\mathsf{D})$ of the diagram

$\mathsf{I}\to\mathsf{D}\hookrightarrow\mathsf{Psh}(\mathsf{D}).$

######  Proposition 4.

[Lur17, Corollary 5.3.5.4] If the small category $\mathsf{D}$ admits
$\kappa$-small colimits, then a functor $F\in\mathsf{Psh}(\mathsf{D})$ belongs
to $\mathsf{Ind}_{\kappa}(\mathsf{D})$ if and only if it preserves
$\kappa$-small colimits.

######  Definition 3.

Let $\mathsf{D}$ be a small $\infty$-category that has finite coproducts.
Then, let $\mathsf{Psh}_{\Sigma}(\mathsf{D})$ be the full subcategory of
$\mathsf{Psh}(\mathsf{D})$ spanned by functors $F$ that preserves finite
products.

######  Proposition 5.

[Lur17, Proposition 5.5.8.10] The inclusion functor
$\mathsf{Psh}_{\Sigma}(\mathsf{D})\to\mathsf{Psh}(\mathsf{D})$ preserves
sifted colimits and is right adjoint. In particular,
$\mathsf{Psh}_{\Sigma}(\mathsf{D})$ is cocomplete.

######  Proposition 6.

[Lur17, Lemma 5.5.8.14] A functor $F\in\mathsf{Psh}(\mathsf{D})$ belongs to
$\mathsf{Psh}_{\Sigma}(\mathsf{D})$ if and only if it obtained as the
geometric realisation in $\mathsf{Psh}(\mathsf{D})$ of a simplicial object $G$
whose simplicies belong to
$\mathsf{Ind}_{\omega}(\mathsf{D})\subseteq\mathsf{Psh}(\mathsf{D})$.

###  1.2. Presentable infinity-category

######  Definition 4.

Let $\kappa$ be an regular cardinal. An $\infty$-category $\mathsf{C}$ is
$\kappa$-presentable if there exists a small $\infty$-category $\mathsf{D}$
and an adjunction

${\mathsf{Psh}(\mathsf{D})}$${\mathsf{C}}$

whose right adjoint is full faithful and preserves $\kappa$-filtered colimits.

######  Remark 1.

The definition of a $\kappa$-presentable category is different from that
[Lur17]. But, the fact that both definitions describe the same notion follows
from [Lur17, Corollary 5.5.7.3].

######  Lemma 1.

[Lur17, Proposition 5.5.4.15] Let $\kappa$ be a regular cardinal. Let
$\mathsf{D}$ be a small category and let $S$ be a small set of morphisms of
$\mathsf{Psh}(\mathsf{D})$ between $\kappa$-small objects. Then the inclusion
of the full subcategory of $\mathsf{Psh}(\mathsf{D})$ spanned by $S$-local
objects into $\mathsf{Psh}(\mathsf{D})$ preserves $\kappa$-filtered colimits
and is right adjoint. Hence, this full subcategory is $\kappa$-presentable.

###### Proof.

From [Lur17, Proposition 5.5.4.15] that uses the small object argument, we get
the left adjoint (together with some accessibility condition on the right
adjoint inclusion). It suffices to show that the full subcategory of $S$-local
objects is stable through $\kappa$-filtered colimits (which follows from the
same arguments that those used to prove the accessibility condition). For any
$\kappa$-filtered diagram $D:K\to\mathsf{Psh}(\mathsf{D})$ which takes values
in the full subcategory of $S$-local objects and for any map $s:X\to Y$ in
$S$, let us consider the following commutative square

${\varinjlim\mathrm{Map}_{\mathsf{Psh}(\mathsf{D})}\left(Y,D\right)}$${\varinjlim\mathrm{Map}_{\mathsf{Psh}(\mathsf{D})}\left(X,D\right)}$${\mathrm{Map}_{\mathsf{Psh}(\mathsf{D})}\left(Y,\varinjlim
D\right)}$${\mathrm{Map}_{\mathsf{Psh}(\mathsf{D})}\left(X,\varinjlim
D\right).}$

Since $D$ takes values in $S$-local objects, the top horizontal arrow is an
equivalence. Since $X$ and $Y$ are $\kappa$-small, the vertical arrows are
equivalences. Hence the bottom horizontal map is also an equivalence. Thus the
colimit of $D$ is $S$-local. This shows that the subcategory of $S$-local
objects is stable through $\kappa$-filtered colimits and so the inclusion
functor preserves these $\kappa$-filtered colimits. ∎

######  Example 1.

Let $\mathsf{D}$ be a small $\infty$-category with finite coproducts. Then
$\mathsf{Psh}_{\Sigma}(\mathsf{D})$ is $\omega$-presentable and the inclusion
functor

$\mathsf{Psh}_{\Sigma}(\mathsf{D})\to\mathsf{Psh}(\mathsf{D})$

is right adjoint and preserves filtered colimits.

######  Theorem 1.

[Lur17, Theorem 5.5.1.1, Corollary 5.5.2.4], [Sim99] A presentable
$\infty$-category is complete and cocomplete and the full subcategory of its
$\lambda$-small objects is small for any small cardinal $\lambda$.

Let us end this subsection by dealing with stable $\omega$-presentable
$\infty$-categories. Let us consider a stable cocomplete $\infty$-category
$\mathsf{C}$, and a small full subcategory $\mathsf{D}$ of $\mathsf{C}$
spanned by $\omega$-small objects. This yields an adjunction

${\mathsf{Psh}(\mathsf{D})}$${\mathsf{C}.}$

whose right adjoint preserves filtered colimits.

Let us consider a reflective full subcategory $\mathsf{E}$ of
$\mathsf{Psh}(\mathsf{D})$ that is stable, that contains the image of
$\mathsf{C}$ and that is stable through limits and filtered colimits. We have
an adjunction

${\mathsf{E}}$${\mathsf{C}}$$\scriptstyle{L}$$\scriptstyle{R}$

which factors the previous adjunction relating $\mathsf{Psh}(\mathsf{D})$ and
$\mathsf{C}$.

######  Lemma 2.

The adjunction $L\dashv R$ is a coreflective localisation in the sense that
the functor $L$ is fully faithful.

###### Proof.

We can notice that $R$ preserves finite limits. Hence it preserves finite
colimits (by stability). Since it preserves filtered colimits, then it
preserves all colimits by [Lur17, Corollary 4.2.3.12]. Any element
$d\in\mathsf{Psh}(\mathsf{D})$ in the image of the Yoneda embedding
$D\to\mathsf{Psh}(\mathsf{D})$ belongs to the image of $\mathsf{C}$ and thus
belongs to $\mathsf{E}$. For any such element, the map $d\to RL(d)$ is an
equivalence by Yoneda’s lemma (since $\mathsf{D}$ is a full subcategory of
$\mathsf{C}$). Thus, since $R$ and $L$ preserves colimits and since
$\mathsf{E}$ is generated by colimits of diagrams made up of such elements
$d$, then for any $e\in\mathsf{E}$, the map $e\to RL(e)$ is an equivalence. ∎

######  Proposition 7.

The adjunction $L\dashv R$ relating $\mathsf{C}$ to $\mathsf{E}$ is an adjoint
equivalence if and only if the right adjoint functor $R$ is conservative.

###### Proof.

It is a direct consequence of Lemma 2. ∎

###  1.3. Strongly monadic functor

######  Definition 5.

A functor $F:\mathsf{A}\to\mathsf{C}$ is strongly monadic if $\mathsf{A}$ has
all geometric realisations and if $F$ is right adjoint and conservative and
preserves geometric realisations. Moreover, an adjunction is strongly monadic
of its right adjoint is strongly monadic.

######  Notation.

A strongly monadic functor from $\mathsf{A}$ to $\mathsf{C}$ will be denoted
$U^{M}$, its left adjoint will be denoted $T_{M}$ and $M$ will be the related
monad $M=U^{M}\circ T_{M}$.

######  Lemma 3.

[RV16] Let $U^{M}:\mathsf{A}\to\mathsf{C}$ be a (strongly) monadic functor,
and let $Q=T_{M}U^{M}$. Then one has one gets a functor

$\displaystyle(\Delta^{\mathrm{op}})^{\triangleright}=\Delta_{+}^{\mathrm{op}}$
$\displaystyle\to\mathrm{Fun}\left(\mathsf{A},\mathsf{A}\right)$
$\displaystyle n$ $\displaystyle\mapsto Q^{n+1},$

so that for any object $A\in\mathsf{A}$, the augmented simplicial object
$n\mapsto Q^{n+1}A$ is the colimit of the underlying simplicial diagram.

