††thanks: Supported by RVO: 67985840 and Praemium Academiæof M. Markl. Brno
Ph.D. Talent Scholarship Holder - Funded by the Brno City Municipality.

18M60<EMAIL_ADDRESS>
# Category-colored operads, internal operads, and Markl $\bO$-operads

Dominik Trnka  Department of Mathematics and Statistics, Masaryk University,
Kotlářská 2, 61137 Brno, and Institute of Mathematics, Czech Academy of
Sciences, Žitná 25, 115 67 Prague 1, Czech Republic

(2023)

###### Abstract

We present a Markl-style definition of operads colored by a small category. In
the presence of a unit these are equivalent to substitudes of Day and Street.
We show that operads colored by a category are internal algebras of a certain
categorical operad of functors. We describe a groupoid-colored quadratic
binary operad, whose algebras are non-unital Markl operads in the context of
operadic categories. As a by-product we describe the free internal operad
construction.

###### keywords:

Colored operad, Internal operad, Operadic category, Markl operad, Hyperoperad

## Introduction

In [vdLaan03] van der Laan proved that non-symmetric Markl operads (i.e.
operads with binary ‘partial compositions’ $\circ_{i}$, cf. [Markl96,
Definition 1.1.]), are algebras for a colored Koszul quadratic operad with
colors the natural numbers. A similar result for symmetric operads was
obtained by Dehling and Vallette in the fascinating paper [DV21], where
symmetric operads were presented as algebras for a linear-quadratic curved
Koszul colored operad. The curved Koszul theory was necessary since the
authors in loc. cit. wanted to resolve, along with the binary structure
operations, also the symmetric group actions.

Our aim is to modify van der Laan’s approach to symmetric operads and, more
generally, to Markl $\bO$-operads over an operadic category $\bO$ [BM23a,
Definition 6.1], considering the group (resp. groupoid) actions as fixed parts
of the structure. The tool of choice should be operads with colors in a
groupoid that are generated, as in the van der Laan case, only by binary
operations satisfying quadratic relations.

Markl $\bO$-operads play an important role in homotopy theory of operad-like
structures. In [BM23a], various kinds of operads are viewed as algebras over
the terminal Markl $\bO$-operad. This includes structures such as the
classical operads, cyclic or modular operads, wheeled properads, dioperads,
$\frac{1}{2}$PROPs, permutads, or pre-permutads. The terminal operads for the
above operad-like structures are binary quadratic. In [BMO23], some of the
terminal operads are resolved, providing strongly homotopy versions of the
corresponding operad-like structures.

An inspiration for us was a result of Ward [Ward22], who showed that modular
operads are algebras of a groupoid-colored quadratic Koszul operad. It turned
out that, in contrast to [Ward22], to present general Markl $\bO$-operads as
algebras of a quadratic operad, we need to work with coloring groupoids, which
have morphisms between different objects. For a clear exposition, we decided
to study operads colored by small categories in general. We introduce new
operations $\otimes_{i}$ on certain functor categories, which allows us to
define category-colored operads explicitly via generators and relations.

### What is a category-colored operad?

For a set $C$, a $C$-colored operad $P$ consists of objects

${P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$

of abstract $n$-ary operations, whose inputs and output have specific types

$c_{1},\ldots,c_{n},c\in C.$

The operad structure tells us how to compose two operations, provided the
output of the second operation matches the type of an input of the first
operation. It also tells us how to permute the inputs of an operation, and
that the composition is associative and compatible with permutations.

When $C$ is a small category, its morphisms act on inputs and output of the
operations, possibly changing their type. Thus, two operations can be composed
even if the input and output does not match, but is connected by a morphism.
Such generalisations of colored operads appeared already in [DS03, Petersen13,
Ward22]. The structure of a category-colored operad is captured geometrically
by labeled non-planar trees, cf. Figure 1. The edges are labeled by morphisms
of the coloring category and vertices are labeled by operations of the operad.
Such a tree determines a composition of operations and $C$-actions of the
operad. Operads colored by categories or groupoids found applications in
deformation theory [DSVV20], or in homotopy theory [BW22, Stoeckl23].

One of the features of category-colored operads is the approach to unary
operations of algebras. Consider the following problem: we are given an
algebraic structure with unary and binary operations. We would like to
describe a homotopy coherent version of this structure, but where only the
binary operations are relaxed, while the unary operations remain strict. The
solution is to encode the structure as an algebra of a category-colored operad
and hide the unary operations into the coloring category. The problem is
solved by resolving the operad in the category of colored operads. An example
of such algebraic structure is a traditional operad. The unary operations are
$\Sigma$-actions and the binary operations are the operad compositions
$\circ_{i}$, cf. Section 7. Another example is a differential graded
associative algebra $A$, cf. Example 6.9. We regard the differential as a
unary operation $\partial\colon A(n)\to A(n-1)$. The coloring category $D$ has
objects the integers. The Hom-spaces $D(n,n-1)$, for any integer $n$, are
generated by an element $\partial_{n}$, while the other Hom-spaces are
trivial, which forces $\partial\partial=0$.

Figure 1: Composition scheme for category-colored operads.

### Approaches to category-colored operads:

Classically, there are four equivalent approaches to unital operads:

  1. (a)

May’s definition is based on composition maps $\gamma$, which combine $n$
operations with an $n$-ary operation,

  2. (b)

Markl’s definition is based on partial compositions $\circ_{i}$, which plug
one operation into the $i$-th input of another,

  3. (c)

operads are algebras of the monad of trees, and

  4. (d)

operads are monoids in the monoidal category of $\Sigma$-modules.

Recent approaches to operads and operad-like structures involve Feynman
categories of Kaufmann and Ward [KW17], polynomial monads [BB17], or operadic
categories of Batanin and Markl [BM15]. For example, classical operads are
operads over the operadic category $\it{Fin}$, the skeletal category of finite
sets.

