# Operads in Derived Deformation Theory

Ramkumar Ramachandra
Université Paris-Cité
<EMAIL_ADDRESS>
###### Abstract

A theorem by Pridham and Lurie provides an equivalence between formal moduli
problems and Lie algebras in characteristic zero. In his work, Lurie has
distilled the axioms that the algebras appearing in the formal moduli problem
need to satisfy, and worked out the case of $\mathbb{E}_{\8}$-algebras using
an incarnation of the Koszul duality, in the setting of -operads. The more
recent work of Calaque-Campos-Nuiten extends Lurie’s work to obtain an
equivalence between formal moduli problem parameterized by a colored operad,
and algebras over its Koszul dual operad. This manuscript is both, a
pedagogical exposition, and a questioning of their work, with modest, but
original, supporting lemmas.

###### Contents

  1. $\emptyset$ Introduction
  2. I Operads
    1. I.1 Non-symmetric operads
    2. I.2 Symmetric operads
    3. I.3 Modern definition of an operad
    4. I.4 Algebra over operad
    5. I.5 The differential graded setting
    6. I.6 Bar and cobar construction
    7. I.7 Koszul duality
    8. I.8 $\mathcal{P}_{\infty}$-algebras
  3. II Derived deformation theory
    1. II.1 Classical deformation theory of associative algebras
    2. II.2 Motivation from classical geometric deformation theory
    3. II.3 The Lurie-Pridham equivalence
    4. II.4 The tangent complex
  4. III Operadic deformation theory
    1. III.1 The Goldman-Millson theorem
    2. III.2 Main results
    3. III.3 Relationship to Lie algebras
  5. A Filtration in Category Theory
  6. B Algebra
  7. C Monoids and Monads
  8. D Trees
  9. E -categories

##  $\emptyset$ Introduction

Derived deformation theory nests under a larger body of mathematics known as
_derived algebraic geometry_. In order to elaborate on the scope of our work,
let us first provide a “big picture” view of derived geometry.

### Derived geometry

Derived geometry is a theory which generalizes the classical counterpart,
while faithfully embedding the classical version at non-singular points in a
space. We work over a _moduli space_ , which is, informally, a space that
classifies something, say solutions to a set of equations, or birational
classification of curves. Examples of moduli spaces include affine schemes,
smooth manifolds, stacks, but perhaps the simplest example is
$\mathbb{CP}^{n}$. The singularities in these spaces are of two different
kinds.

  1. (1)

_Quotient singularities_ appear in the quotient of a group action when a point
has a stabilizer under the action.

Figure 1: The space on the left has zeroth order quotient singularity, while
the space on the right has higher order quotient singularities.

  2. (2)

_Intersection singularities_ appear when intersections are not transverse.

Figure 2: The space on the left has zeroth order intersection singularity,
which is to say that the intersection is transverse, while the space on the
right has higher order intersection singularities.

A singularity can be characterized by saying that not all tangent vectors can
be integrated into a curve. In the pursuit to study higher-order
singularities, the tangent space was then enhanced to a _tangent complex_ ,
whose homology vanishes at non-singular points. At non-singular points, the
structure of the tangent complex is trivial (Abelian), and at singular points,
it encodes a lot of information about the singularity. It turns out that the
tangent complex neatly breaks up into two parts: intersection singularities
correspond to the positive part, and are studied in _derived intersection
theory_ using _derived schemes_ , while the quotient singularities correspond
to the negative part, and are studied in _derived deformation theory_ using
_derived stacks_.

The central role of _Lie algebras_ in this picture derives from the fact that
for a tangent complex $\mathbb{T}$, $\mathbb{T}[-1]$ is always equipped with
some kind of Lie algebra structure.

### Subject of this manuscript

The topic of this manuscript interests itself in the following central fact:
the entire _formal neighborhood_ can be reconstructed from the tangent complex
equipped with its Lie algebra structure. Research in the field today has led
to a seminal result called the _Lurie-Pridham equivalence_. Our work is based
on a variation of this result by _Calaque-Campos-Nuiten_ [OpFMP] done in the
setting of ordinary operads.

Derived deformation theory is a highly algebraic theory, and this has two
consequences for us.

  1. (a)

We will not concern ourselves with either derived schemes or stacks. Indeed,
the only “geometry” that appears in our manuscript is the -category of spaces.

  2. (b)

Much of the modern literature on the subject, and the work of _Calaque-Campos-
Nuiten_ in particular, is written in the language of _operads_. For this
reason, we dedicate a significant portion of our manuscript exclusively to the
study of operads.

### Motivation

The objective of this manuscript is to understand the motivation behind the
development of the machinery of -operads in the context of derived deformation
theory.

### Our contribution

Our contribution is two-fold. For the more pedagogical contribution, we
provide a firm grounding to understand III, by giving the reader a thorough
understanding to the theory of operads in I and a motivated storyline leading
up to Lurie’s results in II.

We present the less pedagogical and more original contribution in III, where
we fill the reader in on several details, and obtain modest supporting
results.

### Prerequisites

The audience for this graduate-level manuscript is expected to have a fair bit
of mathematical background, up to an understanding of -categories [HTT]. No
exposure to the theory of operads is assumed.

### Acknowledgements

The author would like to thank Ricardo Campos 111CNRS Researcher, Université
Paul Sabatier, Toulouse for his continued guidance and support.

##  I Operads

An algebra can be characterized by an abstract multiplication operator $\mu$,
that satisfies certain properties. For example, tensor algebra is
characterized by the properties of the tensor product, of either vector spaces
or modules. We generalize this statement by saying that vector spaces or
modules form a symmetric monoidal category with respect to the tensor product,
$\otimes$. The theory of _operads_ is a further generalization, whence once
considers multiplication operators with multiple inputs and one output. Once
sufficient machinery has been built, we can discuss algebras that are
“homotopy coherent”, or $\mathcal{P}_{\infty}$-algebras.

[$1$-operadic theory versus -operadic theory] _Colored operads_ consider
operads with multiple inputs and multiple outputs: the ordinary theory of
operads nests within this theory, as $1$-colored operads. Finally, -operads
generalize colored operads, to yield a theory in which we can define an
$\mathbb{E}_{\8}$-algebra, and do non-commutative derived deformation theory.

The theory of -operads is developed in detail in [HA], and involves a fair bit
of machinery. Fortunately, the theory of $1$-operads is sufficient for our
purposes, and we provide an exposition to this theory, concluding with
$\mathcal{P}_{\infty}$-algebras. Even though the notation might suggest
otherwise, $\mathcal{P}_{\infty}$-algebras and -quasi-isomorphisms are notions
of $1$-operadic theory.

We will always assume the ground field, $\mathbb{K}$, to be of characteristic
zero. Unless indicated otherwise, tensor products will be over $\mathbb{K}$.

All operads $\mathcal{P}$, and cooperads $\mathcal{C}$ will be assumed to be
trivial in degrees $0$ and $1$: $\mathcal{P}_{0}=0$, $\mathcal{P}_{1}=\I$,
$\mathcal{C}_{0}=0$, and $\mathcal{C}_{1}=\I$, where $\I$ denotes the unit of
the underlying monoidal category.

[A high-level summary] Given operad $\mathcal{P}$ and its _Koszul dual_
$\mathcal{P}^{!}$, the following adjunction

${\mathbf{B}:\mathcal{P}\textsf{-algebras}}$${\textsf{conilpotent
}\mathcal{P}^{!}\textsf{-coalgebras}:\mathbf{\Omega}}$${\bot}$

is a Quillen equivalence. Let $A$ be a $\mathcal{P}$-algebra. Then,

$\displaystyle\mathbf{B}A$ $\displaystyle=\mathcal{P}^{\text{!`}}\circ A$
$\displaystyle\textsf{where }\mathcal{P}^{!}(n)$
$\displaystyle:=\Hom_{\mathbb{K}}(\mathcal{P}^{\text{!`}}(n),\mathbb{K})[1-n]$

Now,

$\begin{matrix}&\Map(\mathbf{\Omega}\mathcal{P}^{\text{!`}},\End{A})&=&\mathsf{MC}(\mathfrak{g})&\\\
&\cap&&\cap&\\\ &\Map_{\mathsf{non-
dg}}(\mathbf{\Omega}\mathcal{P}^{\text{!`}},\End{A})&=&\Hom_{\mathbb{K}}(\mathcal{P}^{\text{!`}},\End{A})&:=\mathfrak{g}\end{matrix}$

where $\mathsf{MC}(\mathfrak{g})$ are the _Maurer-Cartan elements_ of
$\mathfrak{g}$. Moreover, there exists a operad morphism, which induces
"restriction of scalars" functor at the level of algebras:

$\mathsf{L}_{\8}\rightarrow\mathcal{P}\otimes\mathcal{P}^{!}$

where $\mathsf{L}_{\8}$ is the cofibrant resolution of the Lie operad.

###  I.1 Non-symmetric operads

An algebraic operad $\mathcal{P}$ is a functor from $\Vect$ to , that sends
algebras over the source vector space to _free algebras_ over the destination
vector space. $\mathcal{P}$ consists of the data of the triple
$(\mathcal{P},\gamma,\eta)$, where
$\gamma:\mathcal{P}\circ\mathcal{P}\rightarrow\mathcal{P}$ and
$\eta:\I\rightarrow\mathcal{P}$, forming a _monoid_ [C.1] in a strict monoidal
category. In particular, a monoid in a strict monoidal category of
endofunctors is called a _monad_ [C.2].

Non-symmetric operads, or _ns-operads_ , are a simple setting we can use to
initiate the study of operads. This setting will then allow us to study the
theory of symmetric operads, which we otherwise just call operads.

######  Definition I.1 (Schur functor).

Let $M_{\bullet}=\\{M_{n}\\}_{n\geq 0}$ be an arity-graded vector space. The
_Schur functor_ $M:\Vect\rightarrow\Vect$ associated to $M_{\bullet}$, is
defined as:

$\mathcal{M}(V):=\bigoplus_{n}\mathcal{M}_{n}\otimes V^{\otimes n}$

######  Definition I.2 (Non-symmetric operad).

An ns-operad $(\mathcal{P},\gamma,\alpha)$ is defined as a Schur functor that
is compatible with the monoidal structure

$\mathcal{P}(n):=\bigoplus_{n}\mathcal{P}_{n}\otimes V^{\otimes n}$

In other words, an ns-operad, $(P,\gamma,\alpha)$, is an arity-graded vector
space $\mathcal{P}=\\{\mathcal{P}_{n}\\}_{n\geq 0}$ equipped with the
composition maps

$\gamma_{i_{1},\ldots,i_{k}}:\mathcal{P}_{k}\otimes\mathcal{P}_{i_{1}}\otimes\ldots\otimes\mathcal{P}_{i_{k}}\rightarrow\mathcal{P}_{i_{1}+\ldots+i_{k}}$

and an element $\id\in\mathcal{P}_{1}$, such that the transformation functors
$\gamma:\mathcal{P}\circ\mathcal{P}\rightarrow\mathcal{P}$ and
$\gamma:\I\rightarrow\mathcal{P}$ make $(\mathcal{P},\gamma,\alpha)$ into a
monoid.

We use the formalism of planar rooted trees to give an alternative definition
of an ns-operad.

######  Definition I.3 (Combinatorial definition of ns-operad).

The category of graded vector spaces is denoted by $\mathbb{N}\text{-mod}$.
Let $M$ be an arity-graded vector space with $M_{0}=0$. For any _planar rooted
tree_ [D] $t$,

$M_{t}:=\bigotimes_{v\in\textsf{vert}(t)}M_{|v|}$

We then get a functor

$\displaystyle\mathsf{P}\mathbb{T}$
$\displaystyle:\mathbb{N}\text{-mod}\rightarrow\mathbb{N}\text{-mod}$
$\displaystyle\mathsf{P}\mathbb{T}(M)_{n}$
$\displaystyle:=\bigoplus_{t\in\mathsf{P}\mathbb{T}_{n}}M_{t}$

The corolla enables us to define
$\eta:\I_{\mathbb{N}\text{-mod}}\rightarrow\mathsf{P}\mathbb{T}$, and the
_substitution of trees_ , given by grafting trees with matching inputs,
enables us to define
$\alpha:\mathsf{P}\mathbb{T}\circ\mathsf{P}\mathbb{T}\rightarrow\mathsf{P}\mathbb{T}$.

Then, the definition of an ns-operad is an arity-graded module $\mathcal{P}$,
together with a map $\mathsf{P}\mathbb{T}(\mathcal{P})\rightarrow\mathcal{P}$
compatible with $\alpha$ and $\eta$ in the usual sense.

###  I.2 Symmetric operads

######  Definition I.4 (Non-dg $\mathbb{S}$-module).

A non-dg $\mathbb{S}$-module is defined as the family

$M:=\\{M(0),M(1),\ldots,M(n),\ldots\\}$

where each $M(n)$ is a right $\mathbb{K}$[$\mathbb{S}_{n}$] module. Since,
$\mathbb{K}$ is a field [I], an $\mathbb{S}$-module is an arity-graded vector
space.

######  Definition I.5 (Schur functor).

The Schur functor is an endofunctor over

$\displaystyle\widetilde{M}$ $\displaystyle:\Vect\rightarrow\Vect$
$\displaystyle\widetilde{M(V)}$
$\displaystyle:=\bigoplus_{n}M(n)\otimes_{\mathbb{S}_{n}}V^{\otimes n}$

######  Lemma I.6 (Constructions on the Schur functor).

The direct sum, tensor product, and composition of two Schur functors
$F,G:\Vect\rightarrow\Vect$ are given by

$\displaystyle(F\oplus G)(V)$ $\displaystyle:=F(V)\oplus G(V)$
$\displaystyle(F\otimes G)(V)$ $\displaystyle:=F(V)\otimes G(V)$
$\displaystyle(F\circ G)(V)$ $\displaystyle:=F(V)\circ G(V)$

######  Lemma I.7 (Direct sum of non-dg $\mathbb{S}$-modules).

$(M\oplus N)(n):=M(n)\oplus N(n)$

######  Lemma I.8 (Tensor product of non-dg $\mathbb{S}$-modules).

$(M\otimes N)(n):=\bigoplus_{i+j=n}M_{i}\otimes
N_{j}\otimes\mathbb{K}[\mathsf{Sh}(i,j)]$

where $\mathsf{Sh}(i,j)$ is the $ij$-_shuffle_ [B.3].