###### Proof.

Let $\mathsf{Adj}$ be the 2-category that encodes adjunctions. It has two
objects $0$ and $1$ and is generated by morphisms $l:0\to 1$ and $r:1\to 0$
and 2-morphisms $\eta:\mathrm{Id}_{0}\to rl$ and
$\epsilon:lr\to\mathrm{Id}_{1}$ that satisfy the same relations than those
defining adjunctions. We know from [RV16], that the adjunction $T_{M}\dashv
U^{M}$ induces a functor of categories enriched in $\infty$-categories

$\mathsf{Adj}\to\mathsf{QuasiCats}.$

that maps $l$ to the left adjoint and $r$ to the right adjoint. This, gives us
the functor

$\displaystyle\Delta_{+}^{\mathrm{op}}\simeq\mathsf{Adj}(1,1)$
$\displaystyle\to\mathrm{Fun}\left(\mathsf{A},\mathsf{A}\right),$

that sends $n\in\Delta_{+}^{\mathrm{op}}$ to $Q^{n+1}$. Besides, composition
with $r$ in the 2-category $\mathsf{Adj}$ recovers the canonical inclusion

$\Delta_{+}^{\mathrm{op}}\simeq\mathsf{Adj}(1,1)\xrightarrow{r\circ-}\mathsf{Adj}(1,0)\simeq\Delta_{\infty}^{\mathrm{op}}.$

Subsequently, the image through $U^{M}\circ-$ of the augmented simplicial
object $n\mapsto Q^{n+1}$ splits. In particular, for any object
$A\in\mathsf{A}$ the augmented simplicial object in $\mathsf{A}$

$n\mapsto Q^{n+1}A$

is $U^{M}$-split. Hence, by monadicity it is colimiting. ∎

######  Lemma 4.

Let $D:\mathsf{I}\times\mathsf{J}\to\mathsf{E}$ be a functor of
$\infty$-categories. The two following assertions are equivalent

  1. (1)

$D$ admits a left Kan extension with respect to the inclusion

$\mathsf{I}\times\mathsf{J}\to\mathsf{I}^{\triangleright}\times\mathsf{J};$

  2. (2)

the diagram

$D_{j}:\mathsf{I}\simeq\mathsf{I}\times\\{j\\}\hookrightarrow\mathsf{I}\times\mathsf{J}\to\mathsf{E}$

admits a colimit for any $j\in\mathsf{J}$.

Moreover, in the case these are true, the restriction of the left Kan
extension to $\mathsf{I}^{\triangleright}\times\\{j\\}$ is the colimit of
$D_{j}$ for any $j\in\mathsf{J}$.

###### Proof.

Let us denote $\ast$ the final object of $\mathsf{I}^{\triangleright}$. From
[Lur17, Lemma 4.3.2.13], we know that the assertion (1) is equivalent to the
fact that the functor

$(\mathsf{I}\times\mathsf{J})_{/(\ast,j)}\to\mathsf{I}\times\mathsf{J}\to\mathsf{E}$

admits a colimit for any $j\in\mathsf{J}$. Then this colimit will give the
value of the left Kan extension of $D$ on the object $(\ast,j)$.

To conclude, it suffices to notice that the functor

$\mathsf{I}\simeq\mathsf{I}\times\\{j\\}\to(\mathsf{I}\times\mathsf{J})_{/(\ast,j)}$

is cofinal. ∎

######  Lemma 5.

[Lur17, Corollary 4.3.2.16 and Proposition 4.3.2.17] Let
$i:\mathsf{K}\to\mathsf{J}$ be an inclusion functor. If every functor $F$ from
$\mathsf{I}$ to an $\infty$-category $\mathsf{E}$ admits a left Kan extension,
then the functor

$\mathrm{Fun}\left(\mathsf{J},\mathsf{E}\right)\to\mathrm{Fun}\left(\mathsf{K},\mathsf{E}\right)$

given by precomposition with $i$ admits a left adjoint whose value on any
functor is given by its left Kan extension.

######  Proposition 8.

Let $U^{M}:\mathsf{A}\to\mathsf{C}$ be a strongly monadic functor. If the
$\infty$-category $\mathsf{C}$ is cocomplete, then $\mathsf{A}$ is also
cocomplete.

###### Proof.

Let us denote $Q=T_{M}\circ U^{M}$. Let us consider a diagram functor
$D:K\to\mathsf{A}$. Let us show that $D$ admits a colimit. From Lemma 3 we get
a functor

$\displaystyle D^{\Delta_{+}}:K\times\Delta_{+}^{\mathrm{op}}$
$\displaystyle\to\mathsf{A}$ $\displaystyle(k,n)$ $\displaystyle\mapsto
Q^{n+1}D(k).$

We know from Lemma 6, that the functor $D^{\Delta_{+}}$ admits a left kan
extension $X$ with respect to the inclusion

$K\times\Delta_{+}^{\mathrm{op}}\to
K^{\triangleright}\times\Delta_{+}^{\mathrm{op}}.$

Then by Lemma 4, the functor

$K\simeq K\times\\{-1\\}\hookrightarrow
K\times\Delta_{+}^{\mathrm{op}}\to\mathsf{A}$

which is just $D$ admits a colimit. ∎

######  Lemma 6.

The functor $D^{\Delta_{+}}$ described in the proof of Proposition 8 admits a
left kan extension $X$ with respect to the inclusion

$K\times\Delta_{+}^{\mathrm{op}}\to
K^{\triangleright}\times\Delta_{+}^{\mathrm{op}}.$

###### Proof.

Let us consider the following commutative square

${K\times\Delta^{\mathrm{op}}}$${K^{\triangleright}\times\Delta^{\mathrm{op}}}$${K\times\Delta_{+}^{\mathrm{op}}}$${K^{\triangleright}\times\Delta_{+}^{\mathrm{op}}.}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{k}$$\scriptstyle{l}$

Let $D^{\Delta}$ be the restriction of $D^{\Delta_{+}}$ onto
$K\times\Delta^{\mathrm{op}}$. Then, by Lemma 4 and Lemma 3, $D^{\Delta_{+}}$
is the left Kan extension of $D^{\Delta}$ along $j$.

Let us show that $D^{\Delta}$ admits a left Kan extension with respect to
$k\circ i\simeq l\circ j$. First, for any $n\in\Delta$, the functor

$K\simeq K\times\\{n\\}\hookrightarrow
K\times\Delta^{\mathrm{op}}\xrightarrow{D^{\Delta}}\mathsf{A}$

admits a colimit since it decomposes as

$K\xrightarrow{U^{M}Q^{n}D}\mathsf{C}\xrightarrow{T_{M}}\mathsf{A}$

and since $\mathsf{C}$ is cocomplete and $T_{M}$ preserves colimits. Hence by
Lemma 4, $D^{\Delta}$ admits a left Kan extension along $i$. Again by Lemma 4
and since $\mathsf{A}$ admits geometric realisations, then this left Kan
extension admits a further left Kan extension along $k$, that we denote $X$.
Then, $X$ is the left Kan extension of $D^{\Delta}$ along $k\circ i\simeq
l\circ j$.

Using Lemma 4, we can prove that the restriction of $X$ to
$K\times\Delta_{+}^{\mathrm{op}}$ is a left Kan extension of $D^{\Delta}$ with
respect $j$, as is $D^{\Delta_{+}}$. Hence, both functors are equivalent
(apply for instance Lemma 5). Moreover, $X$ is a left Kan extension of
$D^{\Delta_{+}}$ ([Lur17, Proposition 4.3.2.8]). ∎

######  Definition 6.

An adjunction is called $\omega$-accessible if the source of its right adjoint
has filtered colimits and if this right adjoint preserves filtered colimits.

######  Corollary 1.

Let $U^{M}:\mathsf{A}\to\mathsf{C}$ be a strongly monadic functor and let us
suppose that $\mathsf{C}$ is cocomplete. Then, the two following assertions
are equivalent.

  1. (1)

the adjunction $T_{M}\dashv U^{M}$ is $\omega$-accessible, that is $U^{M}$
preserve filtered colimits;

  2. (2)

$U^{M}$ preserves small sifted limits.

###### Proof.

Since $\mathsf{C}$ is cocomplete, then by Proposition 8, $\mathsf{A}$ is also
cocomplete. Then, the result is a direct application of the fact that a
functor from a cocomplete category preserves sifted colimits if and only if it
preserves filtered colimits and geometric realisations ([Lur17, Corollary
5.5.8.17]). ∎

###  1.4. Presentations of algebras

####  1.4.1. The context

Let $\mathsf{C}$ be an $\omega$-presentable $\infty$-category. Thus, let us
consider a small $\infty$-category $\mathsf{D}$, a functor
$i:\mathsf{D}\to\mathsf{C}$ and the induced adjunction

${\mathsf{Psh}(\mathsf{D})}$${\mathsf{C}}$$\scriptstyle{i_{!}}$$\scriptstyle{i^{\ast}}$

so that the map $i_{!}i^{\ast}\to\mathrm{Id}$ is an equivalence and so that
$i^{\ast}$ preserves filtered colimits.