In the category-colored case, (a)-type definition was first given by Day and
Street in [DS03], under the name symmetric substitude, (c)-type definition was
worked out by Ward in [Ward22] for groupoids with no morphisms between
different objects, and (d)-type was given by Petersen in [Petersen13]. We
complete the list by giving below Definition 2.2 of type (b). We will also
address the situation when the coloring category is $\mathscr{V}$-enriched,
cf. Definition 4.3. Describing category-colored operads as operads over a
suitable operadic category is the subject of our current research. Category-
colored operads can also be defined as monads in the bicategory of generalised
species [FGHW08]. The formalism of operads colored by dg-categories was
recently developed in [CCN22].

The Definition 2.2 uses operations $\otimes_{i}$ on functor categories

${C}_{\mathscr{V}}(n)=\it{Cat}\big{(}(C^{\it{op}})^{n}\times
C,\mathscr{V}\big{)},$

where $C$ is a small category and $\mathscr{V}$ is a cocomplete symmetric
monoidal closed category. For functors

$(C^{\it{op}})^{\times n}\times C\xrightarrow{P_{n}}\mathscr{V},\hskip
28.45274pt(C^{\it{op}})^{\times m}\times C\xrightarrow{P_{m}}\mathscr{V},$

and $1\leq i\leq n$, the functor
$P_{n}\otimes_{i}P_{m}\in{C}_{\mathscr{V}}(n+m-1)$ is a colimit coequalizing
$C$-actions on the $i$-th input of $P_{n}$ and output of $P_{m}$. The
$C$-operad is then a collection of functors $P_{n}$ with natural
transformations

${P_{n}\otimes_{i}P_{m}}$${P_{m+n-1},}$${\circ_{i}}$

satisfying classically-looking associativity and equivariance axioms. Let us
describe two main results of this article.

### Result I

We give a conceptual explanation of the operations $\otimes_{i}$. In Section 4
(Proposition 4.2) we prove that the operations $\otimes_{i}$ equip the
collection of categories $C_{\mathscr{V}}(n)$ with an operad structure.
However, the associativity of $\otimes_{i}$ holds only up to canonical
isomorphisms. More precisely, $C_{\mathscr{V}}$ is an example of a (non-
strict) categorical operad, that is, an operad with values in the monoidal
category of categories, where the operad axioms hold only up to coherent
natural isomorphisms. Categorical operads will be defined in Section 3 as
pseudo-algebras of a 2-monad of trees.

For a 2-monad $T$, an internal algebra of a pseudo-$T$-algebra $\mathscr{O}$
is a lax morphism from the terminal pseudo-$T$-algebra to $\mathscr{O}$. A
nice example of internal algebras are monoids in monoidal categories: a
monoidal category $M$ is a pseudo-$\mathbb{M}$-algebra of the 2-monad
$\mathbb{M}$ on the 2-category of categories, which is induced from the ‘free
monoid’ monad in sets. A monoid $m$ in $M$ is then an internal algebra in the
pseudo-$\mathbb{M}$-algebra $M$. Note that monoidal categories provide just
enough coherent data for a definition of a monoid. From this point of view,
categorical operads provide the right coherent data for a definition of
operads. Since monoidal categories are categorical operads concentrated in
arity 1, our philosophy is sound with the classical definition of operads in
monoidal categories. Our first result is:

Theorem A (1). $C$-operads are internal operads in the categorical operad
$(C_{\mathscr{V}}(n),\otimes_{i})$.

Further, we describe free internal operads and, using Theorem A, we give an
explicit free $C$-operad construction, which will be used in Result II.
However, the free construction can be read off the monoidal or monadic
definitions of [Petersen13, Ward22] and hence the Result II can be understood
independently of Result I.

### Result II

Our second result is based on ideas of [BMO23], that express operads as
algebras over a groupoid-colored operad (we will use the term hyperoperad),
and also on the above mentioned result of Ward [Ward22]. The coloring groupoid
incorporates the symmetric action and equivariance axiom of operads to the
level of collections. Hence, we do not regard the symmetric actions as a piece
of operadic structure, which simplifies the process of resolving the
hyperoperad significantly. This should be put in contrast with [DV21], which
treats the symmetric actions as unary operations, hence resolving them as
well. For more background and history on resolving hyperoperads, we refer the
reader to the introduction of [BMO23].

The groupoid of colors for the hyperoperad $\mathbb{H}$, whose algebras are
classical non-unital symmetric Markl operads, will be denoted by $\Sigma$. It
has natural numbers as objects and permutations as morphisms. The idea is to
generate the hyperoperad $\mathbb{H}$ by abstract symbols

$*_{i}\in{\mathbb{H}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}n+m-1\\\
n\hskip
14.22636ptm\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$

which represent the operad structure maps

${O(n)\otimes O(m)}$${O({m+n-1}).}$${\circ_{i}}$ (1)

We equip the generating collection with free actions of $\Sigma$ on the inputs
and the output, obtaining formal elements
${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi\\\
{}_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$,
which represent the composites

${O(n)\otimes O(m)}$${O(n+m-1)}$${O(n)\otimes
O(m)}$${O(n+m-1)}$${(-)\sigma\otimes(-)\tau}$${\circ_{i}}$${(-)\pi}$

of symmetric actions and compositions $\circ_{i}$ of a classical operad $O$.
On the level of collections we make the symbols $*_{i}$ equivariant by
identifying

${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt1\\\
{}_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\sim{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
28.0pt\sigma\circ_{i}\tau\\\ {}_{\sigma(i)},1\hskip
10.0pt1\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

It remains to generate the free operad from the constructed collection and
divide by the operadic ideal generated by associators, which ensures
associativity of the symbols $*_{i}$. The result is:

Theorem B1 (1). (Symmetric non-unital) Markl operads are algebras of the
binary quadratic operad $\mathbb{H}$ with colors $\Sigma$.