######  Lemma I.9 (Composition of non-dg $\mathbb{S}$-modules).

$(M\circ N)(n):=\bigoplus_{k\geq 0}M_{k}\otimes_{\mathbb{S}_{k}}N^{\otimes k}$

Compare the above definition with I.2. Replacing $\mathcal{P}_{n}$ with an
_$\mathbb{S}$ -module_, we get the general definition of an operad.

######  Definition I.10 (Algebraic operad, re-stated).

An algebraic operad, $\mathcal{P}$, consisting of the data of the triple
$(\mathcal{P},\gamma,\eta)$, is a Schur functor that is compatible with monad
laws.

$\displaystyle\mathcal{P}(V):=$
$\displaystyle\bigoplus_{n}\mathcal{P}_{n}\otimes V^{\otimes n}$
$\displaystyle=$
$\displaystyle\bigoplus_{n}\mathcal{P}(n)\otimes_{\mathbb{S}_{n}}V^{\otimes
n}$

[$\mathcal{P}_{n}$] We use two different notations to clarify the framework
we’re working in. In the case of symmetric operads,

$\mathcal{P}(n):=\mathcal{P}_{n}\otimes\mathbb{K}[\mathbb{S}_{n}]$

######  Definition I.11 (The tree monad).

Let $M$ be an $\mathbb{S}$-module. We define
$\mathcal{T}:\Vect\rightarrow\Vect$ as

$\displaystyle\mathcal{T}_{0}M$ $\displaystyle:=\I$
$\displaystyle\mathcal{T}_{1}M$ $\displaystyle:=\I\oplus M$
$\displaystyle\mathcal{T}_{2}M$ $\displaystyle:=\I\oplus(M\circ(\I\oplus M))$
$\displaystyle\ldots$ $\displaystyle\mathcal{T}_{n}M$
$\displaystyle:=\I\oplus(M\circ\mathcal{T}_{n-1}M)$

By definition, the tree module $\mathcal{T}M$ over an $\mathbb{S}$-module is

$\mathcal{T}M:=\colim_{n}\mathcal{T}_{n}M$

There is an operad structure $\gamma$ on $\mathcal{T}M$ such that
$\mathcal{T}(M):=(\mathcal{T}M,\gamma,j)$, where $j:M\rightarrow\mathcal{T}M$
sends $M\circ\mathcal{T}_{n-1}M\mapsto\mathcal{T}_{n}M$, and
$\gamma:\mathcal{T}M\rightarrow\mathcal{T}M$. $\mathcal{T}$ is hence a _free
operad_ on $M$.

###  I.3 Modern definition of an operad

To summarize everything we’ve covered so far, let us now present the abstract
definition of an operad, from two different equivalent viewpoints. In
particular, we introduce grafting of trees, which will be used to generalize
operads to the differential graded setting.

######  Definition I.12 (Combinatorial definition of an operad).

Let $M$ be an arity-graded vector space with $M_{0}=0$. For any _rooted tree_
[D] $t$, we define the _treewise tensor product_ $M_{t}$ as

$M_{t}:=\bigotimes_{v\in\textsf{vert}(t)}M_{|v|}$

and a functor

$\displaystyle\mathbb{T}$
$\displaystyle:\mathbb{S}\text{-mod}\rightarrow\mathbb{S}\text{-mod}$
$\displaystyle\mathbb{T}(M)(X)$
$\displaystyle:=\bigoplus_{t\in\mathsf{R}\mathbb{T}(X)}M(t)$

We then construct a monad structure on $\mathbb{T}$ by _substitution of
trees_. In order to do this, we need to define
$\iota:\id_{\mathbb{S}\text{-mod}}\rightarrow\mathbb{T}$ and
$\alpha:\mathbb{T}\circ\mathbb{T}\rightarrow\mathbb{T}$ to obtain
$(\mathbb{T},\iota,\alpha)$.

Given $\mathbb{S}$-module $M$, $\iota$ sends $M(X)\mapsto\mathbb{T}(M)(X)$. In
$\mathsf{R}\mathbb{T}(X)$, we have one particular tree corresponding to the
corolla, so that $M(\mathsf{corolla})=M(X)=\bigoplus\mathbb{T}(M)(X)$.

Next, let us illustrate the substitution of trees that defines $\alpha$: we
graft matching inputs onto the leaves.

$\Biggl{(}$;

,

,

$\Biggr{)}$

=

######  Definition I.13 (Partial definition of an operad).

Let $\mathcal{P}$ be an operad and $\mu\in\mathcal{P}(n)$,
$\nu\in\mathcal{P}(m)$ be two operations. We then define the partial composite
$\circ_{i}$ of $\mu$ and $\nu$ as

$\displaystyle-\circ_{i}-$
$\displaystyle:\mathcal{P}(m)\otimes\mathcal{P}(n)\rightarrow\mathcal{P}(m+n-1)$
$\displaystyle\mu\circ_{i}\nu$
$\displaystyle:=\gamma(\mu;\id,\ldots,\id,\nu,\id,\ldots,\id)$

This can be represented as a grafting of trees, where the root of $\nu$ is
grafted on to the $i$th leaf of $\mu$.

$\mu$$\nu$$i$

In order to describe the action of the symmetric groups and complete the
definition of $\circ_{i}$, we axiomatize the two different cases of two-stage
compositions.

$\begin{matrix}\textbf{(I)}&(\lambda\circ_{i}\mu)\circ_{i+j-1}\nu&=\lambda\circ_{i}(\mu\circ_{j}\nu)&1\leq
i\leq l,1\leq j\leq m\\\
\textbf{(II)}&(\lambda\circ_{i}\mu)\circ_{k+m-1}\nu&=(\lambda\circ_{k}\nu)\circ_{i}\mu&1\leq
i\leq k\leq l\\\ \end{matrix}$

$\lambda$$\mu$$\nu$$i$$j$(I)

$\lambda$$\mu$$\nu$$i$$k$(II)

The partial definition of an operad is then an $\mathbb{S}$-module
$\mathcal{P}$ equipped with partial compositions $\circ_{i}$ satisfying the
compatibility relations (I) and (II) described above. It is also assumed that
there is an element $\id\in\mathcal{P}(1)$ satisfying $\id\circ_{i}\nu=\nu$
and $\mu\circ_{i}\id=\mu$.

###  I.4 Algebra over operad

######  Definition I.14 ($\mathcal{P}$-algebra).

An algebra over operad $\mathcal{P}$, or a $\mathcal{P}$-algebra, is defined
as a vector space $A$ equipped with map $\gamma_{A}:\mathcal{P}(A)\rightarrow
A$ such that the following diagrams commute:

${\mathcal{P}(\mathcal{P}(A))}$${\mathcal{P}(A)}$${(\mathcal{P}\circ\mathcal{P})(A)}$${\mathcal{P}(A)}$${A}$$\scriptstyle{\mathcal{P}(\gamma_{A})}$$\scriptstyle{\gamma_{A}}$$\scriptstyle{\approx}$$\scriptstyle{\gamma(A)}$$\scriptstyle{\gamma(A)}$
${\I(A)}$${\mathcal{P}(A)}$${A}$$\scriptstyle{\eta(A)}$$\scriptstyle{\approx}$$\scriptstyle{\gamma_{A}}$

See also C.3 for the general definition of a monad algebra.

######  Definition I.15 (Endomorphism operad).

For any vector space $V$, the endomorphism operad is given by:

$\End{V}(n):=\Hom(V^{\otimes n},V)$

By convention, $V^{\otimes 0}=\mathbb{K}$. The right action of
$\mathbb{S}_{n}$ on $\End{V}(n)$ is induced by left action on $V^{\otimes n}$.
The composition $\gamma$ is given by composition of endomorphisms:

${V^{\otimes i_{1}}\otimes}$${\ldots}$${\otimes V^{\otimes
i_{k}}}$${=}$${V^{\otimes n}}$${V\otimes}$${\ldots}$${\otimes
V}$${=}$${V^{\otimes
k}}$${V}$${=}$${V}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{k}}$$\scriptstyle{f_{1}\otimes\ldots\otimes
f_{k}}$$\scriptstyle{f}$$\scriptstyle{f}$

Evidently $\End{V}$ is an operad.

######  Lemma I.16 (Alternative definition of $\mathcal{P}$-algebra).

The $\mathcal{P}$-algebra structure on vector space $V$ is equivalent to the
morphism of operads $\mathcal{P}\rightarrow\End{V}$.

We now supply three examples of $\mathcal{P}$-algebras, namely $\mathit{Ass}$,
$\mathit{Com}$, and $\mathit{Lie}$.

######  Example I.17 ($\mathit{Ass}$).

Let $\mathit{Ass}:\Vect\rightarrow\Vect$ given by
$\mathit{Ass}(V):=\overline{T}(V)=\oplus_{n\geq 1}V^{\otimes n}$ (reduced
tensor algebra). As an $\mathbb{S}$-module, we have
$\mathit{Ass}(n)=\mathbb{K}[\mathbb{S}_{n}]$, since
$\mathbb{K}[\mathbb{S}_{n}]\otimes_{\mathbb{S}_{n}}V^{\otimes n}=V^{\otimes
n}$ for $n\geq 1$, and $\mathit{Ass}(0)=0$. The map
$\gamma(V):\mathit{Ass}(\mathit{Ass}(V))\rightarrow\mathit{Ass}(V)$ is given
by “composition of non-commutative polynomials”. This is a symmetric operad
encoding associative algebras, since an algebra over $\mathit{Ass}$ is the
mapping $\overline{T}(A)\rightarrow A$.

######  Example I.18 ($\mathit{Com}$).

Let $\mathit{Com}:\Vect\rightarrow\Vect$ be a Schur functor given by

$\mathit{Com}(V):=\overline{S}(V)=\bigoplus_{n\geq 1}S^{n}V=\bigoplus_{n\geq
1}(V^{\otimes n})_{\mathbb{S}_{n}}$

As an $\mathbb{S}$-module, we have $\mathit{Com}(n)=\mathbb{K}$ with trivial
action since

$\mathbb{K}\otimes_{\mathbb{S}_{n}}V^{\otimes n}=S^{n}V\text{, for $n\geq 1$}$

and $\mathit{Com}(0)=0$. The map
$\gamma(V):\mathit{Com}(\mathit{Com}(V))\rightarrow\mathit{Com}(V)$ is given
by “composition of polynomials”. This is a symmetric operad encoding
commutative algebras, since an algebra over $\mathit{Com}$ is nonunital
commutative.

######  Example I.19 ($\mathit{Lie}$).

Let $\mathit{Lie}:\Vect\rightarrow\Vect$ be a Schur functor such that the
space $\mathit{Lie}(V)\subset\overline{T}(V)$ is generated by $V$ under the
bracketing operation $[x,y]:=xy-yx$. This is the free algebra over $V$. Let
$\mathit{Lie}(n)$ be the multilinear part of degree $n$ in free Lie algebra
$\mathit{Lie}(\mathbb{K}x_{1}\oplus\ldots\oplus\mathbb{K}x_{n})$. It can be
shown that there is an operad structure on Schur functor $\mathit{Lie}$
induced by the operad structure on $\mathit{Ass}$. An algebra over
$\mathit{Lie}$ is a Lie algebra.

###  I.5 The differential graded setting

In this section, we generalize the notions of non-dg $\mathbb{S}$-modules and
operads to include a non-trivial differential. This allows us to continue
discussing operads in full generality, in the following sections.

[Infinitesimal composite of $\mathbb{S}$-modules] Drawing inspiration from the
partial definition of operads, in which $\circ_{i}$ is defined, we generalize
this notation to define $\circ_{(1)}$, denoting the grafting of trees at one
unspecified leaf. More precisely, given $\mathbb{S}$-modules $M$ and $N$, we
define their infinitesimal composite, $M\circ_{(1)}N$ as

$M\circ_{(1)}N:=M\circ(\I;N)$

This can be shown diagrammatically as

MN

Elements of $M\circ_{(1)}N$ are of the form
$(\mu;\id,\ldots,\id,\nu,\id,\ldots,\id)$

[Infinitesimal composite of morphisms] Given two $\mathbb{S}$-module morphisms
$f:M_{1}\rightarrow M_{2}$ and $g:N_{1}\rightarrow N_{2}$, we define
$f\circ^{\prime}g$ to be

$\displaystyle f\circ^{\prime}g$ $\displaystyle:M_{1}\circ N_{1}\rightarrow
M_{2}\circ(N_{1};N_{2})$ $\displaystyle f\circ^{\prime}g$
$\displaystyle:=\sum_{i}f\otimes(\id_{N_{1}}\otimes\ldots\otimes\id_{N_{1}}\otimes
g\otimes\id_{N_{1}}\otimes\ldots\id_{N_{1}})$

This is unambiguously defined for a given $f$ and $g$: $g$ must occupy the
$i$th position in the tensor product.