Besides, let us consider another adjunction

${\mathsf{C}}$${\mathsf{A}}$$\scriptstyle{T_{M}}$$\scriptstyle{U^{M}}$

that is $\omega$-accessible and strongly monadic. Thus, $\mathsf{A}$ is
cocomplete by Proposition 8 and $U^{M}$ preserves sifted colimits.

Let us consider a full subcategory embedding
$j:\mathsf{B}\hookrightarrow\mathsf{A}$ so that $\mathsf{B}$ is small and so
that it contains the image of $\mathsf{D}$, in the sense that for any object
$d$ of $\mathsf{D}$ its image through the functor $T_{M}\circ
i:\mathsf{D}\to\mathsf{A}$ belong essentially to $\mathsf{B}$. We thus get a
functor $t:\mathsf{D}\to\mathsf{B}$ so that the following square of
$\infty$-categories is commutative (up to homotopy).

${\mathsf{B}}$${\mathsf{D}}$${\mathsf{A}}$${\mathsf{C}}$$\scriptstyle{j}$$\scriptstyle{t}$$\scriptstyle{i}$$\scriptstyle{T_{M}}$

From the universal property of preasheaves categories and the fact that
$T_{M}$ preserves colimits, one gets another commutative square

${\mathsf{Psh}(\mathsf{B})}$${\mathsf{Psh}(\mathsf{D})}$${\mathsf{A}}$${\mathsf{C}.}$$\scriptstyle{j_{!}}$$\scriptstyle{t_{!}}$$\scriptstyle{i_{!}}$$\scriptstyle{T_{M}}$

Noticing that all the functors in the square just above are left adjoint, one
gets another square

${\mathsf{Psh}(\mathsf{B})}$${\mathsf{Psh}(\mathsf{D})}$${\mathsf{A}}$${\mathsf{C}.}$$\scriptstyle{t^{\ast}}$$\scriptstyle{U^{M}}$$\scriptstyle{j^{\ast}}$$\scriptstyle{i^{\ast}}$

whose functors are all right adjoints. We can notice that the right adjoint
functor

$\displaystyle t^{\ast}:\mathsf{Psh}(\mathsf{D})$
$\displaystyle\to\mathsf{Psh}(\mathsf{B})$ $\displaystyle F$
$\displaystyle\mapsto F\circ t$

preserves colimits. Actually, it has a further right adjoint given by right
Kan extension $t_{\ast}$ of $t$.

######  Hypothesis 1.

Let us suppose that:

  * $\triangleright$

the full subcategory $\mathsf{B}$ is stable under finite coproducts;

  * $\triangleright$

its objects are $\omega$-small in $\mathsf{A}$; equivalently, the functor
$\mathsf{A}\to\mathsf{Psh}(\mathsf{B})$ preserves filtered colimits.

In particular, the image of the functor
$\mathsf{A}\to\mathsf{Psh}(\mathsf{B})$ lies in
$\mathsf{Psh}_{\Sigma}(\mathsf{B})$.

Moreover, let $\mathsf{E}$ be a full subcategory of
$\mathsf{Psh}_{\Sigma}(\mathsf{B})$ that satisfies the following hypothesis.

######  Hypothesis 2.

We suppose that

  * $\triangleright$

the subcategory $\mathsf{E}$ contains all the images of objects of
$\mathsf{A}$;

  * $\triangleright$

the functor $\mathsf{E}\to\mathsf{Psh}_{\Sigma}(\mathsf{B})$ has a left
adjoint and preserves filtered colimits; equivalently, the functor
$\mathsf{E}\to\mathsf{Psh}(\mathsf{B})$ has a left adjoint and preserves
filtered colimits.

The adjunction $j_{!}\dashv j^{\ast}$ relating $\mathsf{A}$ to
$\mathsf{Psh}(\mathsf{B})$ decomposes into a sequence of two adjunctions

${\mathsf{Psh}(\mathsf{B})}$${\mathsf{E}}$${\mathsf{A}}$$\scriptstyle{p}$$\scriptstyle{l}$$\scriptstyle{q}$$\scriptstyle{r}$

where the functor $l$ is just given by $j_{!}\circ q$.

######  Remark 2.

The hypothesis are redundant. For instance the fact that the functor $U_{M}$
preserves filtered colimits follows from the fact that the composite functor
$\mathsf{A}\to\mathsf{Psh}(\mathsf{B})\to\mathsf{Psh}(\mathsf{D})$ preserves
filtered colimits and the functor $\mathsf{C}\to\mathsf{Psh}(\mathsf{D})$
preserves filtered colimits and is conservative.

####  1.4.2. Presenting algebras

Our goal is to prove the following theorem.

######  Theorem 2.

The following assertions are equivalent.

  1. (1)

the adjunction $l\dashv r$ relating $\mathsf{A}$ to $\mathsf{E}$ is an adjoint
equivalence;

  2. (2)

the composite functor

$\mathsf{E}\xrightarrow{q}\mathsf{Psh}(\mathsf{B})\xrightarrow{t^{\ast}}\mathsf{Psh}(\mathsf{D})$

is conservative and sends elements of $\mathsf{E}$ to the essential image of
the functor $i^{\ast}$ from $\mathsf{C}$ to $\mathsf{Psh}(\mathsf{D})$;

  3. (3)

the composite functor

$\mathsf{E}\xrightarrow{q}\mathsf{Psh}(\mathsf{B})\xrightarrow{t^{\ast}}\mathsf{Psh}(\mathsf{D})\xrightarrow{i_{!}}\mathsf{C}$

is conservative.

###### Proof.

First, let us suppose (1), that is $l\dashv r$ is an adjoint equivalence, then

$t^{\ast}\circ q\simeq t^{\ast}\circ q\circ r\circ l\simeq i^{\ast}\circ
U^{M}$

is conservative and its image lies in the essential image of $i^{\ast}$. This
proves (2).

Let us suppose (2). For any morphism $f:X\to Y$ in $\mathsf{E}$, if
$i_{!}t^{\ast}q(f)$ is an equivalence, then $i\ast i_{!}t^{\ast}q(f)$ is an
equivalence and so $t^{\ast}q(f)$ is an equivalence since the maps $X\to
i^{\ast}i_{!}(X)$ and $Y\to i^{\ast}i_{!}(Y)$ are equivalences. Thus, $f$ is
an equivalence since $t^{\ast}q$ is conservative. So $i_{!}t^{\ast}q$ is
conservative, that is (3) is true.

Finally, let us suppose (3), that is $i_{!}\circ u^{\ast}\circ q$ is
conservative. We can prove (1), that is the adjunction $l\dashv r$ is an
adjoint equivalence by showing that

  1. (1)

the functor $r$ is conservative

  2. (2)

the map $\mathrm{Id}_{\mathsf{E}}\to rl$ is an equivalence.

The first point is Lemma 7 and the second point is Lemma 9 ∎

######  Lemma 7.

If the composite functor $i_{!}\circ t^{\ast}\circ q$ is conservative then the
functor $r$ is conservative.

###### Proof.

One can notice that $U_{M}\simeq i_{!}\circ t^{\ast}\circ q\circ r$. Since
$U_{M}$ and $i_{!}\circ t^{\ast}\circ q$ are conservative, then $r$ is also
conservative. ∎

######  Lemma 8.

The morphism $F\to j^{\ast}j_{!}(F)$ is an equivalence for any element
$F\in\mathsf{Ind}_{\omega}(\mathsf{B})\subseteq\mathsf{Psh}(\mathsf{B})$.

###### Proof.

This is an equivalence for any $F$ that belong to the image of the Yoneda
embedding of $\mathsf{B}$ since $B$ is a full sucategory of $\mathsf{A}$. To
conclude, it suffices to notice that both functors $j_{!}$ and $j^{\ast}$
preserves filtered colimits. ∎

######  Lemma 9.

If the composite functor $i_{!}\circ t^{\ast}\circ q$ is conservative then the
morphism $F\to j^{\ast}j_{!}(F)$ is an equivalence for any element
$F\in\mathsf{E}\subseteq\mathsf{Psh}(\mathsf{B})$.

###### Proof.

Since $F$ belongs in particular to $\mathsf{Psh}_{\Sigma}(\mathsf{B})$, it is
the colimit in $\mathsf{Psh}(\mathsf{B})$ of a simplicial object

$(G_{n})_{n\in\Delta^{op}}$

where each $G_{n}$ belongs to $\mathsf{Ind}_{\omega}(\mathsf{B})$.