We regard units for operads as an extra structure. To encode units
$\mathbb{k}\xrightarrow{u}O(1)$ of a unital operad $O$, one uses constants
($0$-ary operations)

$u\in{\mathbb{H}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}1\\\
\emptyset\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

Notice however, that the unit axioms $u\circ_{1}\alpha=\alpha$ and
$\alpha\circ_{i}u=\alpha$ are not quadratic. We show, that Theorem B1 extends
to the framework of operads in operadic categories and $C$-colored operads by
proving:

Theorem B2 (1). (Non-unital) Markl $\bO$-operads in an operadic category $\bO$
are algebras of the binary quadratic operad $\mathbb{H}_{\bO}$ with colors
$\bO_{\it{iso}}^{\it{op}}$, the category of isomorphisms of $\bO$.

Theorem B3 (1). For a fixed small category $C$, (non-unital) $C$-operads are
algebras of the binary quadratic operad $\mathbb{H}_{C}$ with colors
$Bq^{\Sigma}(C)$.

In Section 6 we observe that algebras of operads with values in a functor
category $\mathscr{V}^{C}$ can be expressed as algebras of $C$-operads in
$\mathscr{V}$. In our future work we shall address Koszulness of the
constructed hyperoperads $\mathbb{H}_{\bO}$ and their resolutions.

### Plan of the paper:

The first two sections lead to a Markl-style definition of $C$-operads and
comparison with symmetric substitudes. The rest of the paper is divided into
two independent parts. The first part consists of Sections 3-5, which are
devoted to Result I. The second part consists of Sections 6 and 7, where we
construct the hyperoperads of Result II. We believe that the reader can
understand the Result II without the details of the free construction from
Section 5, or look at Example 5.7, if necessary.

In more detail, Section 3 recalls the theory of polynomial monads and internal
algebras. We define internal operads in a categorical operad and describe them
explicitly. In Section 4 we prove Theorem A. Section 5 gives the free internal
operad construction. Using the Theorem A, we describe the free $C$-operad.
Section 6 defines algebras of $C$-operads and adapts some standard notions to
the category-colored setting. In the second part of Section 6 we compare
algebras of classical operads with values in functor categories
$\mathscr{V}^{C}$ with algebras of $C$-colored operads in $\mathscr{V}$. An
application of this comparison is a description of differential graded
associative algebras as algebras over a category-colored operad with values in
$\it{Vect_{\mathbb{k}}}$. The last section is devoted to proving Theorems
B1-B3.

### Acknowledgement

I would like to express gratitude to my supervisor Martin Markl for his
initial idea and constant support, and to my colleagues Miloslav Štěpán and
Michael Batanin for their helpful comments. I also thank to an anonymous
referee for valuable comments and suggestions.

### Conventions

The identity morphism of an object $x$ will be denoted by $\mathbb{1}_{x}$ or
only by $\mathbb{1}$, leaving the object implicit. The set of natural numbers
(including 0) will be denoted by $\mathbb{N}$. If not stated otherwise, $C$
denotes a small category and $\mathscr{V}$ a cocomplete symmetric monoidal
closed category. However, some constructions and results may work with weaker
assumptions on $\mathscr{V}$.

##  1 The $\otimes_{i}$ products

We establish some notation related to functors

${\underbrace{C^{\it{op}}\times\cdots\times
C^{\it{op}}}_{n\textit{-times}}\times C}$${\mathscr{V},n\geq 0}$${X_{n}}$

and their natural transformations, where $C$ is a small category and
$\mathscr{V}$ a cocomplete symmetric monoidal closed category. We denote by
$\it{Cat}$ the 2-category of categories.

######  Definition 1.1.

We denote the functor category $\it{Cat}\big{(}(C^{\it{op}})^{n}\times
C,\mathscr{V}\big{)}$ by ${C}_{\mathscr{V}}(n)$. A collection of functors
$\\{X_{n}\in{C}_{\mathscr{V}}(n)\\}_{n\in\mathbb{N}}$ will be called a non-
symmetric $C$-collection in $\mathscr{V}$.

The objects of $(C^{\it{op}})^{n}\times C$ are denoted as schemes
${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
and the maps are

${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
f_{1}\,\cdots\,f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
d_{1}\,\cdots\,d_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).}$

Fixing an index $i\in\\{1,\ldots,n\\}$ and an object $c\in C$, we partially
evaluate a functor $X\in{C}_{\mathscr{V}}(n)$ in the $i$-th input to obtain a
functor

${X^{i}_{c}={X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\bullet\\\
\bullet\,\cdots\,c\,\cdots\,\bullet\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon\underbrace{C^{\it{op}}\times\cdots\times
C^{\it{op}}}_{(n-1)\textit{-times}}\times C}$${\mathscr{V}.}$

Any $f\colon d\rightarrow c$ in $C$ then induces a natural transformation

${X^{i}_{c}}$${X^{i}_{d}.}$${X^{i}_{f}}$

We thus have a functor $X^{i}_{\bullet}$ from $C^{\it{op}}$ into the functor
category

${C^{\it{op}}}$${\it{Cat}(\underbrace{C^{\it{op}}\times\cdots\times
C^{\it{op}}}_{(n-1)\textit{-times}}\times C,\mathscr{V}).}$${X^{i}_{\bullet}}$

Similarly, the partial evaluation
$\prescript{c}{}{X}:={X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
\bullet\,\cdots\,\bullet\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
gives a functor

${C}$${\it{Cat}(\underbrace{C^{\it{op}}\times\cdots\times
C^{\it{op}}}_{n\textit{-times}},\mathscr{V}).}$${\prescript{\bullet}{}{X}}$

For two functors $X\in C_{\mathscr{V}}(n),Y\in C_{\mathscr{V}}(m),$ define a
functor