######  Definition I.20 (Differential graded $\mathbb{S}$-module).

A _graded $\mathbb{S}$-module_ $M$ is an $\mathbb{S}$-module in the category
of graded vector spaces. The arity $n$ is a graded $\mathbb{S}_{n}$-module
$\\{M_{p}(n)\\}_{p\in\Z}$. Equivalently, $M$ can be considered a family of
$\mathbb{S}$-modules. By abuse of notation,

$M:=M_{\bullet}=\ldots\oplus M_{0}\oplus\ldots\oplus M_{p}\oplus\ldots$

A _differential graded $\mathbb{S}$-module_ $(M,d)$ is a graded
$\mathbb{S}$-module $M$ equipped with differential $d$ of
$\mathbb{S}_{n}$-modules, such that $d^{2}=0$. Note that $H_{\bullet}(M)$ of a
differential graded $\mathbb{S}$-module form a graded $\mathbb{S}$-module.

The composite of two differential graded $\mathbb{S}$-modules $(M,d_{M})$ and
$(N,d_{N})$ comes with a differential $d_{M\circ N}$, which can be written
using the notations we have introduced above.

$d_{M\circ N}:=d_{M}\circ\id_{N}+\id_{M}\circ^{\prime}d_{N}$

It is easily checked that the category $(\textsf{dg
}\mathbb{S}\text{-mod},\circ,\I)$ forms a monoidal category. We will now
proceed to define a dg algebraic operad $\mathcal{P}$, and a dg
$\mathcal{P}$-algebra.

######  Definition I.21 (Differential graded operad).

A _differential graded operad_ , denoted $(\mathcal{P},\gamma,\eta)$, is a
monoid in the monoidal category $(\textsf{dg }\mathbb{S}\text{-mod},\circ,\I)$
where $\gamma:\mathcal{P}\circ\mathcal{P}\rightarrow\mathcal{P}$ is the
composite map and $\eta:\I\rightarrow\mathcal{P}$ is the unit map, both of
degree $0$, such that the following diagram commutes.

${\mathcal{P}\circ\mathcal{P}}$${\mathcal{P}}$${\mathcal{P}\circ\mathcal{P}}$${\mathcal{P}}$$\scriptstyle{\gamma}$$\scriptstyle{d_{\mathcal{P}\circ\mathcal{P}}}$$\scriptstyle{d_{\mathcal{P}}}$$\scriptstyle{\gamma}$

where the differential $d_{\mathcal{P}}$ is given by

$\begin{matrix}d_{\mathcal{P}}(\gamma(\mu;\mu_{1},\ldots,\mu_{k})):=&\gamma(d_{\mathcal{P}}(\mu);\mu_{1}\ldots,\mu_{k})+\\\
&\sum_{i=1}^{k}(-1)^{\epsilon_{i}}\gamma(\mu;\mu_{1},\ldots,d_{\mathcal{P}}(\mu_{i}),\ldots,\mu_{k})\\\
\end{matrix}$

and $\epsilon_{i}$ is defined as

$\epsilon_{i}:=|\mu|+|\mu_{1}|+\ldots+|\mu_{i-1}|$

Since this is a chain complex, ${d_{\mathcal{P}}}^{2}=0$. Further, it can be
verified that $H_{\bullet}(\mathcal{P})$ carries a natural operadic structure.

######  Remark I.22.

The category $(\mathcal{P},\gamma,\eta)$ forms a monad, just like the non-dg
counterpart.

######  Definition I.23 (Differential graded cooperad).

A _differential graded cooperad_ , denoted $(\mathcal{C},\Delta,\epsilon)$, is
a comonoid in the monoidal category $(\textsf{dg
}\mathbb{S}\text{-mod},\circ,\I)$ where
$\Delta:\mathcal{C}\rightarrow\mathcal{C}\circ\mathcal{C}$ is the
_decomposition map_ and $\epsilon:\mathcal{C}\rightarrow\I$ is the _counit
map_ , both of degree $0$, such that the following diagram commutes.

${\mathcal{C}}$${\mathcal{C}\circ\mathcal{C}}$${\mathcal{C}}$${\mathcal{C}\circ\mathcal{C}}$$\scriptstyle{\Delta}$$\scriptstyle{d_{\mathcal{C}}}$$\scriptstyle{d_{\mathcal{C}\circ\mathcal{C}}}$$\scriptstyle{\Delta}$

The formula for $\Delta(d_{\mathcal{C}})$ is straightforward to work out.

[Coaugmented cooperad] A cooperad is termed _coaugmented_ when it is endowed
with a map $\eta:\I\rightarrow\mathcal{C}$ of degree $0$.

[Conilpotent cooperad] Conilpotency is a general condition on a cooperad that
will be necessary to state certain facts in the following sections. However,
it is strictly weaker than the convention we’ve adopted [I], whereby
$\mathcal{C}_{0}$ and $\mathcal{C}_{1}$ are trivial, and we ignore the
qualifier altogether.

This concludes our section detailing the machinery required to define the dg
counterpart to operads. Since we have shown that the differential-graded
version is an strict generalization of the non-dg version, and will proceed to
drop the “dg” qualifiers everywhere.

###  I.6 Bar and cobar construction

In classical homological algebra, there are several ways to obtain cofibrant
and fibrant resolutions of modules: projective, injective and free resolutions
make an early appearance. In operadic homological algebra, we use one
canonical resolution called the _Bar-cobar resolution_ , which serves as the
cofibrant resolution of the operadic algebra. We will not concern ourselves
with fibrant resolutions or resolutions of operadic coalgebras, for reasons
that will eventually become evident.

######  Definition I.24 (Convolution operad).

Given operad $\mathcal{P}$ and cooperad $\mathcal{C}$, we make the following
observations.

$\displaystyle(\Hom_{\mathbb{S}}(\mathcal{C},\mathcal{P}),\star,\partial)$
forms a pre-Lie algebra
$\displaystyle(\Hom_{\mathbb{S}}(\mathcal{C},\mathcal{P}),[,],\partial)$ forms
a Lie algebra

Here, $\Hom_{\mathbb{S}}(\mathcal{C},\mathcal{P})$ is called a _convolution
algebra_ , $\star$ is the pre-Lie multiplication operation, and $[,]$ is the
Lie bracket. See [Loday-Vallette] Section 6.4 for more.

[Twisting morphism] The Maurer-Cartan equation in
$\Hom_{\mathbb{S}}(\mathcal{C},\mathcal{P})$ reads as either

$\displaystyle\partial(\alpha)+\alpha\star\alpha=0$
$\displaystyle\partial(\alpha)+\frac{1}{2}[\alpha,\alpha]=0$

depending on whether $\mathcal{C}$ and $\mathcal{P}$ are pre-Lie algebras or
Lie algebras.

A solution $\alpha:\mathcal{C}\rightarrow\mathcal{P}$ of degree $+1$ (in
cohomological convention) is termed an _operadic twisting morphism_ , and is
written $\mathsf{Tw}(\mathcal{C},\mathcal{P})$. See also: B.5.

If $\alpha$ is of degree $-a$, consider the degree of the various terms in the
Maurer-Cartan equation: $\partial(\alpha)$ is of degree $a+1$, and
$\frac{1}{2}[\alpha,\alpha]$ is of degree $2a$. We hence see that it is
natural to define $\mathsf{Tw}$ to be $(+1)$-degree solutions of the Maurer-
Cartan equation.

######  Definition I.25 (Minimal model of operad).

Given operad $\mathcal{P}$, an operad $\mathcal{M}$ that is quasi-isomorphic
to $\mathcal{P}$ is termed a _model_ of operad $\mathcal{P}$. A given
$\mathcal{M}=\mathcal{T}(E)$ is termed _minimal_ when
$d(E)\subset\mathcal{T}(E)^{\geq 2}$.

######  Definition I.26 (Twisted convolution operad).

In any dg-Lie algebra, a $(+1)$-degree solution of the Maurer-Cartan equation
gives rise to a twisted differential $\partial_{\alpha}$ on
$\Hom_{\mathbb{S}}(\mathcal{C},\mathcal{P})$. We can then define

$\displaystyle\partial_{\alpha}(f)$ $\displaystyle:=\partial(f)+[\alpha,f]$
$\displaystyle\Hom_{\mathbb{S}}^{\alpha}(\mathcal{C},\mathcal{P})$
$\displaystyle:=(\Hom_{\mathbb{S}}^{\alpha}(\mathcal{C},\mathcal{P}),\partial_{\alpha})$

######  Definition I.27 ($\mathbf{B}$ and $\mathbf{\Omega}$).

The functors $\mathbf{B}$ and $\mathbf{\Omega}$ operate as follows.

$\displaystyle\mathbf{B}$ $\displaystyle:\text{augmented
operad}\rightarrow\text{cooperad}$ $\displaystyle\mathbf{\Omega}$
$\displaystyle:\text{coaugmented cooperads}\rightarrow\text{augmented
operads}$

These otherwise noted in literature as the bar and cobar _constructions_ ,
which can be explained by the fact that $\mathbf{B}\mathcal{P}$ and
$\mathbf{\Omega}\mathcal{C}$ are chain complexes, that induce differentials
$d_{\mathbf{B}}$ and $d_{\mathbf{\Omega}}$ respectively.

In terms of the tree functor, $\mathbf{B}$ and $\mathbf{\Omega}$ can
concretely be defined as

$\displaystyle\mathbf{B}\mathcal{P}$
$\displaystyle:=\mathcal{T}^{c}(s\overline{\mathcal{P}},d=d_{\mathbf{B}}+d_{\mathcal{P}})$
$\displaystyle\mathbf{\Omega}\mathcal{P}$
$\displaystyle:=\mathcal{T}(s^{-1}\overline{\mathcal{C}},d=d_{\mathbf{\Omega}}+d_{\mathcal{C}})$

where,

$\displaystyle\overline{\mathcal{P}}$
$\displaystyle:=\ker(\mathcal{P}\rightarrow\I)$ termed augmentation ideal
$\displaystyle\overline{\mathcal{C}}$
$\displaystyle:=\coker(\I\rightarrow\mathcal{C})$ termed coaugmentation
coideal

$d_{\mathbf{B}}$ and $d_{\mathbf{\Omega}}$ remain to be explained. Let
$\mathcal{P}=(\mathcal{P},\gamma,\eta,\epsilon)$ be an augmented operad. The
$\mathbb{S}$-module $\mathcal{P}$ is naturally isomorphic to
$\mathcal{P}=\I\oplus\overline{\mathcal{P}}$. The bar construction
$\mathbf{B}\mathcal{P}$ is a dg cooperad, whose underlying space is the cofree
cooperad $\mathcal{T}^{c}(s\overline{\mathcal{P}})$ on the suspension of
$\overline{\mathcal{P}}$. Since $\mathcal{T}^{c}(s\overline{\mathcal{P}})$ is
a cofree cooperad, there exists a unique coderivation extending the
composition on $\mathcal{P}$, which induces the differential $d_{\mathbf{B}}$,
$\mathcal{T}^{c}(s\overline{\mathcal{P}})\rightarrow\mathcal{T}^{c}(s\overline{\mathcal{P}})$.
Similarly, the differential $d_{\mathbf{\Omega}}$ is induced by
$\mathcal{T}(s^{-1}\overline{\mathcal{C}})\rightarrow\mathcal{T}(s^{-1}\overline{\mathcal{C}})$.
Here, $s$ and $s^{-1}$ refer to _degree suspension_ and _degree desuspension_
; refer to [Loday-Vallette] Section 6.5 for more.

######  Theorem I.28 (Relationship between $\mathbf{B}$ and
$\mathbf{\Omega}$).

The functors $\mathbf{B}$ and $\mathbf{\Omega}$ are related by the following
adjunction.

${\mathbf{\Omega}}$${\mathbf{B}}$${\bot}$

Twisting morphisms are exactly what induce maps to bar constructions, and maps
from cobar constructions.

######  Theorem I.29 ($\mathbf{B}-\mathbf{\Omega}$ resolution).

The unit and counit of the above adjunction

$\displaystyle\mathbf{\Omega}\mathbf{B}\mathcal{P}\rightarrow\mathcal{P}$
$\displaystyle\mathcal{C}\rightarrow\mathbf{B}\mathbf{\Omega}\mathcal{C}$

are quasi-isomorphisms of operads and cooperads, respectively.

######  Theorem I.30 (Fundamental theorem of $\mathsf{Tw}$).

Given a $\alpha\in\mathsf{Tw}(\mathcal{C},\mathcal{P})$, the following
assertions are equivalent.

  1. (a)

$\mathcal{P}\circ_{\alpha}\mathcal{C}$ is acyclic.

  2. (b)

$\mathcal{C}\circ_{\alpha}\mathcal{P}$ is acyclic.

  3. (c)

$\mathcal{C}\rightarrow\mathbf{B}\mathcal{P}$ is a quasi-isomorphism.

  4. (d)

$\mathbf{\Omega}\mathcal{C}\rightarrow\mathcal{P}$ is a quasi-isomorphism.

In this case, the morphism is termed _Koszul_.

###  I.7 Koszul duality

In this section, we restrict our attention to studying the common case of
_quadratic operads_ , since it turns out that the operads of interest,
$\mathit{Ass}$, $\mathit{Com}$, and $\mathit{Lie}$, corresponding to
associative, commutative, and Lie algebras, do indeed turn out to be
quadratic. In fact, $\mathit{Ass}$, $\mathit{Com}$, and $\mathit{Lie}$ are
_binary quadratic operads_ , which is to say that they are generated by binary
operations and quadratic relations.

######  Definition I.31 (Quadratic data).

Let us first recall the _minimal model_ of an operad $\mathcal{P}$, as defined
in I.26: a given $\mathcal{M}=\mathcal{T}(E)$ is _minimal_ when
$d(E)\subset\mathcal{T}(E)^{\geq 2}$. To get the definition of a quadratic
operad, we restrict $d(E)$ further to $d(E)\subset\mathcal{T}(E)^{(2)}$. Here,
$\mathcal{T}(E)^{(2)}$ is the sub-$\mathbb{S}$-module of the free operad
$\mathcal{T}$, which is spanned by two elements of $E$.