Then, the morphism $F\to j^{\ast}j_{!}(F)$ decomposes as

${\varinjlim_{n\in\Delta^{op}}G_{n}}$${\varinjlim_{n\in\Delta^{op}}j^{\ast}j_{!}(G_{n})}$${j^{\ast}(\varinjlim_{n\in\Delta^{op}}j_{!}(G_{n}))}$${j^{\ast}j_{!}(\varinjlim_{n\in\Delta^{op}}G_{n})}$${F}$${j^{\ast}j_{!}(F).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Let us show that the three following maps are equivalences:

  1. (1)

$\varinjlim_{n\in\Delta^{op}}G_{n}\to\varinjlim_{n\in\Delta^{op}}j^{\ast}j_{!}(G_{n})$;

  2. (2)

$j^{\ast}(\varinjlim_{n\in\Delta^{op}}j_{!}(G_{n}))\to
j^{\ast}j_{!}(\varinjlim_{n\in\Delta^{op}}G_{n})$;

  3. (3)

$\varinjlim_{n\in\Delta^{op}}j^{\ast}j_{!}(G_{n})\to
j^{\ast}(\varinjlim_{n\in\Delta^{op}}j_{!}(G_{n}))$.

We can notice that since each $G_{n}$ is in
$\mathsf{Ind}_{\omega}(\mathsf{B})$, then the map $G_{n}\to
j^{\ast}j_{!}(G_{n})$ is an equivalence by Lemma 8. Thus, the map (1) is an
equivalence. In particular, its target belongs to $\mathsf{E}$. The map (2) is
also an equivalence since $j_{!}$ is left adjoint.

It remains to show that the map (3) is an equivalence. Since this morphism
belong to the essential image of $q$ (both the target and the source
essentially belong to the full sucategory $\mathsf{E}$), it suffices to show
that its image through $i_{!}\circ t^{\ast}$ is an equivalence (since
$i_{!}\circ t^{\ast}\circ q$ is conservative). Let us consider the composite
map

$\varinjlim_{n\in\Delta^{op}}i_{!}t^{\ast}j^{\ast}j_{!}(G_{n})\to
i_{!}t^{\ast}(\varinjlim_{n\in\Delta^{op}}j^{\ast}j_{!}(G_{n}))\to
i_{!}t^{\ast}j^{\ast}(\varinjlim_{n\in\Delta^{op}}j_{!}(G_{n}))$

Its first part is an equivalence since $i_{!}\circ t^{\ast}$ preserves
colimits. Besides, since $i_{!}\circ t^{\ast}\circ j^{\ast}\simeq U_{M}$
preserves geometric realisations, then the composite map is also an
equivalence. Hence, the second map

$i_{!}t^{\ast}(\varinjlim_{n\in\Delta^{op}}j^{\ast}j_{!}(G_{n}))\to
i_{!}t^{\ast}j^{\ast}(\varinjlim_{n\in\Delta^{op}}j_{!}(G_{n}))$

is also an equivalence. Hence, $F\to j^{\ast}j_{!}(F)$ is an equivalence. ∎

##  2\. Chain complexes and their algebras

Let $R$ be a ring. The goal of this section is to describe two types of
presentations of an $\infty$-categories of algebras in chain complexes.

We first recall some results about the link between combinatorial model
categories and $\infty$-categories. Then we describe a presentation of the
$\infty$-category of chain complexes and of its product over a set $K$. This
allows us to apply Theorem 2 to describe the cellular and the cartesian
presentations of an $\infty$-category related to chain complexes through an
$\omega$-accessible strongly monadic adjunction. We conclude this section by
describing how these presentations fit the context of algebras over a coloured
operad in chain complexes.

###  2.1. Combinatorial model categories

Let $\mathsf{C}$ be a combinatorial model category and let $\mathsf{C}^{c}$ be
its full subcategory of cofibrant objects.

######  Notation.

For any marked simplicial $(X,\mathrm{W})$, we will denote by
$X[\mathrm{W}^{-1}]$ the underlying quasi-category of a fibrant replacement of
$(X,\mathrm{W})$ for the cartesian model structure on marked simplicial sets.
This applies for instance to relative categories.

######  Lemma 10.

[Lur17, Corollary 3.1.4.3.] The cartesian model structure on the category of
marked simplicial sets is monoidal with respect to the cartesian monoidal
structure.

######  Lemma 11.

The functor

$\mathsf{C}^{c}[\mathrm{W}^{-1}]\to\mathsf{C}[\mathrm{W}^{-1}]$

is an equivalence of $\infty$-categories.

###### Proof.

The functor of relative categories from $(\mathsf{C}^{c},\mathrm{W})$ to
$(\mathsf{C},\mathrm{W})$ induces a morphism of marked simplicial sets

$f:(N(\mathsf{C}^{c}),\mathrm{W})\to(N(\mathsf{C}),\mathrm{W}).$

It suffices to show that this morphism $f$ is an equivalence of marked
simplicial sets.

Since the model category $\mathsf{C}$ is combinatorial, it has a cofibrant
replacement functor which gives another morphism of marked simplicial sets

$g:(N(\mathsf{C}),\mathrm{W})\to(N(\mathsf{C}^{c}),\mathrm{W}).$

Let $e$ be the marked simplicial set which consists in a marked arrow. Then
for any marked simplicial set $X$, $X\times e$ is a cylinder object for $X$.
Moreover, the cofibrant replacement functor gives us maps

$(N(\mathsf{C}),\mathrm{W})\times
e\to(N(\mathsf{C}),\mathrm{W});\quad(N(\mathsf{C}^{c}),\mathrm{W})\times
e\to(N(\mathsf{C}^{c}),\mathrm{W})$

which relate on the one hand $fg$ to $\mathrm{Id}$ and on the other hand $gf$
to $\mathrm{Id}$. Hence $f$ is an equivalence. ∎

######  Proposition 9.

[Lur18, Proposition 1.3.4.22] The $\infty$-category
$\mathsf{C}^{c}[\mathrm{W}^{-1}]$ is $\kappa$-presentable for some regular
cardinal $\kappa$.

######  Proposition 10.

[Lur18, Proposition 1.3.4.24] Let us consider a functor
$D:\mathsf{I}\to\mathsf{C}^{c}$ where $\mathsf{I}$ is a small category. Then,
a cocone of $D$ is its homotopy colimit if and only if the induced cocone in
$\mathsf{C}^{c}[\mathrm{W}^{-1}]$ is colimiting.

######  Proposition 11.

[Lur18, Proposition 1.3.4.25] For any small category $\mathsf{I}$, the
category $\mathrm{Fun}\left(\mathsf{I},\mathsf{C}\right)$ admits a
combinatorial model structure whose weak equivalences are objectwise weak
equivalence. Moreover, the functor

$\mathrm{Fun}\left(\mathsf{I},\mathsf{C}^{c}\right)[\mathrm{W}^{-1}]\to\mathrm{Fun}\left(\mathsf{I},{\mathsf{C}^{c}}[\mathrm{W}^{-1}]\right)$

is an equivalence of $\infty$-categories.

######  Lemma 12.

Let $\kappa$ be a regular cardinal and let $F:\mathsf{C}\to\mathsf{D}$ be a
functor between combinatorial model categories that preserves weak
equivalences. Let us suppose weak equivalences in $\mathsf{C}$ and
$\mathsf{D}$ are stable through $\kappa$-filtered colimits and that $F$
preserves $\kappa$-filtered colimits. Then the induced functor

$\mathsf{C}[\mathrm{W}^{-1}]\to\mathsf{D}[\mathrm{W}^{-1}]$

preserves $\kappa$-filtered colimits.

###### Proof.

For any $\kappa$-filtered $\infty$-category $\mathsf{I}$, there exists a
$\kappa$-filtered poset $\mathsf{P}$ and a cofinal map from this poset to
$\mathsf{I}$ ([Lur17, Proposition 5.3.1.18]). Thus, using Proposition 10 and
Proposition 11, it suffices to prove that $F$ preserves homotopy colimits
indexed by $\kappa$-filtered posets. To conclude, such homotopy colimits are
given by colimits in these two model categories. ∎

######  Proposition 12.

[MG16] Let us consider a Quillen adjunction

${\mathsf{C}}$${\mathsf{D}}$$\scriptstyle{L}$$\scriptstyle{R}$

between combinatorial model categories. Let $Q$ be a cofibrant replacement
functor for $\mathsf{C}$ and let $F$ be a fibrant replacement functor for
$\mathsf{D}$. Then the functors derived from $LQ$ and $RF$ that relate
$\mathsf{C}[\mathrm{W^{-1}}]$ and $\mathsf{D}[\mathrm{W^{-1}}]$ form a pair of
a left and a right adjoint functors.

######  Lemma 13.

Let $(X_{i})_{i\in I}$ be a collection of marked simplicial sets. Then the
functor

$(\prod_{i\in I}X_{i})[\mathrm{W}^{-1}]\to\prod_{i\in
I}(X_{i}[\mathrm{W}^{-1}])$

is an equivalence.