$X^{i}_{c}\otimes\prescript{d}{}{Y}\in{C}_{\mathscr{V}}(n+m-1)$

by the formula

${{(X^{i}_{c}\otimes\prescript{d}{}{Y})}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,y_{1}\,\cdots\,y_{m}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right):={X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{i-1}\hskip 5.0ptc\hskip
5.0ptx_{i+1}\,\cdots\,x_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{Y}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
y_{1}\,\cdots\,y_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

This induces a functor

${C^{\it{op}}\times
C}$${C_{\mathscr{V}}(n+m-1).}$${X^{i}_{\bullet}\otimes\prescript{\bullet}{}{Y}}$
(2)

######  Lemma 1.2.

Let $X\in C_{\mathscr{V}}(n)$, $Y\in C_{\mathscr{V}}(m)$, $Z\in
C_{\mathscr{V}}(k)$, and $1\leq i<j\leq n$. There are canonical natural
isomorphisms

$(X^{j}_{\bullet}\otimes\prescript{\bullet}{}{Y})^{i}_{\bullet}\otimes\prescript{\bullet}{}{Z}\cong\begin{cases*}(X^{i}_{\bullet}\otimes\prescript{\bullet}{}{Z})^{j+k-1}_{\bullet}\otimes\prescript{\bullet}{}{Y}&if
$1\leq i<j\leq n$,\\\
X^{j}_{\bullet}\otimes\prescript{\bullet}{}{(Y^{i-j+1}_{\bullet}\otimes\prescript{\bullet}{}{Z})}&if
$j\leq i<j+m$,\\\
(X^{j}_{\bullet}\otimes\prescript{\bullet}{}{Z})^{i-m+1}_{\bullet}\otimes\prescript{\bullet}{}{Y}&if
$j+m\leq i\leq n+m-1$.\end{cases*}$ (3)

It is enough to evaluate both sides and compare the values in $\mathscr{V}$.
The isomorphisms (3) are induced by the symmetry and associativity
isomorphisms of $\mathscr{V}$. Let us recall the definition of cowedges and
coends:

######  Definition 1.3.

Let $F\colon C^{\it{op}}\times C\rightarrow D$ be a functor and
$\gamma_{c}\colon F(c,c)\rightarrow t$ a family of maps in $D$ with a common
target $t\in D$, indexed by objects $c\in C$. The family $\gamma$ is a cowedge
from the functor $F$, if the diagram

${F(d,c)}$${F(d,d)}$${F(c,c)}$${t}$${F(\mathbb{1},f)}$${F(f,\mathbb{1})}$${\gamma_{c}}$${\gamma_{d}}$

commutes for any $f\colon c\rightarrow d$ in $C$.

######  Definition 1.4.

The coend of a functor $F\colon C^{\it{op}}\times C\rightarrow D$ is the
initial cowedge from $F$:

${F(c,c)}$${\displaystyle\int^{c\in C}F(c,c).}$${\iota_{c}}$

If $D$ has colimits, the coend is the colimit of the diagram

${{\displaystyle\bigoplus_{f\colon c^{\prime}\rightarrow
c^{\prime\prime}}F(c^{\prime\prime},c^{\prime})}}$${{\displaystyle\bigoplus\limits_{c\in
C}F(c,c).}}$${\displaystyle\bigoplus_{f\colon c^{\prime}\rightarrow
c^{\prime\prime}}F(f,\mathbb{1})}$${\displaystyle\bigoplus_{f\colon
c^{\prime}\rightarrow c^{\prime\prime}}F(\mathbb{1},f)}$ (4)

The symbol $\bigoplus$ stands for the categorical coproduct.

######  Definition 1.5.

Let $X\in C_{\mathscr{V}}(n),Y\in C_{\mathscr{V}}(m),$ and
$i\in\\{1,\ldots,n\\}$. We define the functor

$X\otimes_{i}Y\in C_{\mathscr{V}}(n+m-1)$

as the coend

$X\otimes_{i}Y:=\displaystyle\int^{c\in C}X^{i}_{c}\otimes\prescript{c}{}{Y}.$

Any two natural transformations $\alpha\colon X^{\prime}\to X^{\prime\prime}$
and $\beta\colon Y^{\prime}\to Y^{\prime\prime}$ induce a natural
transformation $\alpha\otimes_{i}\beta\colon
X^{\prime}\otimes_{i}Y^{\prime}\to
X^{\prime\prime}\otimes_{i}Y^{\prime\prime}$. In fact, $-\otimes_{i}-$ is a
functor

${C_{\mathscr{V}}(n)\times
C_{\mathscr{V}}(m)}$${C_{\mathscr{V}}(n+m-1).}$${\otimes_{i}}$

The operations $\otimes_{i}$ are partial composition analogs of the plethysm
operation, which is the monoidal product of the monoidal category of
$C\mathbb{S}$-modules in $\mathscr{V}$ of [Petersen13]. By definition, a
natural transformation

${X\otimes_{i}Y}$${Z}$${f}$

is a collection of natural transformations

${X^{i}_{c}\otimes\prescript{c}{}{Y}}$${Z,}$${f_{c}}$

for all $c\in C$, which form a cowedge.

######  Lemma 1.6.

The operations $\otimes_{i}$ are associative, i.e. for $X$, $Y$, $Z$, $i$, and
$j$ as in Lemma 1.2, there are canonical natural isomorphisms

$(X\otimes_{j}Y)\otimes_{i}Z\cong\begin{cases*}(X\otimes_{i}Z)\otimes_{j+k-1}Y&if
$1\leq i<j\leq n$,\\\ X\otimes_{j}(Y\otimes_{i-j+1}Z)&if $j\leq i<j+m$,\\\
(X\otimes_{j}Z)\otimes_{i-m+1}Y&if $j+m\leq i\leq n+m-1$.\end{cases*}$ (5)

The proof follows from Lemma 1.2 and formal manipulations with coends.