A quadratic operad or cooperad can be constructed with two pieces of data, say
$(E,R)$, where elements of $E$ are termed _generating operations_ and elements
of $R$ are termed _relators_. The quadraticity hypothesis says that the
relators are made up of elements involving exactly two compositions. We will,
at the end of the section, see the example of $\mathit{Ass}$ and
$\mathit{Com}$, which are _binary quadratic operads_ ; the extra “binary”
qualifier says that there the generating operations consist entirely of binary
operations, in which case, all relations involve three variables.

######  Definition I.32 (Quadratic operad).

A quadratic operad $\mathcal{P}$ is the free operad over $E$ quotiented out by
the ideal generated by $R$.

$\mathcal{P}(E,R):=\mathcal{T}(E)/(R)$

such that the composite

${R}$${\mathcal{T}(E)}$${\mathcal{P}}$

is zero, and the following diagram commutes

${R}$${\mathcal{T}(E)}$${\mathcal{P}}$${\mathcal{P}(E,R)}$$\scriptstyle{\exists!}$

######  Example I.33 (Explication of quadratic operad).

Let $E=\\{0,E(1),E(2),E(3),E(4),\ldots\\}$. Then,
$\mathcal{T}(E)=\oplus_{k}\mathcal{T}(E)^{(k)}$, where

$\begin{matrix}\mathcal{T}(E)^{(0)}&=&(0,\mathbb{K}\id,0,0,\ldots)&=&\I\\\
\mathcal{T}(E)^{(1)}&=&(0,E(1),E(2),E(3),\ldots)&=&E\\\
\mathcal{T}(E)^{(2)}&=&(0,E(1)^{\otimes 2},E(2)^{\otimes 2},E(3)^{\otimes
2},\ldots)&&\\\ \end{matrix}$

The quotient $\mathcal{P}(E,R)$ of $\mathcal{T}(E)$ is such that

$\displaystyle\mathcal{P}(E,R)^{(0)}$ $\displaystyle=\I$
$\displaystyle\mathcal{P}(E,R)^{(1)}$ $\displaystyle=E$
$\displaystyle\mathcal{P}(E,R)^{(2)}$ $\displaystyle=(0,E(1)^{\otimes
2}/R(1),\ldots)$

######  Definition I.34 (Quadratic cooperad).

A quadratic cooperad $\mathcal{C}$ is the sub-cooperad of the co-free cooperad
$\mathcal{T}^{c}(E)$ such that the composite

${\mathcal{C}}$${\mathcal{T}^{c}(E)}$${\mathcal{T}^{c}(E)^{(2)}/R}$

is zero, and the following diagram commutes

${\mathcal{C}}$${\mathcal{T}^{c}(E)}$${\mathcal{T}^{c}(E)^{(2)}/R}$${\mathcal{C}(E,R)}$$\scriptstyle{\exists!}$

######  Example I.35 (Explication of quadratic cooperad).

Contrasting with the explication of a quadratic operad, we see that
$\mathcal{T}^{c}(E)$ has the same underlying $\mathbb{S}$-module as
$\mathcal{T}(E)$, and $\mathcal{C}(E,R)$ is such that

$\displaystyle\mathcal{C}(E,R)^{(0)}$ $\displaystyle=\I$
$\displaystyle\mathcal{C}(E,R)^{(1)}$ $\displaystyle=E$
$\displaystyle\mathcal{C}(E,R)^{(2)}$
$\displaystyle=(0,R(1),R(2),R(3),\ldots)$

We now take a second look at many of notions introduced in the previous
section, in the special case of quadratic operads.

######  Definition I.36 ($\mathcal{P}^{\text{!`}}$).

We define the _Koszul dual cooperad_ of a quadratic operad,
$\mathcal{P}^{\text{!`}}$ 222prononouced ”anti-shriek”, as

$\mathcal{P}^{\text{!`}}(E,R):=\mathcal{C}(E[1],R[2])$

By definition,

$(\mathcal{P}^{\text{!`}})^{\text{!`}}=\mathcal{P}$

######  Definition I.37 ($\mathcal{P}^{!}$).

The _Koszul dual operad_ , $\mathcal{P}^{!}$ 333pronounced ”shriek”, of a
quadratic operad $\mathcal{P}$ is the arity-wise linear dual of
$\mathcal{P}^{\text{!`}}$, where the square brackets indicate degree shift:

$\mathcal{P}^{!}(n):=\Hom_{\mathbb{K}}(\mathcal{P}^{\text{!`}}(n),\mathbb{K})[1-n]$

######  Lemma I.38 (Self-dualizing property of shriek).

For quadratic operad $\mathcal{P}$, generated by a reduced $\mathbb{S}$-module
of finite dimension in each arity,

$(\mathcal{P}^{!})^{!}=\mathcal{P}$

In particular, the finite dimensionality condition ensures that
$(E^{\vee})^{\vee}=E$.

######  Example I.39 (Relationships between $\mathit{Ass}$, $\mathit{Com}$,
and $\mathit{Lie}$).

In summary,

$\displaystyle\mathit{Ass}^{!}$ $\displaystyle=\mathit{Ass}$
$\displaystyle\mathit{Com}^{!}$ $\displaystyle=\mathit{Lie}$
$\displaystyle\mathit{Lie}^{!}$ $\displaystyle=\mathit{Com}$

######  Remark I.40 ($\mathbf{B}$ and $\mathbf{\Omega}$ for quadratic operad).

Since the internal differential of $E$ is trivial,

  1. ($\mathcal{P}$)

The inclusion $\mathcal{P}^{\text{!`}}\rightarrowtail\mathbf{B}\mathcal{P}$
induces an isomorphism $\mathcal{P}^{\text{!`}}\cong
H^{0}(\mathbf{B}^{\bullet}\mathcal{P})$.

  2. ($\mathcal{C}$)

The surjection
$\mathbf{\Omega}\mathcal{C}\twoheadrightarrow\mathcal{C}^{\text{!`}}$ induces
an isomorphism $H_{0}(\mathbf{\Omega}_{\bullet}\mathcal{C})\cong\mathcal{C}$.

######  Remark I.41 (Natural twisting morphism).

Given quadratic data $(E,R)$, since $\mathcal{P}(E,R)^{(1)}=E$ and
$\mathcal{C}(E,R)^{(1)}=E$, we can define a morphism $\kappa$ as the composite

$\mathcal{C}(E[1],R[2])\twoheadrightarrow sE\rightarrow
E\rightarrowtail\mathcal{P}(E,R)$

Notice that $\kappa\star\kappa=0$, and hence $\kappa$ is a twisting morphism.

######  Definition I.42 (Koszul complex).

One complex associated to $\kappa$,

$\mathcal{P}^{\text{!`}}\circ_{\kappa}\mathcal{P}:=(\mathcal{P}^{\text{!`}}\circ\mathcal{P},d_{\kappa})$

is termed as the _(right) Koszul complex_ associated to quadratic operad
$\mathcal{P}$. We can define another completely different complex
corresponding to $\mathcal{P}\circ_{\kappa}\mathcal{P}^{\text{!`}}$, and this
would be called the _left Koszul complex_.

[Koszuality of an operad] A quadratic operad $\mathcal{P}$ is termed _Koszul_
if and only if its associated Koszul complex is acyclic.

######  Lemma I.43 (Minimal model of Koszul operad).

The minimal model of a Koszul operad $\mathcal{P}$ is
$\mathbf{\Omega}\mathcal{P}^{\text{!`}}$.

######  Lemma I.44 (Koszuality and duality).

Finitely-generated operad $\mathcal{P}$ is Koszul if and only if
$\mathcal{P}^{!}$ is Koszul.

[$\mathcal{P}_{\infty}$-algebra] Given Koszul operad $\mathcal{P}$, by
definition, a _homotopy $\mathcal{P}$-algebra_, or a
$\mathcal{P}_{\infty}$-algebra, is an algebra over $\mathcal{P}_{\infty}$,
which is defined as

$\mathcal{P}_{\infty}:=\mathbf{\Omega}\mathcal{P}^{\text{!`}}$

[Canonical cofibrant resolution] Recalling I.30, we infer that for a Koszul
operad $\mathcal{P}$,
$\mathcal{P}^{\text{!`}}\rightarrow\mathbf{B}\mathcal{P}$ and
$\mathbf{\Omega}\mathcal{P}^{\text{!`}}\rightarrow\mathcal{P}$ are quasi-
isomorphisms, and they can be thought of as resolutions of $\mathcal{P}$.

Indeed, with a bit of background on the model structure on $\mathcal{P}$
444Otherwise known as the Hinich model structure [HinichModel]., which we
refrain from discussing here, it turns out that $\mathcal{P}_{\infty}$ is the
_canonical cofibrant resolution_ of $\mathcal{P}$.

######  Example I.45 (Duals of $\mathit{Com}$ and $\mathit{Lie}$).

As mentioned earlier, $\mathit{Com}$ and $\mathit{Lie}$ are binary quadratic
operads. We take this opportunity to work out their duals.

The quadratic operad $\mathit{Com}=\mathcal{P}(E,R)$ is generated by binary
symmetric operations in $\mu$, i.e. $E=\mathbb{K}\mu$. Let us denote by
$\mu_{\mathsf{I}}$, $\mu_{\mathsf{II}}$ and $\mu_{\mathsf{III}}$ the elements
$\mu\circ_{\mathsf{I}}\mu$, $\mu\circ_{\mathsf{II}}\mu$, and
$\mu\circ_{\mathsf{III}}\mu$ in $\mathcal{T}(\mathbb{K}\mu)(3)$. The space of
relations $R$ admits the following $\mathbb{K}$-linear basis made up of two
elements, $\mu_{\mathsf{II}}-\mu_{\mathsf{I}}$ and
$\mu_{\mathsf{III}}-\mu_{\mathsf{II}}$.

The Koszul dual is generated by the space $E^{\vee}\cong\mathbb{K}\nu$, where
$\nu^{(12)}=-\nu$. Again, let us denote by $\nu_{\mathsf{I}}$,
$\nu_{\mathsf{II}}$, and $\nu_{\mathsf{III}}$, the elements of
$\nu\circ_{\mathsf{I}}\nu$, $\nu\circ_{\mathsf{II}}\nu$, and
$\nu\circ_{\mathsf{III}}\nu$ of $\mathcal{T}(\mathbb{K}\nu)(3)$, respectively.
The orthogonal space of $R$ under scalar product $\langle-,-\rangle$ has
dimension with basis $\nu_{\mathsf{I}}+\nu_{\mathsf{II}}+\nu_{\mathsf{III}}$.
Denoting skew-symmetric generating operation $\nu$ by bracket $[-,-]$, the
latter is nothing but the Jacobi identity $[[a,b],c]+[[b,c],a]+[[c,a],b]=0$.
Hence, $\mathit{Com}^{!}=\mathit{Lie}$.

Since all our operads of interest are Koszul, we can safely assume that Koszul
duality works, and indeed that, a homotopy operadic algebra is well-defined.
In the final section, we will shift our attention exclusively to
$\mathcal{P}_{\infty}$-algebras.

###  I.8 $\mathcal{P}_{\infty}$-algebras

All that we’ve studied so far culminate into this final section containing the
seminal results of operadic theory.

The main topic of study is a theorem central to study of homotopy
$\mathcal{P}$-algebras, the _Rosetta Stone_ , which provides isomorphisms
between four different definitions of a $\mathcal{P}_{\infty}$-algebra. We
state it as-is, with some commentary.

######  Theorem I.46 (Rosetta Stone).

The Rosetta Stone establishes that four different explications of the
$\mathcal{P}_{\infty}$\- structure on module $A$ are equivalent.

$\Hom(\mathbf{\Omega}\mathcal{P}^{\text{!`}},\End{A})\cong\mathsf{Tw}(\mathcal{P}^{\text{!`}},\End{A})\cong\Hom(\mathcal{P}^{\text{!`}},\mathbf{B}\End{A})\cong\mathsf{Codiff}(\mathcal{P}^{\text{!`}}(A))$

$\mathbf{\Omega}\mathcal{P}^{\text{!`}}\tilde{\longrightarrow}\mathcal{P}\longrightarrow\End{A}$

can be interpreted in terms of the $\mathsf{Tw}$. A morphism of operads from
$\mathbf{\Omega}\mathcal{P}^{\text{!`}}$ to $\End{A}$ is equivalent to the
twisting morphism in the convolution algebra

$\mathfrak{g}:=\Hom_{\mathbb{S}}(\mathcal{P}^{\text{!`}},\End{A})$

A homotopy $\mathcal{P}$-algebra structure on module $A$ is equivalent to the
twisting morphism $\mathsf{Tw}(\mathcal{P}^{\text{!`}},\End{A})$.

######  Lemma I.47 (Characterization of $\mathcal{P}$-algebras).

$\mathcal{P}$-algebras are characterized among $\mathcal{P}_{\infty}$-algebras
as particular solutions to the Maurer-Cartan equation. In particular, a
$\mathcal{P}_{\infty}$-algebra is a $\mathcal{P}$-algebra if its twisting
morphism is concentrated in weight (number of vertices in the tree) $1$.

######  Definition I.48 (-isomorphism and -quasi-isomorphism).

An -morphism of $\mathcal{P}_{\infty}$ algebras is given by:

$f:(\mathcal{P}^{\text{!`}}(A),d_{\sigma})\rightarrow(\mathcal{P}^{\text{!`}}(B),d_{\psi})$

It is an -isomorphism if its first component $f_{(0)}:A\rightarrow B$ is an
isomorphism.

An -quasi-isomorphism $f$ is called so if its first component
$f_{(0)}:A\rightarrow B$ is a quasi-isomorphism.