###### Proof.

It amounts to show that the morphism of marked simplicial sets

$\prod_{i\in I}X_{i}\to\prod_{i\in I}(X_{i}[\mathrm{W}^{-1}])$

is a weak equivalence. If $I$ is finite, this follows from the fact that the
cartesian model structure of marked simplical sets is monoidal. In the general
case, it suffices to notice that the transfinite composition of weak
equivalences in marked simplicial sets is a weak equivalence. ∎

###  2.2. Chain complexes

######  Definition 7.

Let $\mathsf{Chain}_{R}$ be the category of chain complexes of $R$-modules.
For any $m\in\mathbb{Z}$, we denote $S^{m}$ the chain complex which consists
in $R$ in degree $m$ and zero on any other degree. Moreover, $D^{m}$ is the
chain complex which consists in $R$ in degree $m$ and $m-1$ with differential
given by the identity of $R$ and which is zero in any other degree.

######  Proposition 13.

[Hov99] The category $\mathsf{Chain}_{R}$ of chain complexes of $R$-modules
admits a combinatorial model structure whose

  * $\triangleright$

fibrations are degreewise surjections;

  * $\triangleright$

weak equivalences are quasi-isomorphisms;

  * $\triangleright$

generating cofibrations are given by maps

$S^{n}\to D^{n+1},\ n\in\mathbb{Z};$

  * $\triangleright$

generating acyclic cofibrations are given by maps

$0\to D^{n},n\in\mathbb{Z};$

  * $\triangleright$

cofibrations are degreewise injections whose cokernel is degreewise
projective.

The resulting $\infty$-category of chain complexes
$\mathsf{Chain}_{R}[\mathrm{W}^{-1}]$ is denoted $\mathsf{Ch}_{R}$.

######  Proposition 14.

[Lur18, Proposition 1.3.4.5] Let $\mathsf{Chain}^{\mathsf{dg}}_{R}$ be the
differential graded category of chain complexes of $R$-modules and let
$\mathsf{Chain}^{\mathsf{dg},c}_{R}$ be the full subcategory spanned by
objects that are cofibrant for the model structure described above. The
functor

$\mathsf{Chain}^{c}_{R}\to\mathrm{N}_{\mathrm{dg}}(\mathsf{Chain}^{\mathsf{dg},c}_{R})$

induces an equivalence of $\infty$-categories

$\mathsf{Ch}_{R}\simeq\mathsf{Chain}^{c}_{R}[\mathrm{W}^{-1}]\to\mathrm{N}_{\mathrm{dg}}(\mathsf{Chain}^{\mathsf{dg},c}_{R}).$

######  Proposition 15.

[Lur18, Proposition 1.3.2.10] The $\infty$-category $\mathsf{Ch}_{R}$ is
stable.

######  Definition 8.

Let $n\in\mathbb{Z}\sqcup\\{\infty\\}$. We denote $\mathsf{Loops}_{R,\leq n}$
the full subcategory of the $\infty$-category of chain complexes
$\mathsf{Ch}_{R}$ spanned by the objects

$0,S^{m},m\leq n.$

The $\infty$-category $\mathsf{Loops}_{R,\leq n}$ is pointed with $0$ as zero
object. Moreover, for any $m\leq n$

$S^{m}=\Sigma S^{m-1};`\quad\Omega S^{m}=S^{m-1}.$

######  Definition 9.

Let $\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n})$ be the full
subcategory of $\mathsf{Psh}(\mathsf{Loops}_{R,\leq n})$ spanned by the
functors $F:\mathsf{Loops}_{R,\leq n}^{\mathrm{op}}\to\mathsf{S}$ that sends
$0$ to the final $\infty$-groupoid $\ast$ and so that the map

$F(S^{m})\simeq F(\Sigma\Omega S^{m})\to\Omega(F(\Omega S^{m}))$

is an equivalence for any $m\leq n$. A functor that belongs to
$\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n})$ will be called a
stable functors.

######  Lemma 14.

The $\infty$-category $\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq
n})$ is stable.

###### Proof.

From Lemma 1, it is $\omega$-presentable hence complete. The final object of
$\mathsf{Psh}(\mathsf{Loops}_{R,\leq n})$ is the image through the Yoneda
embedding of $0\in\mathsf{Loops}_{R,\leq n}$. It belongs to the full
subcategory of stable functors and then is also the final object of this
subcategory. Moreover, for any stable functor $F$, we have equivalences

$\mathrm{Map}_{\mathsf{Psh}(\mathsf{Loops}_{R,\leq
n})}\left(\ast,F\right)\simeq F(0)\simeq\ast.$

Hence, this final object is a zero object. Moreover, the loop functor is an
equivalence whose pseudo inverse is the precomposition with the loop functor
of $\mathsf{Loops}_{R,\leq n}$. ∎

######  Lemma 15.

The inclusion functor

$\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq
n})\hookrightarrow\mathsf{Psh}(\mathsf{Loops}_{R,\leq n})$

is right adjoint and preserves filtered colimits.

###### Proof.

This is just an application of Lemma 1. ∎

######  Lemma 16.

Any object of the form $S^{m}$ for $m\in\mathbb{Z}$ is $\omega$-small in
$\mathsf{Ch}_{R}$.

###### Proof.

First, the equivalence $\Sigma$ and thus its inverse $\Omega$ preserves
$\omega$-small objects since $0$ is $\omega$-small and finite colimits of
$\omega$-small objects are $\omega$-small. Hence, it suffices to prove that
$S^{0}$ is small.

The Dold-Kan correspondence gives a Quillen adjunction relating the Kan
Quillen model category of simplicial sets to the model category
$\mathsf{Chain}_{R}$. Proving that $S^{0}\in\mathsf{Ch}_{R}$ is $\omega$-small
amounts to prove that the right derived functor

$\mathsf{Chain}[\mathrm{W}^{-1}]\to\mathsf{sSet}[\mathrm{W}^{-1}]$

preserves filtered colimits. This follows from Lemma 12. ∎

######  Proposition 16.

Let $n\in\mathbb{Z}\sqcup\\{\infty\\}$. The functor

$\mathsf{Ch}_{R}\to\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n})$

induced by the inclusion of $\mathsf{Loops}_{R,\leq mn}$ into
$\mathsf{Ch}_{R}$ is an equivalence.

###### Proof.

We have a sequence of adjunctions

${\mathsf{Psh}(\mathsf{Loops}_{R,\leq
n})}$${\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq
n})}$${\mathsf{Ch}_{R}.}$

whose right adjoint functors preserve filtered colimits, since the objects of
$\mathsf{Loops}_{R,\leq n}$ are all $\omega$-small in $\mathsf{Ch}_{R}$.
Moreover, the $\infty$-categories
$\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n})$ and
$\mathsf{Ch}_{R}$ are both stable. To conclude using Proposition 7, it
suffices to notice that the functors

$X\in\mathsf{Ch}_{R}\mapsto\mathrm{Map}_{\mathsf{Ch}_{R}}\left(S^{m},X\right)\in\mathsf{S},\
m\leq n$

are jointly conservative. ∎

###  2.3. Products of chain complexes

Let $K$ be a set. We will call the elements of $K$ colours.

######  Definition 10.

We denote

$\mathsf{Chain}_{R}^{K}=\mathrm{Fun}\left(K,\mathsf{Chain}_{R}\right);\quad\mathsf{Ch}_{R}^{K}=\mathrm{Fun}\left(K,\mathsf{Ch}_{R}\right).$

Moreover, for any $k\in K$, and any $m\in\mathbb{Z}$, let us denote
$S^{m}_{k}$ the object of $\mathsf{Chain}_{R}^{K}$ given by the functor that
sends $k$ to $S^{m}$ and any other $k^{\prime}\neq k$ to $0$. We will denote
also $S^{m}_{k}$ its image in the $\infty$-category $\mathsf{Ch}_{R}^{K}$.
Finally, we define similarly $D^{m}_{k}$.

######  Proposition 17.

The product category $\mathsf{Chain}_{R}^{K}$ admits a combinatorial model
structure whose

  * $\triangleright$

fibrations are colourwise degreewise surjections;

  * $\triangleright$

weak equivalences are colourwise quasi-isomorphisms;

  * $\triangleright$

generating cofibrations are given by maps

$S_{k}^{n}\to D_{k}^{n+1},\ n\in\mathbb{Z},k\in K;$

  * $\triangleright$

generating acyclic cofibrations are given maps

$0\to D_{k}^{n},n\in\mathbb{Z};$

  * $\triangleright$

cofibrations are colourwise degreewise injections whose cokernel is colourwise
degreewise projective.