######  Example 1.7.

For $\mathscr{V}=\it{Vect_{\mathbb{k}}}$, the category of vector spaces over a
field ${\mathbb{k}}$, the $\otimes_{i}$ operations can be described as
follows.

Let $X\in C_{\mathscr{V}}(n)$, $Y\in C_{\mathscr{V}}(m)$. Pick two vectors

$\alpha\in{X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\ x_{1}\,\cdots\,c\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$ | $\beta\in{Y}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\ y_{1}\,\cdots\,y_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$
---|---

and a map $f\colon d\rightarrow c$ of $C$. The map $f$ acts on the $i$-th
input of $X$ by

${X{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}_{x}\\\
\mathbb{1}_{x_{1}}\,\cdots\,f\,\cdots\,\mathbb{1}_{x_{n}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon
X{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,c\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${X{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,d\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).}$

Similarly, $f$ acts on the output of $Y$ by

${Y{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
\mathbb{1}_{y_{1}}\,\cdots\,\mathbb{1}_{y_{m}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon{Y}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
y_{1}\,\cdots\,y_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{Y}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
y_{1}\,\cdots\,y_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).}$

Denote by $\alpha\cdot f$ and $f\cdot\beta$ the values

$X{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\
\mathbb{1}\,\cdots\,f\,\cdots\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\alpha)\in{X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,d\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\text{
and
}Y{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
\mathbb{1}\,\cdots\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\beta)\in{Y}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
y_{1}\,\cdots\,y_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

Then each space

${X\otimes_{i}Y}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,y_{1}\,\cdots\,y_{m}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$

is the direct sum of spaces

$\displaystyle\bigoplus_{c\in
C}{X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,c\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{Y}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
y_{1}\,\cdots\,y_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$

divided by the ideal generated by elements $(\alpha\cdot
f\otimes\beta-\alpha\otimes f\cdot\beta)$ for all $\alpha$, $\beta$ and $f$.

##  2 Category-colored operads

Before we define operads colored by a small category $C$, we introduce their
underlying (symmetric) $C$-collections. Let $\mathbb{S}C$ be the free
symmetric monoidal category on $C$. The objects of $\mathbb{S}C$ are ordered
lists $\underline{c}=(c_{1}\,\cdots\,c_{n})$ of objects of $C$ and maps
$\underline{c}\rightarrow\underline{d}$ are tuples
$(\sigma\,f_{1}\,\cdots\,f_{n})$, where $\sigma\in\Sigma_{n}$ and each
$f_{i}\colon c_{i}\longrightarrow d_{\sigma(i)}$ is a map in $C$, cf. Figure
2.

Figure 2: A map of $\mathbb{S}C$.

######  Definition 2.1.

A $C$-collection in $\mathscr{V}$ is a functor

${(\mathbb{S}C)^{\it{op}}\times C}$${\mathscr{V}.}$${P}$ (6)

Morphisms of $C$-collections are their natural transformations.

The value
${P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
of a $C$-collection $P$ is interpreted as a collection of abstract $n$-ary
operations whose inputs are of types $c_{1},\ldots,c_{n}$ and the output is of
type $c$. Such an operation is depicted in Figure 3.

Figure 3: Abstract $n$-ary operation.

The morphisms of $(\mathbb{S}C)^{\it{op}}\times C$ are

${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0ptf\\\
\sigma\,f_{1}\,\cdots\,f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
d_{1}\,\cdots\,d_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$

and the value
${P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0ptf\\\
\sigma\,f_{1}\,\cdots\,f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
is interpreted as a permutation of inputs, together with an action of maps of
$C$ on inputs and output of operations. The actions thus change the type of an
operation, cf. Figure 4.

Figure 4: Acting on an operation.

Any morphism $(\sigma\,f_{1}\,\cdots\,f_{n})$ of $\mathbb{S}C$ can be
decomposed as

${{(d_{\sigma(1)}\,\cdots\,d_{\sigma(n)})}}$${{(c_{1}\,\cdots\,c_{n})}}$${{(d_{1}\,\cdots\,d_{n})}.}$${{(c_{\sigma^{-1}(1)}\,\cdots\,c_{\sigma^{-1}(n)})}}$${(\sigma\,f_{1}\,\cdots\,f_{n})}$${(\sigma\,\mathbb{1}\,\cdots\,\mathbb{1})}$${(\mathbb{1}\,f_{1}\,\cdots\,f_{n})}$${(\sigma\,\mathbb{1}\,\cdots\,\mathbb{1})}$${(\mathbb{1}\,f_{\sigma^{-1}(1)}\,\cdots\,f_{\sigma^{-1}(n)})}$

A $C$-collection $P$ is therefore a non-symmetric $C$-collection of functors
$P_{n}\in C_{\mathscr{V}}(n)$, for every $n\geq 0$, equipped with natural
transformations

${{P_{n}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{P_{n}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{\sigma(1)}\,\cdots\,c_{\sigma(n)}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),}$${(-)\sigma}$

for every permutation $\sigma\in\Sigma_{n}$, such that
$((-)\tau)\sigma=(-)\tau\sigma$ and
$(-)\mathbb{1}_{n}=\mathbb{1}_{C_{\mathscr{V}}(n)}$. We will sometimes omit
the subscripts of the functors $P_{n}$, when they are clear from the context.
For any $\sigma\in\Sigma_{n}$, $\tau\in\Sigma_{m}$ and $c\in C$, the actions
$(-)\sigma\text{ and }(-)\tau$ on functors $X\in C_{\mathscr{V}}(n)$, $Y\in
C_{\mathscr{V}}(m)$, induce a natural transformation

${X^{\sigma(i)}_{c}\otimes\prescript{c}{}{Y}}$${X^{i}_{c}\otimes\prescript{c}{}{Y},}$${(-)\sigma\otimes(-)\tau}$

which extends to colimits

${X\otimes_{\sigma(i)}Y}$${X\otimes_{i}Y.}$${(-)\sigma\otimes(-)\tau}$

In the following definition, the symbol ‘$\cdot$’ denotes ordinary composition
of morphisms.