######  Theorem I.49 (Homotopy Transfer Theorem).

When working over a field, the homotopy $H(A)$ can be made a deformation
retract of $A$. It enables us to transfer the $\mathcal{P}_{\infty}$-algebra
structure from $A$ to $H(A)$.

Let $(V,d_{V})$ be the homotopy retract of the data of $(W,d_{W})$:

${(W,d_{W})}$${(V,d_{V})}$$\scriptstyle{h}$$\scriptstyle{p}$$\scriptstyle{i}$

Let $\mathcal{P}$ be a Koszul operad, and let $(V,d_{V})$ be the homotopy
retract of $(W,d_{W})$. Any $\mathcal{P}_{\infty}$-algebra structure on $W$
can be transferred to a $\mathcal{P}_{\infty}$-algebra structure on $V$ such
that $i$ extends to an -quasi-isomophism.

To prove it, we use the third definition of a $\mathcal{P}_{\infty}$-algebra
supplied in the Rosetta Stone.

$\Hom(\mathbf{\Omega}\mathcal{P}^{\text{!`}},\End{A})\cong\mathsf{Tw}(\mathcal{P}^{\text{!`}},\End{A})\cong\Hom(\mathcal{P}^{\text{!`}},\mathbf{B}\End{A})$

##  II Derived deformation theory

In the first two sections, we introduce and motivate derived deformation
theory. The final section introduces the Lurie-Pridham equivalence, and can be
read as an introduction to derived deformation theory, as it appears in [SAG].

###  II.1 Classical deformation theory of associative algebras

In order to understand precisely what a “deformation” is, we provide some
intuitions from its historical roots: the classical deformation theory of
associative algebras.

######  Definition II.1 (Hochschild cohomology).

The Hochschild cohomology of an associative algebra $A$ with coefficients in
module $M$ is given by:

${0}$${M}$${C^{1}_{\mathsf{Hoch}}(A,M)}$${\ldots}$${C^{n}_{\mathsf{Hoch}}(A,M)}$${\ldots}$$\scriptstyle{\delta_{\mathsf{Hoch}}}$$\scriptstyle{\delta_{\mathsf{Hoch}}}$$\scriptstyle{\delta_{\mathsf{Hoch}}}$$\scriptstyle{\delta_{\mathsf{Hoch}}}$$\scriptstyle{\delta_{\mathsf{Hoch}}}$

where $C^{n}_{\mathsf{Hoch}}(A,M)$ is the space of $n$-multilinear maps from
$A$ to $M$.

$C^{n}_{\mathsf{Hoch}}(A,M):=\Hom_{\mathbb{K}}(A^{\otimes n},M)$

######  Definition II.2 (Formal deformation).

A formal deformation of associative algebra $A$ is defined as a deformation
along the formal power series $\mathbb{K}\llbracket t\rrbracket$, given by the
family

$\\{\mu_{i}:A\otimes A\rightarrow A\mid i\in\mathbb{N}\\}$

where each $\mu_{i}$ corresponds to the $t^{i}$ deformation, satisfying the
following:

  1. ($D_{0}$)

$\mu_{0}(a,b)=ab$.

  2. ($D_{1}$)

$\displaystyle\mu_{1}(a,\mu_{0}(b,c))$
$\displaystyle+\mu_{0}(a,\mu_{1}(b,c))=$
$\displaystyle\mu_{1}(\mu_{0}(a,b),c)$ $\displaystyle+\mu_{0}(\mu_{1}(a,b),c)$

  3. ($D_{k}$)

$\Sigma_{i+j=k}\mu_{i}(a,\mu_{j}(b,c))=\Sigma_{i+j=k}\mu_{i}(\mu_{j}(a,b),c)$

where $a,b,c\in A$.

######  Remark II.3 (An observation on $D_{1}$).

Observe that $D_{1}$ reads as

$a\mu_{1}(b,c)-\mu_{1}(ab,c)+\mu_{1}(a,bc)-\mu_{1}(a,b)c=0$

This says that $\mu_{1}\in\Hom_{\mathbb{K}}(A^{\otimes 2},A)$ is a Hochschild
cocycle. In other words, $\delta_{\mathsf{Hoch}}(\mu_{1})=0$. This motivates
our next definition.

######  Definition II.4 (Infinitesimal deformation).

An infinitesimal deformation $\epsilon$, concentrated only at degree $0$, of
associative algebra $A$ over field $\mathbb{K}$ is a $D$-deformation of $A$,
where

$D:=\mathbb{K}[\epsilon]/\epsilon^{2}$

is a _local Artin ring_ [B.1] of dual numbers.

######  Theorem II.5 (Significance of Hochschild cochain).

There is a one-to-one correspondence between equivalence classes of
infinitesimal deformations of $A$ and Hochschild cohomology
$H_{\mathsf{Hoch}}^{2}(A,A)$.

[Maurer-Cartan equation in associative algebras] Let $A$ be an associative
algebra and multiplication operator $\mu:A\otimes A\rightarrow A$ and $\nu\in
C_{\mathsf{Hoch}}(A,A)$ be a Hochschild 2-cochain. Then, $\mu+\nu$ is
associative if and only if

$\delta_{\mathsf{Hoch}}(\nu)+\frac{1}{2}[\nu,\nu]=0$

This is indeed an incarnation of the Maurer-Cartan equation, and this gives us
the first hint that there is a Lie algebra hiding in every deformation
problem.

###  II.2 Motivation from classical geometric deformation theory

We now present the journey from (classical) deformation theory in algebraic
geometry to derived deformation theory, in derived algebraic geometry.

In algebraic geometry, the deformation functor

$\mathsf{Def}_{X_{0}}:\mathsf{Art}_{\mathbb{K}}\rightarrow\Set$

sends an augmented Artin algebra $\mathbf{A}$ to isomorphism classes $(X,u)$,
where $X$ is a scheme and $u:X\otimes_{\mathbf{A}}\mathbb{K}\cong X_{0}$ is an
isomorphism of _algebraic varieties_.

It was then realized that this deformation problem was controlled by a dg-Lie
algebra $L$, and we form the set $\mathsf{MC}(L\otimes_{\mathbb{K}}m)$, where
$m$ is the maximal ideal of an augmented Artin algebra $\mathbf{A}$.

$X_{L}:\mathsf{Art}_{\mathbb{K}}\rightarrow\Set$

The ring of functions formed by $X_{L}$ can be identified with
$H^{0}(\hat{C}^{*}(L))$ where
$\hat{C}^{*}:\op{\mathsf{dgLie}}_{\mathbb{K}}\rightarrow\mathsf{cdga}_{\mathbb{K}}$
is termed _Chevalley complex_.

In the above picture, the _tangent space_ is a simple $H^{1}(L)$, and
$H^{2}(L)$ gives us one obstruction to the formal smoothness. $H^{i}(L)$ for
$i\leq 0$ gives us increasing orders of symmetry, and the information
contained in $H^{i}(L)$ for $i\geq 2$ is most interesting, and marks the
beginning of derived deformation theory; indeed, it gives higher order
defects, and there is no way to extract this information from the classical
picture.

We proceed to refine the above picture.

$\F:\mathsf{Art}_{\mathbb{K}}\rightarrow\Set$

Functor $\F$ sends (augmented) Artin algebras to _equivalence classes_ of
moduli spaces. The tangent space of $\F$, formerly given by
$\F(\mathbb{K}[\epsilon]/\epsilon^{2})$, can now be thought of as a _tangent
vector_ , $\F(\mathbb{K}[x]/x^{3})\rightarrow\F(\mathbb{K}[x]/x^{2})$. It then
determines the following _homotopy cartesian_ square

${\mathbb{K}[x]/x^{3}}$${\mathbb{K}}$${\mathbb{K}[x]/x^{2}}$${\mathbb{K}[\epsilon_{-1}]/{\epsilon_{-1}}^{2}}$

where $\F(\mathbb{K}[\epsilon_{-1}]/{\epsilon_{-1}}^{2})$ corresponds to the
classical deformation problem, and $\epsilon_{-1}$ is in cohomological degree
$-1$.

If $\F$ comes from the restriction of a functor defined on the dg-Artin
algebra, we can study the smoothness defects of $\F$ at $x$ using the above
homotopy cartesian square.

The _equivalence classes_ mentioned in the above picture is also vague; using
machinery we will detail in III.1, the equivalence classes are then identified
as weak equivalences of simplicial sets. Further, the functor $\F$ can be
identified as the _formal spectrum_ of augmented Artin algebra $\mathbf{A}$,
denoted $\mathsf{Spf}$($\mathbf{A}$).

$\mathsf{Spf}(\mathbf{A}):\mathsf{dgArt}_{\mathbb{K}}\rightarrow\SSet$

Having to work with weak equivalences of simplicial sets suggests that the
picture can be lifted to the level of abstraction of -categories, which we
describe in the next section.

###  II.3 The Lurie-Pridham equivalence

The Lurie-Pridham equivalence, in its elaborated form, involves a lot of
machinery. We supply the formal definitions as-is, and supply intuitions.

Lurie’s work is done in the setting of -operads, and in particular, all
algebras mentioned are -operadic algebras. However, we will not concern
ourselves with the details of this.

######  Definition II.6 (Deformation context).

A deformation context is a pair $(\mathbf{A},\\{E_{\alpha}\\}_{\alpha\in T})$,
where $\mathbf{A}$ is a _presentable -category_ [E.2], and
$\\{E_{\alpha}\\}_{\alpha\in T}$ is a set of _spectrum objects_ of
$\mathbf{A}$, denoted $\mathsf{Sp}(\mathbf{A})$ [E.6].

[$\\{E_{\alpha}\\}_{\alpha\in T}$] This can be thought of as
$\mathbb{K}[\epsilon_{i}]/({\epsilon_{i}}^{2})$ in the classical setting,
where $\epsilon$ is concentrated in non-negative degrees; each $\epsilon_{i}$
is a shift by $\mathsf{[i]}$, corresponding to the loop functor $\Omega$ in
stable homotopy theory.

######  Definition II.7 (Formal moduli problem).

Let $(\mathbf{A},\\{E_{\alpha}\\}_{\alpha\in T})$ be a deformation context. A
formal moduli problem is defined as a functor from an _Artin algebra_ [E.1] to
the _-category of spaces_ [E.3].

$\mathbf{A}^{\mathsf{Art}}\rightarrow\mathcal{S}$

satisfying:

  1. (a)

Space $X(*)$ is contractible, where $*$ denotes final object of $\mathbf{A}$.

  2. (b)

Let $\sigma$:

${A^{\prime}}$${B^{\prime}}$${A}$${B}$$\scriptstyle{\phi}$

be a diagram in $\mathbf{A}^{\mathsf{Art}}$. If $\sigma$ is a pullback diagram
and $\phi$ is small, then $X(\sigma)$ is a pullback diagram in $\mathcal{S}$.

######  Definition II.8 (Deformation problem).

A deformation problem is a functor
$\mathfrak{D}:\op{\mathbf{A}}\rightarrow\mathbf{B}$, where $\mathbf{A}$ and
$\mathbf{B}$ are some algebras, satisfying the following list of axioms.

  1. (D1)

$\mathbf{B}$ is a _presentable -category_ [E.2].

  2. (D2)

$\mathfrak{D}$ admits a left adjoint
$\mathfrak{D}^{\prime}:\mathbf{B}\rightarrow\op{\mathbf{A}}$.

  3. (D3)

There exists a full subcategory $\mathbf{B}_{0}\subseteq\mathbf{B}$
satisfying:

    1. (a)

For every $K\in B_{0}$ the unit map
$K\rightarrow\mathfrak{D}\mathfrak{D}^{\prime}K$ is an equivalence.

    2. (b)

$\mathbf{B}_{0}$ contains the initial object $\0\in\mathbf{B}$. Then, it
follows from (a) that
$\0\simeq\mathfrak{D}\mathfrak{D}^{\prime}\0\simeq\mathfrak{D}(*)$ where $*$
is the initial object of $\mathbf{A}$.

    3. (c)

For every $\alpha\in T$, and every $n\geq 1$, there exists an object
$K_{\alpha,n}\in B_{0}$, and an equivalence
$\Omega^{\8-n}E_{\alpha}\simeq\mathfrak{D}^{\prime}K_{\alpha,n}$.

    4. (d)

For every pushout:

${K_{\alpha,n}}$${K}$${\0}$${K^{\prime}}$

where $\alpha\in T$, $n\in\mathbb{N}$, if $K\in B_{0}$, then $K^{\prime}\in
B_{0}$.

  4. (D4)

For every $\alpha\in T$,
$e_{\alpha}:\mathbf{B}\rightarrow\mathsf{Sp}(\mathcal{S})$ preserves _small
sifted colimits_ [E.3]. Moreover, a morphism $f\in\mathbf{B}$ is an
equivalence if and only if each $e_{\alpha}(f)$ is an equivalence of spectra.

[Interpretation of the axioms] Let us try to approximate what (D1)-(D4) mean.

  1. (1)

$\mathfrak{D}$ and $\mathfrak{D}^{\prime}$ form an adjoint pair. They’re
almost equivalences, but for two caveats:

    1. (a)

The counit map is not an equivalence.

    2. (b)

The unit map is an equivalence only for some “good” subcategory
$\mathbf{B}_{0}\subseteq\mathbf{B}$.

  2. (2)

$\mathfrak{D}^{\prime}$ is homotopy inverse to $\mathfrak{D}$ on some “good”
subcategory $\mathbf{B}_{0}\subseteq\mathbf{B}$. In particular, (D3d) is dual
to the condition on Artin algebras [III.8].

  3. (D4)

$e_{\alpha}$ admits a left adjoint, and
$\mathbf{B}\simeq\mathsf{LMod}_{U}(Sp^{T})$ is an equivalence of -categories.
In other words, we can think of objects of $\mathbf{B}$ being determined by a
collection of spectra indexed by $T$ equipped with the additional structure of
_left action by monad U_ [E.3].