###### Proof.

This proposition is just an incarnation of the fact that the product of model
categories inherits a model structure whose fibrations, cofibrations and weak
equivalences are componentwise respectively fibrations, cofibrations and weak
equivalences. ∎

######  Proposition 18.

The functor

$\mathsf{Chain}_{R}^{K}\to\mathsf{Ch}_{R}^{K}$

induces an equivalence

$\mathsf{Chain}_{R}^{K}[\mathrm{W}^{-1}]\to\mathsf{Ch}_{R}^{K}.$

###### Proof.

This is just an application of Lemma 13. ∎

######  Definition 11.

For any $n\in\mathbb{Z}\sqcup\\{\infty\\}$, let $\mathsf{Loops}_{R,\leq
n}^{(K)}$ be the full subcategory of $\mathsf{Chain}_{R}^{K}[\mathrm{W}^{-1}]$
spanned by the objects, $0$ and $S^{m}_{k}$ for $m\leq n$ and $k\in K$.

######  Definition 12.

For any $n\in\mathbb{Z}\sqcup\\{\infty\\}$, let
$\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n}^{(K)})$ the full
subcategory of $\mathsf{Psh}(\mathsf{Loops}_{R,\leq n}^{(K)})$ spanned by
functors $F$ so that

  * $\triangleright$

$F(0)\simeq\ast$;

  * $\triangleright$

the map

$F(S^{m}_{k})\simeq F(\Sigma\Omega S^{m-1}_{k})\to\Omega F(\Omega
S^{m-1}_{k})$

is an equivalence for any $m\leq n$ and any $k\in K$.

A functor that satisfies such conditions is called stable.

######  Proposition 19.

For any $n\in\mathbb{Z}\sqcup\\{\infty\\}$, the inclusion

$\mathsf{Loops}_{R,\leq
n}^{(K)}\to\mathsf{Chain}_{R}^{K}\left[\mathrm{W}^{-1}\right]$

induces and equivalence of $\infty$-categories between
$\mathsf{Chain}_{R}^{K}[\mathrm{W}^{-1}]$ and the full subcategory
$\mathsf{Psh}_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n}^{(K)})$ of
$\mathsf{Psh}(\mathsf{Loops}_{R,\leq n}^{(K)})$.

###### Proof.

This follows from the same arguments as in the proof of Proposition 16.
Moreover, the ingredients of this proof (stability, smallness of objects
$S^{m}_{k}$) are straightforward consequences of their counterparts in the
case $K=\ast$. ∎

######  Remark 3.

One can also prove that one has the following equivalence of
$\infty$-categories

$\mathsf{Loops}_{R,\leq n}^{(K)}\simeq\mathsf{Loops}_{R,\leq
n}\coprod_{K}\ast.$

This leads then to the fact that the functor

$Psh_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n}^{(K)})\to\prod_{k\in
K}Psh_{\mathsf{stab}}(\mathsf{Loops}_{R,\leq n})$

is an equivalence.

###  2.4. Algebras in chain complexes

Let $K$ be a set called the set of colours. Let us consider an adjunction of
$\infty$-categories

${\mathsf{Ch}_{R}^{K}}$${\mathsf{A}.}$$\scriptstyle{T_{M}}$$\scriptstyle{U^{M}}$

that is is strongly monadic and $\omega$-accessible. Our goal is to describe
several presentations of the $\infty$-category $\mathsf{A}$

####  2.4.1. The cellular presentation

Let $n\in\mathbb{Z}\sqcup\\{\infty\\}$.

######  Definition 13.

A $n$-elementary morphism $f:A\to B$ in $\mathsf{A}$ is a morphism so that
there exists an integer $m\leq n-1$, a colour $k\in K$ and a pushout

${T_{M}(S_{k}^{m})}$${A}$${T_{M}(0)}$${B.}$

######  Definition 14.

A finitely $n$-cellular morphism $f:A\to B$ in $\mathsf{A}$ is the composition
of a finite collection of $n$-elementary morphisms.

######  Definition 15.

An object $A\in\mathsf{A}$ is finitely $n$-cellular if the morphism
$\emptyset=T_{M}(0)\to A$ is finitely $n$-cellular. We denote
$\mathsf{Cell}_{M,\leq n}$ the full subcategory of $\mathsf{A}$ spanned by
finitely $n$-cellular objects.

######  Lemma 17.

The full subcategory $\mathsf{Cell}_{M,\leq n}$ of $\mathsf{A}$ is stable
through coproducts and its objects are $\omega$-small in $\mathsf{A}$.

###### Proof.

Let us first show that $\mathsf{Cell}_{M,\leq n}$ is stable through
coproducts. Let $A$ and $B$ be finitely $n$-cellular objects and let us
consider a sequence of $n$-elementary morphisms

$\emptyset=A_{0}\xrightarrow{f_{0}}A_{1}\xrightarrow{f_{1}}\cdots\xrightarrow{f_{l-1}}A_{l}=A.$

Then, let us consider the following sequence of morphisms from $B$ to $A\sqcup
B$

$B=A_{0}\sqcup B\xrightarrow{f_{0}\sqcup\mathrm{Id}}A_{1}\sqcup
B\xrightarrow{f_{1}\sqcup\mathrm{Id}}\cdots\xrightarrow{f_{l-1}\sqcup\mathrm{Id}}A_{l}\sqcup
B=A\sqcup B.$

Any of these maps $f_{i}\sqcup\mathrm{Id}$ is $n$-elementary as we have two
pushouts that yield another pushout as follows

${T_{M}(S^{m}_{k})}$${A_{i}}$${A_{i}\sqcup
B}$${T_{M}(0)}$${A_{i+1}}$${A_{i+1}\sqcup B.}$

Let us show now that finitely $n$-cellular objects are $\omega$-small. Since
the functor $U^{M}$ preserves filtered colimits, then its adjoint $T_{M}$
sends $\omega$-small objects to $\omega$-small objects. To conclude, finite
colimits of $\omega$-small objects are $\omega$-small. ∎

######  Theorem 3.

The $\infty$-category $\mathsf{A}$ is canonically equivalent to the full
subcategory of $\mathsf{Psh}(\mathsf{Cell}_{M,\leq n})$ spanned by functors
$F$ so that

  1. (1)

$F(\emptyset)\simeq\ast$;

  2. (2)

for any pushout square in $\mathsf{C}$ of the form

${T_{M}(S^{m}_{k})}$${A}$${T_{M}(0)}$${B}$

its image through $F$ is a pullback.

###### Proof.

The following diagram

${\mathsf{Cell}_{M,\leq n}}$${\mathsf{Loops}_{R,\leq
n}}$${\mathsf{A}}$${\mathsf{Ch}_{R}^{K}.}$

is commutative. To check that we are in the context of Theorem 2, we need to
check that the full subcategory $\mathsf{Cell}_{M,\leq n}$ of $\mathsf{A}$ is
stable through coproducts and that its objects are $\omega$-small in
$\mathsf{A}$; this follows from Lemma 17. We need moreover to check that the
full subcategory of $\mathsf{Psh}(\mathsf{Cell}_{M,\leq n})$ described in the
theorem is reflective and is stable through filtered colimits; this is given
by Lemma 1. We notice that this full subcategory contains the image of
$\mathsf{A}$. Finally, using arguments similar as those used to prove Lemma
17, one can check that any functor in the full subcategory of
$\mathsf{Psh}(\mathsf{Cell}_{M,\leq n})$ described in the theorem preserves
finite products.

Thus it suffices to check that condition (2) of Theorem 2 is satisfied, which
is straightforward. ∎

####  2.4.2. The cartesian presentation

Let $n\in\mathbb{Z}\sqcup\\{\infty\\}$.

######  Definition 16.

Let $\mathsf{Cart}_{M,\leq n}$ be the smallest full subcategory of
$\mathsf{A}$ that contains $T_{M}{S_{k}^{m}}$ for any $m\leq n$, and $k\in K$
and that is stable through finite coproducts.

######  Theorem 4.

The $\infty$-category $\mathsf{A}$ is canonically equivalent to the full
subcategory of $\mathsf{Psh}(\mathsf{Cart}_{M,\leq n})$ spanned by functors
$F$ so that

  1. (1)

$F$ preserves finite products;

  2. (2)

for any pushout square in $\mathsf{Cart}_{M,\leq n}$ of the form

${T_{M}(X)}$${T_{M}(0)}$${T_{M}(0)}$${T_{M}(\Sigma X)}$

its image through $F$ is a pullback.