######  Definition 2.2.

A (non-unital) $C$-operad in $\mathscr{V}$ is a $C$-collection $P$ in
$\mathscr{V}$, together with natural transformations

${P_{n}\otimes_{i}P_{m}}$${P_{m+n-1},}$${\circ_{i}}$ (7)

for any $n\geq 1,m\geq 0,i\in\\{1,\ldots,n\\}$, called the composition maps,
which are required to satisfy the following two axioms.

  * •

Equivariance: Let $\sigma\circ_{i}\tau$ be the permutation given by inserting
the permutation $\tau$ at the $i$-th place in $\sigma$ (see [MSS02,
II.1.3(1.8)]). For every $\sigma\in\Sigma_{n}$ and $\tau\in\Sigma_{m}$,

$\circ_{i}\cdot((-)\sigma\otimes(-)\tau)=(-)(\sigma\circ_{i}\tau)\cdot\circ_{\sigma(i)}.$
(8)

  * •

Associativity: For any $1\leq j\leq n,0\leq m$ and $0\leq k$,

$\circ_{i}\cdot(\circ_{j}\otimes_{i}\mathbb{1})=\begin{cases*}\circ_{j+k-1}\cdot(\circ_{i}\otimes_{j+k-1}\mathbb{1})&if
$1\leq i<j\leq n$,\\\ \circ_{j}\cdot(\mathbb{1}\otimes_{j}\circ_{i-j+1})&if
$j\leq i<j+m$,\\\ \circ_{j}\cdot(\circ_{i-m+1}\otimes_{j}\mathbb{1})&if
$j+m\leq i\leq n+m-1$.\end{cases*}$ (9)

In the associativity axiom we ignored the canonical isomorphisms of sources of
morphisms (9), given by Lemma 1.6. The first and the third cases of (9) are
called the parallel associativity axioms and the second case is called the
sequential associativity axiom.

######  Example 2.3.

When $C$ is discrete, the operations

$X\otimes_{i}Y=\displaystyle\bigoplus\limits_{c\in C}X^{i}_{c}\otimes Y^{c}$

reduce to plain coproducts. The maps

${{P_{n}\otimes_{i}P_{m}}}$${P_{n+m-1}}$${\circ_{i}}$

are then composition maps of classical operads colored by the set $C$.

######  Definition 2.4.

A unit for a $C$-operad $P$ in $\mathscr{V}$ is a collection of maps

${I}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}b\\\
a\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),}$${u_{f}}$ (10)

from the monoidal unit $I$ of $\mathscr{V}$, for any $f\colon a\rightarrow b$
of $C$, such that

${I}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
b\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
a\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${u_{g}}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}h\\\
f\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${u_{hgf}}$
(11)

commutes for all maps $f,g$ and $h$, for which the diagram makes sense, and
the unit axioms are satisfied:

$\circ_{1}\cdot(u_{f}\otimes\mathbb{1})={P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
\mathbb{1}\,\cdots\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\text{
and }\circ_{i}\cdot\,(1\otimes
u_{f})={P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\
\mathbb{1}\,\cdots\,f\,\cdots\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$
(12)

A $C$-operad with a unit is called a unital $C$-operad.

In equation (12) we ignored the unit isomorphisms

$X\otimes I\cong X\cong I\otimes X$

of the monoidal category $\mathscr{V}$. The unit maps $u_{f}$ interpret
morphisms of $C$ as actual unary operations of $P$.

######  Definition 2.5.

A morphism of $C$-operads (resp. unital $C$-operads) is a morphism of the
underlying $C$-collections, which preserves the compositions (resp.
compositions and unit). The category of $C$-operads in $\mathscr{V}$ is
denoted by $\it{Op}^{C}(\mathscr{V})$ and the category of unital $C$-operads
in $\mathscr{V}$ is denoted by $\it{Op}_{u}^{C}(\mathscr{V})$.

We can view the maps (7) as a cowedge

${\big{\\{}(P_{n})^{i}_{c}\otimes\prescript{c}{}{(P_{m})}}$${P_{m+n-1}\big{\\}}_{c\in
C}.}$${\circ_{i}}$ (13)

In the presence of a unit we have:

######  Lemma 2.6.

For a unital $C$-operad $P$, the condition on the maps (13) to form a cowedge
is redundant.

Consider the diagram

${{{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
\Gamma\,c\,\Delta\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
\Lambda\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}}$${{{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
\Gamma\,c\,\Delta\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes
I\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
\Lambda\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}}$${{{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
\Gamma\,c\,\Delta\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
d\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
\Lambda\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}}$${{{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
\Gamma\,c\,\Delta\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
\Lambda\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}}$${{{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
\Gamma\,d\,\Delta\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
\Lambda\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}}$${{{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
\Gamma\Lambda\Delta\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}.}$${\cong}$${\mathbb{1}\otimes
u_{f}\otimes\mathbb{1}}$${\circ_{i}\otimes\mathbb{1}}$${\mathbb{1}\otimes\circ_{1}}$${\circ_{i}}$${\circ_{i}}$${\mathbb{1}\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
\mathbb{1}\,\cdots\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\
\mathbb{1}\,\cdots\,f\,\cdots\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes\mathbb{1}}$

The capital Greek letters stand for lists of objects of $C$. It is enough to
show that the outer diagram commutes for every map $f\colon d\rightarrow c$ of
$C$. But the top right part and left part commute by unitality and the bottom
right part commutes by the associativity of $\circ_{i}$. The next proposition
gives an alternative description of $C$-operads, using more general
composition maps $\circ_{i}^{f}$. These maps combine two abstract operations
along a morphism $f$ of $C$, which connects their input and output.