We will return to this once again, in the III.

######  Theorem II.9 (Lurie-Pridham).

Given a deformation problem
$\mathfrak{D}:\op{\mathbf{A}}\rightarrow\mathbf{B}$, it is related to the
class to the class of formal moduli problems $\mathsf{FMP}$, as follows.

$\mathsf{FMP}^{\mathbf{A}}\simeq\mathbf{B}$

is an equivalence of -categories.

###  II.4 The tangent complex

The tangent complex is an enhancement of the tangent vector space into a chain
complex, such that it is defined at every point in the space and globally as a
bundle. For smooth points, tangent complexes are quasi-isomorphic to tangent
spaces. The most important property of tangent complexes is that they have a
natural Lie algebra structure; at smooth points, it is Abelian. The whole
formal neighborhood of a singular point can be reconstructured from this
structure.

Let $X$ be an algebraic variety over the field $\mathbf{C}$ of complex
numbers, and let $x:X\rightarrow\mathsf{Spec}(\mathbf{C})$ be a point of $X$.
Then a tangent vector at point $x$ is the dotted arrow rendering the following
diagram commutative.

${\mathsf{Spec}(\mathbf{C})}$${X}$${\mathsf{Spec}(\mathbf{C}[\epsilon]/\epsilon^{2})}$${\mathsf{Spec}(\mathbf{C})}$$\scriptstyle{x}$

The collection of tangent vectors at $x$ comprise a vector space
$\mathbb{T}_{x}$. Our goal is to generalize the construction of
$\mathbb{T}_{x}$ in the setting of an arbitrary formal moduli problem.

######  Definition II.10 (Tangent space).

Let $(\mathbf{A},\\{E_{\alpha}\\}_{\alpha\in T})$ be a deformation context and
let $Y:\mathbf{A}^{\mathsf{Art}}\rightarrow\mathcal{S}$ be a formal moduli
problem. For each $\alpha\in T$, the tangent space of $Y$ at $\alpha$ is the
space $Y(\Omega^{\8}E_{\alpha})$.

######  Lemma II.11 (Characterization of spectrum objects).

If $\mathscr{C}$ is an -category which admits finite limits, the spectrum
objects of $\mathscr{C}$, or $\mathsf{Sp}(\mathscr{C})$, is defined to be the
full subcategory of $\Hom(\mathcal{S}_{*}^{\mathsf{fin}},\mathscr{C})$,
spanned by those functors which are reduced and excisive. [E]

Let $(\mathbf{A},\\{E_{\alpha}\\}_{\alpha\in T})$ be a deformation context.
Then, we identify, for each $\alpha\in T$,
$E_{\alpha}\in\mathsf{Sp}(\mathscr{C})$ with the corresponding functor
$\mathcal{S}_{*}^{\mathsf{fin}}\rightarrow\mathscr{C}$. It can be shown that,
for pointed space $K$, $E_{\alpha}(K)$ is Artinian. It can further be shown
that the composite functor

$\mathcal{S}_{*}^{\mathsf{fin}}\overset{E_{\alpha}}{\rightarrow}\mathbf{A}^{\mathsf{Art}}\overset{Y}{\rightarrow}\mathcal{S}$

is reduced and excisive.

These observations motivate the following definition.

######  Definition II.12 (Tangent complex).

Let $(\mathbf{A},\\{E_{\alpha}\\}_{\alpha\in T})$ be a deformation context,
and $Y:\mathbf{A}^{\mathsf{Art}}\rightarrow\mathcal{S}$ be a formal moduli
problem. Then, the composite functor

$\mathcal{S}_{*}^{\mathsf{fin}}\overset{E_{\alpha}}{\rightarrow}\mathbf{A}^{\mathsf{Art}}\overset{Y}{\rightarrow}\mathcal{S}$

denoted by $Y(E_{\alpha})$, is an object of -category
$\mathsf{Sp}(\mathcal{S})$. We will call $Y(E_{\alpha})$ the _tangent complex_
of $Y$ at $\alpha$.

######  Remark II.13 (Tangent spectrum).

The terminology introduced in the above definition is potentially misleading,
as $Y(E_{\alpha})$ is a spectrum. However, for instance, when
$(\mathbf{A},\\{E_{\alpha}\\}_{\alpha\in
T})=(\mathsf{CAlg}^{\mathsf{aug}}_{\mathbb{K}},\\{E\\})$, for field
$\mathbb{K}$, the tangent spectrum $Y(E)$ admits the structure of a
$\mathbb{K}$-module spectrum, and can hence be identified with chain complexes
of vector spaces over $\mathbb{K}$. See also: [E.5]

##  III Operadic deformation theory

This part is dedicated to re-interpreting II in the setting of operads, for
which the reader should be fully prepared, after having digested I. All
original results will be marked with a $\dagger$.

###  III.1 The Goldman-Millson theorem

The aim of this interlude is to understand the mechanics of _simplicial
enrichment_ of a Lie algebra $\mathfrak{g}$, to better equip ourselves to
understand the next sections.

We begin with the well-known concept of the Maurer-Cartan set of elements of
Lie algebra $\mathfrak{g}$, $\mathsf{MC}$($\mathfrak{g}$) [B.5]. It has the
defining property that, a field $\mathfrak{g}^{1}\rightarrow\mathfrak{g}^{1}$
with a _flow_ , preserves the Maurer-Cartan set of elements of $\mathfrak{g}$.
We say that $\mathsf{MC}(\mathfrak{g})$ is _flow-invariant_ , and we obtain
_gauge equivalences_ in $\mathfrak{g}$, which we denote as
$\sim_{\mathsf{gauge}}$. We can now define

######  Definition III.1 (Maurer-Cartan moduli space).

The _moduli space_ of Maurer-Cartan elements is defined as

$\overline{\mathsf{MC}}(\mathfrak{g}):=\mathsf{MC}(\mathfrak{g})/\sim_{\mathsf{gauge}}$

In fact,

$\overline{\mathsf{MC}}:\\{\mathfrak{g}\in\mathit{Lie}\mid\text{$\mathfrak{g}$
is complete with respect to its canonical filtration [\ref{term:complete-
canonical}]}\\}\rightarrow\Set$

######  Theorem III.2 (Goldman-Millson).

Let $\mathfrak{g}$, $\mathfrak{h}$ be _complete Lie algebras_ [B.7], and
$\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ be a _filtered quasi-isomorphism_
[A.6]. Then,

$\overline{\mathsf{MC}}(\phi):\overline{\mathsf{MC}}(\mathfrak{g})\rightarrow\overline{\mathsf{MC}}(\mathfrak{h})$

is a bijection. See [NicoudSurvey] for more on this.

The Goldman-Millson theorem is but an avatar of a “higher” result. This result
encodes higher equivalences into $\mathsf{MC}$, not just gauge equivalences.

Consider a nilpotent Lie algebra $\mathfrak{g}$, and tensor it with the
_Sullivan algebra_ $\Omega_{\bullet}$ [B.8]. Notice that
$\mathfrak{g}\otimes\Omega_{n}$ is again a Lie algebra by

$[x\otimes\alpha,y\otimes\beta]=[x,y]\otimes\alpha\beta$

######  Definition III.3 (Simplicial enrichment of Lie algebra).

Given a complete Lie algebra $(\mathfrak{g},F_{\bullet}\mathfrak{g})$, we can
_simplicially enrich_ it by tensoring it with the Sullivan commutative algebra
$\Omega_{\bullet}$, as follows:

$\mathfrak{g}\otimes\Omega_{\bullet}:=\lim_{n}(\mathfrak{g}/F_{n}\mathfrak{g}\otimes\Omega_{\bullet})$

######  Definition III.4 (Hinich-Getzler -groupoid).

Let $\mathfrak{g}$ be a complete Lie algebra. The _Hinich-Getzler -groupoid_
is defined as the simplicial set

$\displaystyle\mathsf{MC}_{\bullet}$ $\displaystyle:\textsf{dg-
Lie}\rightarrow\SSet$ $\displaystyle\mathsf{MC}_{\bullet}(\mathfrak{g})$
$\displaystyle:=\mathsf{MC}(\mathfrak{g}\otimes\Omega_{\bullet})$

As hinted earlier, indeed

$\pi_{0}\mathsf{MC}_{\bullet}(\mathfrak{g})=\overline{\mathsf{MC}}(\mathfrak{g})$

######  Theorem III.5 (Fibrancy of $\mathsf{MC}_{\bullet}$).

The functor $\mathsf{MC}_{\bullet}$ has image in the full subcategory of . See
[HinichFibrancy] for more.

Finally, we state the more modern version of the Goldman-Millson theorem.

######  Theorem III.6 (Dolgushev-Rogers).

Let $\mathfrak{g}$, $\mathfrak{h}$ be complete Lie algebras, and
$\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ be a filtered quasi-isomorphism.
Then,

$\mathsf{MC}_{\bullet}(\phi):\mathsf{MC}_{\bullet}(\mathfrak{g})\rightarrow\mathsf{MC}_{\bullet}(\mathfrak{h})$

is a homotopy equivalence of simplicial sets. See [DolgushevRogers] for more.

The functor $\mathsf{MC}$ has several concrete definitions, and its image is
an -category, in its most general form. To avoid encumbering ourselves with
notation, we will henceforth refer to all variations as $\mathsf{MC}$, letting
the context resolve the ambiguity.

###  III.2 Main results

The setting is homotopy invariant, and in particular, $\mathcal{P}$-algebras
and $\mathcal{P}_{\infty}$-algebras are indistinguishable in this setting.

All (co)operads are assumed to be (co)augmented, and (filtered) cofibrant as
left $\mathbb{K}$-modules.

The paper [OpFMP] is written in the setting of _colored operads_. In order to
avoid encumbering ourselves with notation and fine detail, we will drop the
colors, which must be drawn from $\mathbb{K}$, everywhere.

[$\mathcal{P}$-algebras form an -category] The category of
$\mathcal{P}$-algebras, has image in the full subcategory of .

######  Proof III.7.

In the setting of -operads, the machinery of which is detailed in [HA], this
is true by construction. However, we are working in the setting of ordinary
operads (with color implicit), there is only a model theoretic structure on
it, and there is a simplicial enrichment possible on a ordinary
$\mathcal{P}$-algebra $\mathbf{A}$just like the case of Lie algebras in field
of characteristic $0$ given by III.3, by tensoring with simplicial commutative
_Sullivan algebra_ $\Omega_{\bullet}$:

$\mathbf{A}\otimes\Omega_{\bullet}$

######  Definition III.8 (Artin $\mathcal{P}$-algebra).

The -category $\mathsf{Art}_{\mathcal{P}}$ of _Artin $\mathcal{P}$-algebras_
is the smallest full subcategory of the -category of $\mathcal{P}$-algebras
such that

  1. (i)

Every trivial algebra $\mathbb{K}[n]$ is Artin, for all $n\in\mathbb{N}$.

  2. (ii)

Any given Artin $\mathcal{P}$-algebra $\mathbf{A}$, the homotopy pullback

${\mathbf{A}^{\prime}}$${0}$${\mathbf{A}}$${\mathbb{K}[n]}$

is also Artin, for all $n\geq 1$.

######  Definition III.9 (Formal moduli problem over operad).

A formal moduli problem is defined as a functor from Artin
$\mathcal{P}$-algebras to the -category of spaces.

$\mathsf{FMP}_{\mathcal{P}}:\mathsf{Art}_{\mathcal{P}}\rightarrow\mathcal{S}$

satisfying the following conditions:

  1. (i)

$F(0)\simeq*$, where $0$ is the zero-algebra.

  2. (ii)

$F$ sends a pullback diagram in $\mathsf{Art}_{\mathcal{P}}$ of the form

${\mathbf{A}^{\prime}}$${0}$${\mathbf{A}}$${\mathbb{K}[n]}$

to pullback squares of spaces, for each $n\geq 1$.

######  Example III.10 (Formal spectrum of $\mathcal{P}$-algebra).

To supply an example of a formal moduli problem, we can associate to every
$\mathcal{P}$-algebra $\mathbf{B}$, its _formal spectrum_ defined as

$\displaystyle\mathsf{Spf}(\mathbf{B}):\mathsf{Art}_{\mathcal{P}}$
$\displaystyle\rightarrow\mathcal{S}$ $\displaystyle\mathbf{A}$
$\displaystyle\mapsto\Map_{\mathcal{P}}(\mathbf{B},\mathbf{A})$

Applying condition (ii) from the definition of a formal moduli problem, for
the case $\mathbf{A}=0$, the formal spectrum can be interpreted as the
following sequence of equivalences.

$\F(\mathbb{K})\tilde{\longrightarrow}\Omega\F(\mathbb{K}[1])\tilde{\longrightarrow}\Omega^{2}\F(\mathbb{K}[2])\tilde{\longrightarrow}\ldots$

The sequence of spaces forms an $\Omega$-cohomological-spectrum
$\mathbb{T}_{\F}($).

[Homotopy category of formal spectrum] The space of morphisms between
$\mathsf{Art}_{\mathcal{P}}$ and $\mathcal{S}$ is defined as the formal
spectrum $\mathsf{Spf}$. The homotopy category of this particular formal
spectrum (as opposed to the spectrum defined by Lurie, using the machinery of
-operads) does not possess a triangulated structure [E.2].

######  Proof III.11.

$\mathsf{Spf}$ is an $\Omega$-cohomological spectrum, in which the $\pi_{*}$
information is lost. It is not a stable homotopy category [E.5]; in
particular, $\Sigma$ and $\Omega$ are not homotopy inverses to each other.
Hence, its homotopy category has loss of spatial information, and does not
possess a triangulated structure.