###### Proof.

It follows from the same arguments as those used to prove Theorem 3. ∎

###  2.5. Algebras over an operad

Let $\mathsf{P}$ be a coloured operad enriched in chain complexes and let $K$
be its set of colours. We have an adjunction relating the product over $K$ of
the category of chain complexes to the category of $\mathsf{P}$-algebras

${\mathsf{Chain}_{R}^{K}}$${\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}.}$$\scriptstyle{T_{P}}$$\scriptstyle{U^{P}}$

######  Definition 17.

The operad $\mathsf{P}$ is called admissible if there exists a combinatorial
model structure on the category of $\mathsf{P}$-algebras transferred from that
of $\mathsf{Chain}_{R}^{K}$, that is

  * $\triangleright$

the weak equivalences are the morphisms that are colourwise quasi-
isomorphisms;

  * $\triangleright$

the fibrations are the morphisms that are colourwise and degreewise
surjections.

Then, the adjunction $T_{P}\dashv U^{P}$ becomes a Quillen adjunction.

######  Definition 18.

The operad $\mathsf{P}$ is $\Sigma$-cofibrant if the underlying $K$-coloured
symmetric sequence is cofibrant in the projective model structure on
$K$-coloured symmetric sequences in chain complexes.

######  Remark 4.

If $R$ is rational or if $\mathsf{P}$ is planar, then $\mathsf{P}$ is
admissible (see for instance [PS18] and [BM07]). Moreover, in [Hin97], Hinich
proves that a monochromatic $\Sigma$-split operad is admissible. However, to
the best of my knowledge, the case of $\Sigma$-cofibrant operads in chain
complexes is a blindspot in the literature about admissible operads.

######  Proposition 20.

Let us suppose that $\mathsf{P}$ is admissible. Then, the functor

$\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]\to\mathsf{Chain}_{R}^{K}[\mathrm{W}^{-1}]\simeq\mathsf{Ch}_{R}^{K}$

induced by $U^{P}$ is conservative.

###### Proof.

Let us consider the following diagram of $\infty$-categories

${\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]}$${\mathsf{Chain}_{R}^{K}[\mathrm{W}^{-1}]}$${\mathrm{Ho}(\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg})}$${\mathrm{Ho}(\mathsf{Chain}_{R}^{K}),}$

where the notation $\mathrm{Ho}$ stands for the homotopy category of a model
category. The two vertical functors are conservative. Hence proving that the
top horizontal functor is conservative amounts to prove that the bottom
horizontal functor is conservative.

Let $f:A\to B$ be a morphism in
$\mathrm{Ho}(\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg})$ that is sent to an
isomorphism in $\mathrm{Ho}(\mathsf{Chain}_{R}^{K})$. We can suppose that $A$
is cofibrant and that $B$ is fibrant. Then $f$ proceeds from a morphism
$f^{\prime}:A\to B$ in $\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}$. Then,
$U^{M}(f^{\prime})$ becomes an isomorphism in
$\mathrm{Ho}(\mathsf{Chain}_{R}^{K})$. Thus, general properties of model
categories ensures us that $U^{M}(f^{\prime})$ is a weak equivalence. Hence
$f^{\prime}$ is a weak equivalence. So $f$ is an isomorphism. Thus the right
derived functor from $\mathrm{Ho}(\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg})$
to $\mathrm{Ho}(\mathsf{Chain}_{R}^{K})$ is conservative. ∎

######  Proposition 21.

[HNP19, Appendix A] Let us suppose that $\mathsf{P}$ is $\Sigma$-cofibrant and
admissible. Then, the functor

$\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]\to\mathsf{Ch}_{R}^{K}$

induced by $U^{P}$ preserves sifted colimits.

######  Corollary 2.

In the case where $\mathsf{P}$ is $\Sigma$-cofibrant and admissible, then the
adjunction

${\mathsf{Ch}^{K}_{R}}$${\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W^{-1}}]}$

induced by the Quillen adjunction $T_{P}\dashv U^{P}$ is strongly monadic and
$\omega$-accessible.

###### Proof.

This follows from Proposition 20 and Proposition 21 ∎

Thus, in that case where $\mathsf{P}$ is $\Sigma$-cofibrant and admissible,
one can apply Theorem 3 or Theorem 4 to get presentations of the
$\infty$-category
$\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]$. One can notice
that for any $n\in\mathbb{Z}\sqcup\\{\infty\\}$:

  1. (1)

an object $A\in\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]$ is
finitely $n$-cellular if and only if it is equivalent to the image of an
element $A^{\prime}$ in the category
$\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}$ so that there exists a finite
sequence of morphisms in this category

$A_{0}\xrightarrow{f_{0}}A_{1}\xrightarrow{f_{1}}\cdots\xrightarrow{f_{l-1}}A_{l}=A^{\prime}$

so that any morphism $f_{i}$ is a pushout of the form

${T_{M}(S^{m}_{k})}$${A_{i}}$${T_{M}(D^{m+1}_{k})}$${A_{i+1}}$

for some $m\leq n-1$ and $k\in K$;

  2. (2)

an object $A\in\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]$
belongs to $\mathsf{Cart}_{P,\leq n}$ if and only if it is equivalent to the
image of an object in the category $\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}$
of the form $T_{P}(X)$ where $X$ is the sum of a finite set of objects
$S^{m}_{k}$ for some $m\leq n$ and some $k\in K$.

###  2.6. A word about formal moduli problems

We end this section by informally describing relations between the cellular
presentation introduced above and formal moduli problems. Let $\mathsf{P}$ be
an admissible $\Sigma$-cofibrant $K$-coloured operad in chain complexes.

The operad $\mathsf{P}$ is called augmented if the unit morphism of operads
$1_{K}\to\mathsf{P}$ has a left inverse $\epsilon$. This yields an adjunction

${{\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]}}$${\mathsf{Ch}_{R}^{K}.}$$\scriptstyle{\epsilon_{!}}$$\scriptstyle{\epsilon^{\ast}}$

Koszul duality techniques provide us with an operad $\mathsf{P}^{!}$ so that
the left adjoint functor $\epsilon_{!}$ factorises through a functor from
$\mathsf{P}$-algebras to $\mathsf{P}^{!}$-coalgebras called the Bar
construction. Using also linear duality that sends $\mathsf{P}^{!}$-coalgebras
to $\mathsf{P}^{!}$-algebras, we obtain the following sequence of functors

${\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]}\xrightarrow{\mathrm{Bar}}\mathsf{\mathsf{P}^{!}}\mathrm{-}\mathsf{cog}[\mathrm{W}^{-1}]\to\mathsf{\mathsf{P}^{!}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]^{\mathrm{op}}.$

In some good cases described in [Lur11], [Pri07], [BM19] and [CCN19], the
composite functor induces a contravariant equivalence between the full
subcategory $\mathsf{Cell}_{P,\leq O}$ of
${\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]}$ spanned by
finitely $0$-cellular objects and a full subcategory $\mathsf{Art}_{P^{!}}$ of
$\mathsf{\mathsf{P}^{!}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]$ spanned by
objects that are called artinian and that are obtained using finitely many
square zero extensions starting from the zero algebra.

If $R$ is a characteristic zero field, such results may also be proven using
techniques specific to operads in chain complexes over such a field as
described for instance in [LV12].

##  Appendix A Chain complexes in non negative degrees

In this appendix, we describe a presentation of the $\infty$-category of chain
complexes in non negative degrees and of some $\infty$-categories of algebras
in such chain complexes.

###  A.1. Projectively generated infinity categories

We recall here a result of Lurie about projectively generated
$\infty$-categories that is a special case of the Theorem 2.

Let $\mathsf{D}$ be a small $\infty$-category with finite coproducts and let
us consider an adjunction

${\mathsf{Psh}_{\Sigma}(\mathsf{D})}$${\mathsf{A},}$$\scriptstyle{T_{M}}$$\scriptstyle{U^{M}}$

that is strongly monadic and $\omega$-accessible. Moreover, let $\mathsf{B}$
be the full subcategory of $\mathsf{A}$ spanned by objects that are images of
the functor
$\mathsf{D}\hookrightarrow\mathsf{Psh}_{\Sigma}(\mathsf{D})\to\mathsf{A}$. In
particular $\mathsf{B}$ is stable under finite coproducts.

######  Proposition 22.

[Lur18, Corollary 4.7.3.18] The functor
$\mathsf{A}\to\mathsf{Psh}_{\Sigma}(\mathsf{B})$ is an equivalence.