######  Proposition 2.7.

A $C$-operad in $\mathscr{V}$ is equivalently a $C$-collection $P$ in
$\mathscr{V}$, together with maps $\circ_{i}^{f}$ in $\mathscr{V}$

${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{i}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
d_{1}\,\cdots\,d_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,d_{1}\,\cdots\,d_{m}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),}$${\circ^{f}_{i}}$

for any $n\geq 1,m\geq 0,1\leq i\leq n$, objects
$c,c_{1},\ldots,c_{n},d,d_{1},\ldots,d_{m}$ and $f\colon d\rightarrow c_{i}$
in $C$, which are

  * •

$C$-equivariant: for any two maps

${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0ptf\\\
\sigma\,f_{1}\,\cdots\,f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
d_{1}\,\cdots\,d_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),}$
${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0ptg\\\
\tau\,g_{1}\,\cdots\,g_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}b\\\
b_{1}\,\cdots\,b_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}a\\\
a_{1}\,\cdots\,a_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),}$

in $(\mathbb{S}C)^{\it{op}}\times C$ and a map $h\colon a\rightarrow c_{i},$

$\circ_{i}^{h}\cdot\big{(}{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0ptf\\\
\sigma\,f_{1}\,\cdots\,f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0ptg\\\
\tau\,g_{1}\,\cdots\,g_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\big{)}={P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
35.0ptf\\\
(\sigma\circ_{i}\tau)\,f_{1}\,\cdots\,g_{1}\,\cdots\,g_{m}\,\cdots\,f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\cdot\circ_{\sigma(i)}^{f_{i}hg},$
(14)

  * •

$C$-associative: for any $1\leq j\leq n,0\leq m$ and $0\leq k$,

$\circ^{g}_{i}\cdot(\circ^{f}_{j}\otimes\mathbb{1})=\begin{cases*}\circ^{f}_{j+k-1}\cdot(\circ^{g}_{i}\otimes\mathbb{1})&if
$1\leq i<j\leq n$,\\\
\circ^{f}_{j}\cdot(\mathbb{1}\otimes\circ^{g}_{i-j+1})&if $j\leq i<j+m$,\\\
\circ^{f}_{j}\cdot(\circ^{g}_{i-m+1}\otimes\mathbb{1})&if $j+m\leq i\leq
n+m-1$.\end{cases*}$ (15)

The proof is a matter of rewriting the data and axioms of a $C$-operad. We now
relate unital $C$-operads to symmetric substitudes of [DS03]. The
correspondence is analogous to the classical correspondence of May operads and
unital Markl operads, see [Markl08, Proposition 13].

Substitudes were originally defined for a small $\mathscr{V}$-enriched
category $C$. We consider only $\mathscr{V}$-categories which arise from
ordinary small categories. That is, if $\mathscr{V}$ has coproducts and the
product $\otimes$ is distributive, any small category $C$ is
$\mathscr{V}$-enriched via

$C(c,d):=\displaystyle\bigoplus_{c\xrightarrow{f}d}I,$ (16)

the coproduct of the monoidal unit $I$ of $\mathscr{V}$. The
$\mathscr{V}$-functors $C\rightarrow\mathscr{V}$ and their
$\mathscr{V}$-natural transformations then agree with ordinary functors
$C\rightarrow\mathscr{V}$ and their natural transformations. We reformulate
the definition of a symmetric $\mathscr{V}$-substitude:

######  Definition 2.8.

Let $C$ be a small category, viewed as a $\mathscr{V}$-category. A
$\mathscr{V}$-substitude $P$ is a functor

${(\mathbb{S}C)^{\it{op}}\times C}$${\mathscr{V},}$${P}$

together with:

  * •

a family of morphisms

${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x_{1}\\\
x_{11}\,\cdots\,x_{1m_{1}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes\cdots\otimes{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x_{n}\\\
x_{n1}\,\cdots\,x_{nm_{n}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{11}\,\cdots\,x_{nm_{n}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${\mu}$

natural in $x_{11},\ldots,x_{nm_{n}}$ and $x$, called substitution, and

  * •

a family of morphisms

${C(x,y)}$${{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}y\\\
x\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${\eta}$

natural in $x$ and $y$, called the unit,

subject to obvious equivariance, associativity and unit axioms (cf. [DS03]).

Note, that naturality of the substitution in the connecting variables
$x_{1},\ldots,x_{n}$ is not assumed explicitly, but follows from the unit and
the associativity axioms, similarly as in Lemma 2.6.

######  Proposition 2.9.

Let $C$ be a small category. The categories of $\mathscr{V}$-substitudes over
$C$ and unital $C$-operads in $\mathscr{V}$ are isomorphic.

The underlying collections of the two structures are the same. Since $C(c,d)$
is defined by (16), the data of a unit for both structures also agree. Due to
Lemma 2.6 we are left to show the correspondence of the compositions
$\circ_{i}$ and the substitution $\mu$, which is standard. The operation $\mu$
is built up from consecutive $\circ_{i}$ operations and conversely, the
operations $\circ_{i}$ are obtained from $\mu$ by inserting units into the
inputs of $\mu$.

######  Remark 2.10.

As noted in [BW22, Remark 5.2.1], $C$-collections are the same as
$C\mathbb{S}$-modules of [Petersen13], up to variance and order of variables,
and thus also the same as $\mathbb{V}$-sequences of [Ward22]. A
$\mathscr{V}$-substitude over a small category $C$ corresponds to a
$C$-colored operad in $\mathscr{V}$ of [Petersen13], and hence to a $C$-operad
of [Ward22], when $C$ is a groupoid. However, it seems that [Ward22] works
only with skeletal groupoids, i.e. groupoids with $C(a,b)=\emptyset$ when
$a\neq b$. For us, it is essential to allow morphisms between different
objects of the coloring groupoid, since we encode Markl $\bO$-operads, which
are $\bO_{\it{iso}}$-collections in the first place. An example of an operadic
category with isomorphisms between different objects is the operadic category
Tr of rooted trees (cf. Section 3 and Example 4.8 of [BM23a], and the
introduction of [BMO23]). Algebras over the terminal Tr-operad are classical
symmetric operads.