[Motivation for the machinery of -operads] The motivation for the machinery of
-operads arises from the fact that they built out of simplicial sets, the
homotopy category of whose spectra possess a triangulated structure [E.2]
(which endow them with spatial information), and whose spectra are homotopy
categories of stable -categories [E.5, E.6].

######  Theorem III.12 (Calaque-Campos-Nuiten).

Given a formal moduli problem $\mathsf{FMP}$, the main result of [OpFMP] can
be stated as the equivalence of the -categories

$\displaystyle\mathsf{FMP}_{\mathcal{P}}$
$\displaystyle\simeq\mathsf{Alg}_{\mathcal{P}^{!}}$ $\displaystyle F$
$\displaystyle\mapsto\mathbb{T}_{F}([)-1]$

where $\mathbb{T}_{F}($) is the tangent complex of $F$.

###  III.3 Relationship to Lie algebras

[Formulating the deformation problem] In order to prove the main result, we
start with the adjunction

${\mathfrak{D}:\mathsf{Alg}_{\mathcal{P}}}$${\op{\mathsf{Alg}}_{\mathfrak{D}(\mathcal{P})}:\mathfrak{D}^{\prime}}$${\bot}$

where $\mathfrak{D}(\mathcal{P})$ is the $\mathbb{K}$-linear dual of the bar
construction

$\displaystyle\mathfrak{D}_{\mathbb{K}}(\mathcal{P}):=\mathbf{B}_{\mathbb{K}}(\mathcal{P})^{\vee}$

Moreover, the Koszul dual property of $\mathcal{P}$ asserts that there are
weak equivalences (using square brackets to denote degree shift):

$\mathfrak{D}(\mathcal{P})\tilde{\longrightarrow}\mathcal{P}^{!}[1]$

The above adjunction is never an equivalence, because both $\mathfrak{D}$ and
$\mathfrak{D}^{\prime}$ send an algebra to its module of derivations with
coefficients in $\mathbb{K}$. To prove the main result, we impose conditions
on $\mathfrak{D}$, so that it satisfies the axioms of a formal moduli problem,
and obtain an equivalence between $\mathfrak{D}(\mathcal{P})$ and
$\mathsf{FMP}_{\mathcal{P}}$.

######  Theorem III.13 (Deformation problem in the operadic setting).

Let $\mathcal{P}$ be an $\mathbb{K}$-operad. There is then an equivalence of
-categories

$\displaystyle\mathsf{MC}:\mathsf{Alg}_{\mathfrak{D}(\mathcal{P})}$
$\displaystyle\tilde{\longrightarrow}\mathsf{FMP}_{\mathcal{P}}$
$\displaystyle\mathfrak{g}$
$\displaystyle\mapsto\Map_{\mathfrak{D}(\mathcal{P})}(\mathfrak{D}(-),\mathfrak{g})$

if the following conditions are satisfied.

  1. (A)

For every Artin $\mathcal{P}$-algebra $\mathbf{A}$, the unit map
$\mathbf{A}\rightarrow\mathfrak{D}\mathfrak{D}^{\prime}(\mathbf{A})$ is an
equivalence.

  2. (B)

For every trivial algebra $\mathbb{K}[n]$ generated by a single element of
degree $n\geq 0$, the $\mathfrak{D}$($\mathcal{P}$)-algebra
$\mathfrak{D}(\mathbb{K}[n])$ is freely generated by $\mathbb{K}[n]^{\vee}$.

  3. (C)

The functor $\mathfrak{D}$ sends every pullback square of Artin algebras

${\mathbf{A}^{\prime}}$${0}$${\mathbf{A}}$${\mathbb{K}[n]}$

with $n\geq 1$ to a pushout square of $\mathfrak{D}$($\mathcal{P}$)-algebras.

When $\mathcal{P}$ satisfies suitable finite-dimensionality conditions, the
formal moduli problem $\mathsf{MC}_{\mathfrak{g}}$ can be described in terms
of Maurer-Cartan simplicial sets of Lie algebras.

[The degenerate case of $\mathsf{MC}_{\mathfrak{g}}$] The
$\mathbb{K}$-operadic version of the result says that, for a suitable
augmented $\mathbb{K}$-operad $\mathcal{P}$, every algebra $\mathfrak{g}$ over
the dual operad $\mathfrak{D}$($\mathcal{P}$) determines the formal moduli
problem

$\mathsf{MC}_{\mathfrak{g}}:\mathsf{Art}_{\mathcal{P}}\rightarrow\mathcal{S}$

where

$\mathsf{MC}_{\mathfrak{g}}(\mathbf{A})=\Map(\mathfrak{D}(\mathbf{A}),\mathfrak{g})$

This can be understood as Maurer-Cartan elements of nilpotent
$\mathsf{L}_{\8}$-algebras.

To recover the classical result, set $\mathcal{P}=\mathit{Com}$ and
$\mathfrak{g}=\mathit{Lie}$ to obtain

$\mathsf{MC}_{\mathfrak{g}}(\mathbf{A})=\mathsf{MC}(\mathbf{A}\otimes\mathfrak{g}\otimes\Omega_{\bullet})$

[Deformation functor and the Koszul duality] The minimum required condition on
operads is detailed in the _Splendid operads_ section of [OpFMP]. However, it
is convenient to use the Koszul duality to construct the $\mathbf{B}$ algebra
from the $\mathbf{A}$ algebra. An incarnation of the Koszul duality exists in
the setting of -operads, which is what Lurie uses in his work.

##  Appendix A Filtration in Category Theory

######  Definition A.1 (Filtered category).

A non-empty small category $\mathscr{C}$ is termed _filtered_ if

  1. (i)

For every $x,y\in\mathscr{C}$, there exists $z\in\mathscr{C}$, and morphisms
$z\rightarrow x$ and $z\rightarrow y$.

  2. (ii)

For every $f,g:a\rightarrow b$ in $\mathscr{C}$, there exists $h:c\rightarrow
b$ such that $fh=gh$.

######  Definition A.2 ().

The category $\ind(\mathscr{C})$ of _ind-objects_ in $\mathscr{C}$ is a
category that has all diagrams $\F:\D\rightarrow\mathscr{C}$ where $\D$ is a
filtered category. Moreover, for two objects in ($\mathscr{C}$),
$\F:\D\rightarrow\mathscr{C}$ and $\G:\E\rightarrow\mathscr{C}$,

$\Hom_{\ind(\mathscr{C})}(\F,\G):=\lim_{d\in\D}\colim_{e\in\E}\Hom_{\mathscr{C}}(\F(d),\G(e))$

where (co)limits are taken in .

######  Definition A.3 ().

The category $\pro(\mathscr{C})$ of _pro-objects_ in $\mathscr{C}$ is a
category that has all diagrams $\F:\D\rightarrow\mathscr{C}$ where $\D$ is a
cofiltered category. Moreover, for two objects in ($\mathscr{C}$),
$\F:\D\rightarrow\mathscr{C}$ and $\G:\E\rightarrow\mathscr{C}$,

$\Hom_{\pro(\mathscr{C})}(\F,\G):=\lim_{e\in\E}\colim_{d\in\D}\Hom_{\mathscr{C}}(\F(d),\G(e))$

where (co)limits are taken in .

######  Lemma A.4 (Equivalence and ).

Let $\D$ be a full subcategory of category $\mathscr{C}$, such that

  1. (i)

$\mathscr{C}$ is cocomplete.

  2. (ii)

There exist functors

$\displaystyle\delta$ $\displaystyle:\mathscr{C}\rightarrow\ind(\D)$
$\displaystyle\colim$ $\displaystyle:\ind(\D)\rightarrow\mathscr{C}$

such that $\colim\delta=\id_{\mathscr{C}}$.

  3. (iii)

Every object in $\D$ is compact in $\mathscr{C}$.

Then, functors $\colim$ and $\delta$ induce an equivalence of categories.

######  Example A.5 (Relationship between $\Set$ and ).

Let $\mathscr{C}\in\Set$, and let $\D\in\fSet$ be the full subcategory of
finite sets. Every set is the colimit of its finite sets, so taking the
diagram of all finite subsets yields

$\delta:\Set\rightarrow\ind(\fSet)$

A.4 tells us that $\Set$ is equivalent to ().

[$F_{\bullet}$] The notation $F_{\bullet}\mathbf{A}$ is used to denote the
_filtrations_ of $\mathbf{A}$. Without additional qualifiers, a filtration of
$\mathbf{A}$ is a sequence of morphisms

$\ldots\rightarrow\mathbf{A}_{n}\rightarrow\ldots\rightarrow\mathbf{A}_{2}\rightarrow\mathbf{A}_{1}\rightarrow\mathbf{A}_{0}\rightarrow\mathbf{A}$

or

$\mathbf{A}\rightarrow\mathbf{A}_{0}\rightarrow\mathbf{A}_{1}\rightarrow\mathbf{A}_{2}\rightarrow\ldots\rightarrow\mathbf{A}_{n}\rightarrow\ldots$

######  Definition A.6 (Filtered quasi-isomorphism).

A _filtered quasi-isomorphism_ is a quasi-isomorphism that is compatible with
filtration. Concretely, given algebras $\mathbf{A}$ and $\mathbf{B}$, a
filtered quasi-isomorphism between them is a quasi-isomorphism such that
morphisms

$\mathbf{A}/F_{\bullet}\mathbf{A}\rightarrow\mathbf{B}/F_{\bullet}\mathbf{B}$

are also quasi-isomorphisms.

##  Appendix B Algebra

######  Definition B.1 (Artin ring).

A ring is said to be _Artin_ if it satisfies the descending chain condition on
ideals. The ring of dual numbers $\mathbb{K}[\epsilon]/\epsilon^{2}$ is a
classic example of an Artin ring.

######  Definition B.2 ($\mathbb{S}_{n}$).

The _symmetric group_ , $\mathbb{S}_{n}$, is defined as the group of
automorphisms on the set $\\{1,\ldots,n\\}$. Members of this group, written as
$[\sigma(1),\ldots,\sigma(n)]$, are sometimes referred to as _permutations_.

######  Definition B.3 (Shuffle).

A shuffle, denoted $\mathsf{Sh}(p,q)$, is a subgroup of $\mathbb{S}_{p+q}$
that satisfies

$\displaystyle\sigma(1)$ $\displaystyle<\ldots<\sigma(p)$
$\displaystyle\sigma(p+1)$ $\displaystyle<\ldots<\sigma(p+q)$

For example, the three shuffles in $\mathsf{Sh}(1,2)$ are $[1,2,3]$ (the
identity shuffle), $[2,1,3]$, and $[2,3,1]$.

######  Definition B.4 (Lie algebra).

An ordinary Lie algebra is a vector space $\mathfrak{g}$ equipped with a
bilinear skew-symmetric map
$[-,-]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$, which
satisfies the _Jacobi identity_.

$\forall x,y,z\in\mathfrak{g}:[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0$

######  Definition B.5 (Maurer-Cartan element of Lie algebra).

A _Maurer-Cartan element_ of Lie algebra $\mathfrak{g}$ is defined as a degree
$1$ element $x\in\mathfrak{g}$ satisfying the Maurer-Cartan equation.

$dx+\frac{1}{2}[x,x]=0$

The set of Maurer-Cartan elements of $\mathfrak{g}$ is represented as
$\mathsf{MC}(\mathfrak{g})$.

######  Definition B.6 (Canonical filtration of Lie algebra).

The _canonical filtration_ of Lie algebra $\mathfrak{g}$ is defined as

$F^{\mathsf{Lie}}_{n}\mathfrak{g}:=\\{x\in\mathfrak{g}\mid\text{$x$ can be
obtained as a bracketing of at least $n$ elements of $\mathfrak{g}$}\\}$

Notice that $F^{\mathsf{Lie}}_{n}\mathfrak{g}$ is both a sub-Lie algebra and a
Lie ideal of $\mathfrak{g}$.

[Complete with respect to a filtration] A Lie algebra $\mathfrak{g}$ is termed
_complete with respect to its canonical filtration_ if

$\mathfrak{g}\cong\lim_{n}\mathfrak{g}/F^{\mathsf{Lie}}_{n}\mathfrak{g}$

######  Definition B.7 (Complete Lie algebra).

A (complete) filtration of Lie algebra $\mathfrak{g}$ is the sequence of
subspaces

$\mathfrak{g}:=F_{0}\mathfrak{g}\supseteq F_{1}\mathfrak{g}\supseteq
F_{2}\mathfrak{g}\supseteq\ldots$

such that

  1. (a)

$\mathfrak{g}\cong\lim_{n}\mathfrak{g}/F_{n}\mathfrak{g}$.

  2. (b)

$[F_{i}\mathfrak{g},F_{j}\mathfrak{g}]=F_{i+j}\mathfrak{g}$.

  3. (c)

$d(F_{n}\mathfrak{g})\subseteq F_{n}\mathfrak{g}$.

A _complete Lie algebra_ is defined as the pair
$(\mathfrak{g},F_{\bullet}\mathfrak{g})$.

######  Definition B.8 ($\Omega_{\bullet}$).

The _Sullivan algebra_ 555Sullivan algebra has roots in rational homotopy
theory. The book [FelixRational] is an especially good resource.
$\Omega_{\bullet}$ denotes the simplicial commutative algebra of polynomial
differential forms on an $n$-simplex,

$\Omega_{n}:=\frac{\mathbb{K}[t_{0},\ldots,t_{n},dt_{0},\ldots,dt_{n}]}{\sum_{i=1}^{n}t_{i}=1,\sum_{i=1}^{n}dt_{i}=0}$

endowed with a square-zero differential.