###### Proof.

We are in a the context of Theorem 2 where
$\mathsf{C}=\mathsf{Psh}_{\Sigma}(\mathsf{D})$ and
$\mathsf{E}=\mathsf{Psh}_{\Sigma}(\mathsf{B})$. Thus it suffices to prove that
the functor
$\mathsf{Psh}_{\Sigma}(\mathsf{B})\to\mathsf{Psh}_{\Sigma}(\mathsf{D})$ is
conservative. This functor fits in the following square diagram

${\mathsf{Psh}(\mathsf{B})}$${\mathsf{Psh}(\mathsf{D})}$${\mathsf{Psh}_{\Sigma}(\mathsf{B})}$${\mathsf{Psh}_{\Sigma}(\mathsf{D}).}$

Since the functor $\mathsf{D}\to\mathsf{B}$ is surjective on objects, then the
induced functor $\mathsf{Psh}(\mathsf{B})\to\mathsf{Psh}(\mathsf{D})$ is
conservative. Moreover, the two vertical functors are conservative. Thus the
functor
$\mathsf{Psh}_{\Sigma}(\mathsf{B})\to\mathsf{Psh}_{\Sigma}(\mathsf{D})$ is
also conservative. ∎

###  A.2. Chain complexes

######  Definition 19.

Let $\mathsf{Chain}_{R}^{\geq 0}$ be the category of chain complexes in non
negative degrees and let $\mathsf{Chain}_{R}^{\geq 0,K}$ be its product over a
set $K$

$\mathsf{Chain}_{R}^{\geq 0,K}=\prod_{k\in K}\mathsf{Chain}_{R}^{\geq 0}.$

We denote $R^{n}_{k}$ for $n\geq 0$ and $k\in K$ the object of this category
whose $k$-component is $R^{n}$ and whose $k^{\prime}$-component is zero for
any other $k^{\prime}\neq k$.

######  Proposition 23.

[Hov99] The category $\mathsf{Chain}_{R}^{\geq 0,K}$ admits a model structure
so that

  * $\triangleright$

weak equivalences are given by colourwise quasi-isomorphisms;

  * $\triangleright$

fibrations are colourwise degreewise surjections in positive degrees;

  * $\triangleright$

cofibrations are morphisms whose image in $\mathsf{Chain}_{R}^{K}$ is a
cofibration.

######  Definition 20.

Let $\mathsf{Ch}_{R}^{\geq 0,K}$ the localisation of the category
$\mathsf{Chain}_{R}^{\geq 0,K}$ at weak equivalences:

$\mathsf{Ch}_{R}^{\geq 0,K}=\mathsf{Chain}_{R}^{\geq 0,K}[\mathrm{W}^{-1}].$

From Lemma 13, we know that $\mathsf{Ch}_{R}^{\geq 0,K}$ is canonically
equivalent to the product over $K$ of the $\infty$-category
$\mathsf{Ch}_{R}^{\geq 0}=\mathsf{Ch}_{R}^{\geq 0,\ast}$.

######  Definition 21.

Let $\mathsf{Cart}^{(K)}_{R}$ be the full subcategory of
$\mathsf{Chain}_{R}^{\geq 0,K}$ spanned by finite sums of the objects
$R^{n}_{k}$ for $n>0$ and $k\in K$. In particular, it is stable through finite
coproducts.

######  Proposition 24.

The $\infty$-category $\mathsf{Ch}_{R}^{\geq 0}$ is equivalent to
$\mathsf{Psh}_{\Sigma}(\mathsf{Cart}^{(K)}_{R})$.

###### Proof.

Let
$\mathsf{Fun}_{\Sigma}\left((\mathsf{Cart}^{(K)}_{R})^{\mathrm{op}},\mathsf{sSet}\right)$
the full subcategory of
$\mathrm{Fun}\left((\mathsf{Cart}^{(K)}_{R})^{\mathrm{op}},\mathsf{sSet}\right)$
spanned by functors that preserves finite products. We know from [Qui67] that
it admits a combinatorial model structure whose weak equivalences are
objectwise Kan-Quillen weak equivalence and whose fibrations are objectwise
Kan-Quillen fibrations. Moreover, we know from [Lur17, Corollary 5.5.9.3] that
the resulting $\infty$-category is equivalent to
$\mathsf{Psh}_{\Sigma}(\mathsf{Cart}^{(K)}_{R})$.

To conclude, the Dold Kan correspondence, gives us equivalences of model
categories

$\mathsf{Fun}_{\Sigma}\left((\mathsf{Cart}^{(K)}_{R})^{\mathrm{op}},\mathsf{sSet}\right)\simeq{R}\mathrm{-}\mathsf{mod}^{\Delta^{\mathrm{op}}\times
K}\simeq\mathsf{Chain}_{R}^{\geq 0,K}.$

∎

###  A.3. Algebras

Let us consider an adjunction.

${\mathsf{Ch}_{R}^{\geq
0,K}}$${\mathsf{A}.}$$\scriptstyle{T_{M}}$$\scriptstyle{U^{M}}$

that is $\omega$-accessible and strongly monadic, and let $\mathsf{Cart}_{M}$
be the smallest full subcategory $\mathsf{A}$ that contains the objects
$T_{M}(R_{k})$ and that is stable through finite coproducts.

######  Corollary 3.

The $\infty$-category $\mathsf{A}$ is canonically equivalent to
$\mathsf{Psh}_{\Sigma}(\mathsf{Cart}_{M})$.

###### Proof.

This is just an application of Proposition 22. ∎

Moreover, given a $K$-coloured operad in chain complexes in non negative
degrees $\mathsf{P}$, one has an adjunction between categories

${\mathsf{Chain}_{R}^{\geq
0,K}}$${\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}.}$$\scriptstyle{T_{P}}$$\scriptstyle{U^{P}}$

We say that $\mathsf{P}$ is connectively admissible if the category of
$\mathsf{P}$-algebras admits a model structure transferred from that of
$\mathsf{Chain}_{R}^{\geq 0,K}$, that is whose weak equivalences and
fibrations are morphisms whose image through $U^{P}$ is respectively a weak
equivalence and a fibration. In this case, the adjunction $T_{P}\dashv U^{P}$
becomes a Quillen adjunction.

######  Proposition 25.

If $\mathsf{P}$ is $\Sigma$-cofibrant and connectively admissible, then the
adjunction between the localised $\infty$-categories $\mathsf{Chain}_{R}^{\geq
0,K}[\mathrm{W}^{-1}]$ and
$\mathsf{\mathsf{P}}\mathrm{-}\mathsf{alg}[\mathrm{W}^{-1}]$ resulting from
the Quillen adjunction $T_{P}\dashv U^{P}$ is $\omega$-accessible and strongly
monadic.

###### Proof.

The fact the functor induced by $U^{P}$ is conservative follows from the same
arguments as those used in the proof of Proposition 20. The fact that it
preserves sifted colimits follows from [HNP19, Appendix A]. ∎

## References

  * [BM07] Clemens Berger and Ieke Moerdijk, _Resolution of coloured operads and rectification of homotopy algebras_ , Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 31–58.
  * [BM19] Lukas Brantner and Akhil Mathew, _Deformation theory and partition lie algebras_ , ArXiv e-prints (2019).
  * [CCN19] D. Calaque, R. Campos, and J. Nuiten, _Moduli problems for operadic algebras_ , ArXiv e-prints (2019).
  * [Hin97] Vladimir Hinich, _Homological algebra of homotopy algebras_ , Comm. Algebra 25 (1997), no. 10, 3291–3323.
  * [HNP19] Yonatan Harpaz, Joost Nuiten, and Matan Prasma, _Tangent categories of algebras over operads_ , Israel Journal of Mathematics (2019), 691–742.
  * [Hov99] Mark Hovey, _Model categories_ , Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999.
  * [Lur11] Jacob Lurie, _Derived algebraic geometry x: Formal moduli problems_ , September 2011, https://www.math.ias.edu/ lurie/papers/DAG-X.pdf.
  * [Lur17] by same author, _Higher topos theory_ , April 2017, https://www.math.ias.edu/ lurie/papers/HTT.pdf.
  * [Lur18] by same author, _Higher algebra_ , September 2018, https://www.math.ias.edu/ lurie/papers/HA.pdf.
  * [LV12] Jean-Louis Loday and Bruno Vallette, _Algebraic operads_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer-Verlag, Berlin, 2012.
  * [MG16] Aaron Mazel-Gee, _Quillen adjunctions induce adjunctions of quasicategories_ , New York Journal of Mathematics 22 (2016), 57–93.
  * [Pri07] J. Pridham, _Unifying derived deformation theories_ , Advances in Mathematics 224 (2007).
  * [PS18] Dimitri Pavlov and Jakob Scholbach, _Admissibility and rectification of colored symmetric operads_ , Journal of Topology 11 (2018), no. 3, 559–601.
  * [Qui67] Daniel G. Quillen, _Homotopical algebra_ , Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin, 1967.
  * [RV16] Emily Riehl and Dominic Verity, _Homotopy coherent adjunctions and the formal theory of monads_ , Advances in Mathematics 286 (2016), 802 – 888\.
  * [Sim99] Carlos Simpson, _A giraud-type characterization of the simplicial categories associated to closed model categories as infinity-pretopoi_ , ArXiv e-prints (1999).