##  Appendix C Monoids and Monads

######  Definition C.1 (Monoid).

A monoid $\mathcal{M}$ is an object $\mathcal{M}$ in monoidal category
$\mathscr{C}$, endowed with a composition operation $\circ$, which along with
two additional pieces of data,
$\gamma:\mathcal{M}\circ\mathcal{M}\rightarrow\mathcal{M}$ and
$\eta:\I\rightarrow\mathcal{M}$, makes the following diagrams commute.

${\mathcal{M}\circ(\mathcal{M}\circ\mathcal{M})}$${(\mathcal{M}\circ\mathcal{M})\circ\mathcal{M}}$${\mathcal{M}\circ\mathcal{M}}$${\mathcal{M}\circ\mathcal{M}}$${\mathcal{M}}$$\scriptstyle{\alpha(\mathcal{M},\mathcal{M},\mathcal{M})}$$\scriptstyle{\id\circ\gamma}$$\scriptstyle{\gamma\circ\id}$$\scriptstyle{\gamma}$$\scriptstyle{\gamma}$
${\I\circ\mathcal{M}}$${\mathcal{M}\circ\mathcal{M}}$${\mathcal{M}\circ\I}$${\mathcal{M}}$$\scriptstyle{\eta\circ\mathcal{M}}$$\scriptstyle{\lambda(\mathcal{M})}$$\scriptstyle{\gamma}$$\scriptstyle{\mathcal{M}\circ\eta}$$\scriptstyle{\rho(\mathcal{M})}$

where

$\displaystyle\alpha(\mathcal{M},\mathcal{M},\mathcal{M})$
$\displaystyle:\mathcal{M}\circ(\mathcal{M}\circ
M)\rightarrow(\mathcal{M}\circ M)\circ\mathcal{M}$
$\displaystyle\lambda(\mathcal{M})$
$\displaystyle:\I\circ\mathcal{M}\rightarrow\mathcal{M}$
$\displaystyle\rho(\mathcal{M})$
$\displaystyle:\mathcal{M}\circ\I\rightarrow\mathcal{M}$

are natural isomorphisms in the monoidal category 666In a _strict_ monoidal
category, this natural isomorphism is an equality 777In a _symmetric_ monoidal
category, $\lambda$ and $\rho$ encode the same natural isomorphism.

######  Definition C.2 (Monad).

A monad $\mathcal{M}$ is a monoid in the strict monoidal category of
endofunctors, $(\End{\mathscr{C}},\circ,\I_{\mathscr{C}})$.

######  Definition C.3 (Monad algebra).

An algebra over a monad $\mathcal{M}$, or a monad algebra, is an object
$A\in\mathscr{C}$ endowed with
$\gamma_{A}:\mathcal{M}(A)\rightarrow\mathcal{M}$ that makes the following
diagrams commute:

${\mathcal{M}\circ\mathcal{M}(A)}$${\mathcal{M}(A)}$${\mathcal{M}(A)}$${A}$$\scriptstyle{\gamma(A)}$$\scriptstyle{\mathcal{M}(\gamma_{A})}$$\scriptstyle{\gamma_{A}}$$\scriptstyle{\gamma_{A}}$
${\id_{\mathscr{C}}(A)}$${\mathcal{M}(A)}$${A}$$\scriptstyle{\gamma(A)}$$\scriptstyle{\gamma_{A}}$

##  Appendix D Trees

######  Definition D.1 (Rooted tree).

A _rooted tree_ is a tree with a choice of a univalent (external) vertex.

Rooted trees are denoted $\mathsf{R}\mathbb{T}$.

######  Definition D.2 (Planar tree).

A tree is termed _planar_ if the ordering of the leaves is significant. Since,
in our study of operads, the leaves are never labeled, this amounts to saying
that the tree cannot be reflected about the $y$-axis. For example, the
following two planar rooted trees are distinct.

[$\mathsf{P}\mathbb{T}_{n,k}$] A _planar rooted tree_ is denoted
$\mathsf{P}\mathbb{T}_{n}$, where $n$ is the number of leaves. One can further
classify them as $\mathsf{P}\mathbb{T}_{n,k}$, where $k$ is the number of
vertices. Obviously,
$\mathsf{P}\mathbb{T}_{n}=\bigcup_{k}\mathsf{P}\mathbb{T}_{n,k}$.

$\mathsf{P}\mathbb{T}_{3}:=\Biggl{\\{}$,

,

,

$\Biggr{\\}}$

[$k$-corolla] $\mathsf{P}\mathbb{T}_{n,k}$ is termed the $k$-_corolla_ of
$\mathsf{P}\mathbb{T}_{n}$.

[|v|] $|v|$ is used to denote the number of inputs to the vertex $v$.

[vert] Given tree $t$, $\textsf{vert}(t)$ is used to denote the set of
vertices of the tree.

##  Appendix E -categories

######  Definition E.1 (Artin algebra).

An $\mathbb{E}_{\8}$-algebra, $\mathbf{A}$ will be said to be _Artin_ if it is
connective, and $\pi_{*}\mathbf{A}$ is finite-dimensional.

######  Definition E.2 (Triangulated category).

A triangulated category consists of the following pieces of data:

  1. (1)

Additive category .

  2. (2)

A translation functor $\D\rightarrow\D$, which is an equivalence of
categories. We denote this functor as $X\mapsto X[1]$.

  3. (3)

A collection of distinguished triangles $X\rightarrow Y\rightarrow
Z\rightarrow X[1]$.

These data are required to satisfy the following axioms:

  1. (TR1)

(a) Every morphism $f:X\rightarrow Y$ in $\D$ can be extended to a
distinguished triangle in . (b) The collection of distinguished triangles is
stable under isomorphism. (c) Given an object $X\in\D$, the diagram
$X\rightarrow X\rightarrow 0\rightarrow X[1]$ is a distinguished triangle.

  2. (TR2)

A diagram

$X\rightarrow Y\rightarrow Z\rightarrow X[1]$

is a distinguished triangle if and only if the rotated diagram is a
distinguished triangle:

$Y\rightarrow Z\rightarrow X[1]\rightarrow Y[1]$

  3. (TR3)

Given a commutative diagram

${X}$${Y}$${Z}$${X[1]}$${X^{\prime}}$${Y^{\prime}}$${Z^{\prime}}$${X^{\prime}[1]}$

in which both horizontal rows are distinguished triangles, there exists a
dotted arrow rendering the entire diagram commutative.

  4. (TR4)

Suppose given three distinguished triangles:

${X}$${Y}$${Y/X}$${X[1]}$${Y}$${Z}$${Z/Y}$${Y[1]}$${X}$${Z}$${Z/X}$${X[1]}$

There exists a fourth distinguished triangle

${Y/X}$${Z/X}$${Z/Y}$${Y/X[1]}$

such that the diagram

${X}$${Z}$${Z/Y}$${Y/X[1]}$${Y}$${Z/X}$${Y[1]}$${Y/X}$${X[1]}$

The homotopy category of a stable -category $\mathscr{C}$, $h\mathscr{C}$
possesses a triangulated structure.

[Presentable -category] A presentable -category is an _accessible_ -category,
that admits all small colimits. An -category is termed accessible if it has a
good supply of filtered colimits and compact objects. What this concretely
means for us are the following properties.

  1. (a)

Accessible -categories are stable under homotopy fiber products.

  2. (b)

Let $\F:\mathscr{C}\rightarrow\op{\mathcal{S}}$ be a functor. If $\mathscr{C}$
is presentable, it implies that $\F$ preserves colimits.

######  Definition E.3 (-category of spaces).

Let $\Kan$ denote the full subcategory of $\SSet$ spanned by Kan complexes.
Then, the simplicial nerve of $\Kan$ is termed as the -category of spaces, and
denoted $\mathcal{S}$.

[$\mathcal{S}_{*}^{\mathsf{fin}}$] Let $\mathcal{S}_{*}$ denote the -category
of pointed spaces in $\mathcal{S}$. That is, $\mathcal{S}_{*}$ is the full
subcategory of $\Hom(\Delta^{1},\mathcal{S})$ spanned by those morphisms
$X\rightarrow Y$ for which $X$ is a final object of $\mathcal{S}$. Let
$\mathcal{S}^{\mathsf{fin}}$ denote the smallest subcategory of $\mathcal{S}$
which contains the final object and is stable under finite colimits. Then
$\mathcal{S}_{*}^{\mathsf{fin}}\subseteq\mathcal{S}_{*}$ is the -category of
pointed spaces in $\mathcal{S}^{\mathsf{fin}}$.

[Sifted colimits] In ordinary category theory, sifted colimits can “almost” be
characterized as combinations of filtered colimits and reflexive coequalizers.
In -categories, of particular consequence to us, is the property that colimits
of simplicial objects in an -category are sifted; this can otherwise be stated
as “simplicial -colimits are sifted”.

[Barr-Beck theorem] Given functor $\F:\mathscr{C}\rightarrow\D$, which admits
a left adjoint $\G$, the -categorical analogue of the Barr-Beck theorem tells
us that $\G\circ\F=\Hom(\mathscr{C},\mathscr{C})$ can be promoted to a monad
$T\in\mathsf{Alg}(\End{\mathscr{C}})$ on $\mathscr{C}$, and that $\G$ factors
through $\G^{\prime}:\D\rightarrow\mathsf{LMod}_{T}(\mathscr{C})$ of left
$T$-module objects of $\mathscr{C}$.

[Triangle] Let $\mathscr{C}$ be a pointed -category. A _triangle in
$\mathscr{C}$_ is a diagram in
$\Simplex{1}\times\Simplex{1}\rightarrow\mathscr{C}$ depicted as

${X}$${Y}$${0}$${Z}$

######  Definition E.4 (Stable -category).

An -category $\mathscr{C}$ is termed _stable_ if it satisfies the following
conditions.

  1. (a)

There exists a zero object $0\in\mathscr{C}$.

  2. (b)

Every morphism in $\mathscr{C}$ admits a kernel and cokernel.

  3. (c)

Every triangle in $\mathscr{C}$ is exact if and only if it is coexact.

######  Theorem E.5 (Dold-Kan correspondence).

The Dold-Kan correspondence establishes an equivalence between simplicial sets
and (homologically) non-negatively graded chain complexes of an Abelian
category. The -categorical analogue states that, if $\mathscr{C}$ is a stable
-category, then the -categories $\Hom(N(\Z_{\geq 0}),\mathscr{C})$ and
$\Hom(N(\op{\Delta}),\mathscr{C})$ are equivalent. Intuitively, it means that,
to every simplicial object $X_{\bullet}$ of stable -category $\mathscr{C}$, we
associate the filtered object

$D(0)\rightarrow D(1)\rightarrow D(2)\rightarrow\ldots$

where $D(k)$ is the $k$-skeleton of $X_{\bullet}$. In particular, $\colim
D(j)$ can be identified with geometric realizations of $X_{\bullet}$.

[Background from classical stable homotopy theory] The study of _stable
homotopy theory_ [Spectra] arose from multiple pain points, but we will place
emphasis on one, in particular: the parallel between homology and homotopy is
incomplete in the following sense. In homology, the suspension functor induces
the following isomorphism.

$\displaystyle\tilde{H}_{n+k}\Sigma^{k}X\cong\tilde{H}_{n}X$

Now, the Freudenthal suspension theorem says that

$\displaystyle\pi_{n+k}\Sigma^{k}X\cong\pi_{n+k+1}\Sigma^{k+1}X$

given a large enough, finite, $n$. We would achieve the parallel that we
wanted if we could smash everything with $S^{-k}$, a “negative sphere”. For
this to work, we need to define $\Sigma^{\8}$. Let us start with the following
observation. Let $A$ and $B$ be finite CW-complexes with basepoints.

$\displaystyle[A,B]\xrightarrow{\Sigma}[\Sigma A,\Sigma
B]\xrightarrow{\Sigma}[\Sigma^{2}A,\Sigma^{2}B]\xrightarrow{\Sigma}\ldots$

Except for the first two terms, this is a sequence of homomorphisms between
Abelian groups. By Freudenthal’s suspension theorem, after a finite number of
terms, the sequence turns into a sequence of isomorphisms. We define

$\displaystyle[\Sigma^{\8}A,\Sigma^{\8}B]:=\lim_{N}[\Sigma^{N}A,\Sigma^{N}B]$

to capture this convenient term, and define the category of spectra 888To be
precise, the Spanier-Whitehead category of finite spectra to have objects
$\Sigma^{\8+n}A$ and morphisms

$\displaystyle[\Sigma^{\8+n}A,\Sigma^{\8+m}B]:=\lim_{N}[\Sigma^{N+n}A,\Sigma^{N+m}B]$

Here, $m$ and $n$ are integers. In this category, $\Sigma$ induces a self-
equivalence.

The category of spectra has a _triangulated structure_.

######  Definition E.6 ($\mathsf{Sp}$).

For any -category that admits finite limits, consider -category
$\mathsf{Sp}(\mathscr{C})$ of _spectrum objects_ of $\mathscr{C}$. In the
special case where $\mathscr{C}$ is the -category of spaces, we recover
classical stable homotopy theory. If -category $\mathscr{C}$ is pointed, the
spectrum objects of $\mathscr{C}$ can be defined as the homotopy inverse limit
of the tower of -categories:

${\ldots}$${\mathscr{C}}$${\mathscr{C}}$${\mathscr{C}}$$\scriptstyle{\Omega_{\mathscr{C}}}$$\scriptstyle{\Omega_{\mathscr{C}}}$

The -category $\mathsf{Sp}$ is stable and presentable, whose homotopy category
recovers the classical stable homotopy category, and in particular possesses a
triangulated structure (see [HA], Chapter 1). $\mathsf{Sp}$ can also be
thought of as an -category whose objects are cohomology theories. An
alternative viewpoint is to think of $\mathsf{Sp}$ as the -category of loop
spaces. $\mathsf{Sp}$ bears the same relationship to the -category of spaces
as the $\Ab$ bears to $\Set$.

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