##  3 Categorical operads and internal operads

Recall from [BB17, Section 9] the polynomial monad $T$ on the slice category
$\textit{Set}/\mathbb{N}$ (the category of families of sets indexed by natural
numbers), whose category of algebras is isomorphic to the category of
symmetric operads in Set. The monad is generated by the polynomial

$\mathbb{N}\xleftarrow{s}\texttt{ORTrees}^{*}\xrightarrow{p}\texttt{ORTrees}\xrightarrow{t}\mathbb{N},$

where ORTrees is a set of isomorphism classes of rooted trees, which have a
linear order on the set of leaves and also on the sets of incoming edges for
each vertex of the tree. $\texttt{ORTrees}^{*}$ stands for ordered rooted
trees with one marked vertex. The map $p$ forgets the marking, the target map
$t$ sends a tree to the number of its leaves, and the source map $s$ gives a
number of inputs of the marked vertex. The monad $T$ is defined by the formula

$TX_{n}=\displaystyle\sum_{a\in t^{-1}(n)}\displaystyle\prod_{b\in
p^{-1}(a)}X_{s(b)}.$

The multiplication is given by substituting a tree into the marked vertex, as
in Figure 5.

Figure 5: Ordered trees and substitution.

The polynomial monad $T$ generates a cartesian 2-monad (also denoted by $T$)
on the 2-category $\textit{Cat}/\mathbb{N}$ of small categories over the
discrete category of natural numbers.

######  Definition 3.1.

A unital categorical operad is a pseudo-$T$-algebra. The category of unital
categorical operads is denoted by $\it{Op}_{u}$.

######  Definition 3.2.

Let $\mathscr{O}$ be a unital categorical operad. An internal unital operad
$P$ in $\mathscr{O}$ is a lax morphism from the terminal pseudo-$T$-algebra to
$\mathscr{O}$. The category of internal unital operads in $\mathscr{O}$ and
$T$-natural transformations will be denoted by $\it{IOp}_{u}(\mathscr{O})$.

We consider variants of the monad $T$ which describe non-unital and non-
symmetric categorical operads. Let $T_{nu}$ be the polynomial monad generated
by the polynomial

$\mathbb{N}\xleftarrow{s}\texttt{ORTrees}_{+}^{*}\xrightarrow{p}\texttt{ORTrees}_{+}\xrightarrow{t}\mathbb{N},$

where $\texttt{ORTrees}_{+}$ is the set of isomorphism classes of ordered
rooted trees, without the tree $|$ (the free living edge). Consequently,
algebras of the monad $T_{nu}$ have no units for their composition operations.
Let $T_{\it{ns}}$ be generated by the polynomial

$\mathbb{N}\xleftarrow{s}\texttt{PRTrees}_{+}^{*}\xrightarrow{p}\texttt{PRTrees}_{+}\xrightarrow{t}\mathbb{N},$

where $\texttt{PRTrees}_{+}$ is the set of planar rooted trees, see [BB17,
Section 14], without the tree $\,|\,$.

######  Definition 3.3.

A categorical operad is a pseudo-$T_{nu}$-algebra. The category of categorical
operads is denoted by $\it{Op}$. A non-symmetric categorical operad is a
pseudo-$T_{\it{ns}}$-algebra. The category of non-symmetric categorical
operads is denoted by $\it{Op}_{\it{ns}}$.

######  Definition 3.4.

An internal operad in a categorical operad $\mathscr{O}$ is a lax morphism
from the terminal categorical operad into $\mathscr{O}$. The category of
internal operads in $\mathscr{O}$ will be denoted $\it{IOp}(\mathscr{O})$. An
internal non-symmetric operad in a non-symmetric categorical operad
$\mathscr{O}$ is a lax morphism from the terminal non-symmetric categorical
operad into $\mathscr{O}$. The category of internal non-symmetric operads in
$\mathscr{O}$ will be denoted $\it{IOp}_{\it{ns}}(\mathscr{O})$.

The identity functor $\mathbb{1}_{\it{Cat}/\mathbb{N}}$ on the category
$\it{Cat}/\mathbb{N}$ is a monad induced by a constant polynomial and
$\mathbb{1}_{\it{Cat}/\mathbb{N}}$-algebras are just objects of
$\it{Cat}/\mathbb{N}$. An internal algebra $x$ in a
$\mathbb{1}_{\it{Cat}/\mathbb{N}}$-algebra $X$ is a collection of object
$\\{x_{n}\in X(n)\\}_{n\geq 0}$. We will refer to it as to internal collection
in $X$. The category of internal collections in $X$ will be denoted by
$\it{IColl}(X)$.

Any planar tree can be regarded as an ordered tree, for which the global order
on its leaves agrees with the order induced by local orders of vertices and
planarity. We have inclusions

${\mathbb{N}}$${{\texttt{PRTrees}_{+}}}$${{\texttt{ORTrees}_{+}}}$${{\texttt{ORTrees}}.}$

The leftmost inclusion interprets a natural number $n$ as an $n$-corolla (i.e.
a planar rooted tree with one vertex and $n$ leaves). The inclusions induce
maps of polynomial monads and hence functors between pseudo-algebras:

${\it{Cat}/\mathbb{N}}$${\it{Op}_{\it{ns}}}$${\it{Op}}$${\it{Op}_{u}.}$

In other words, we can view any unital categorical operad as a non-unital/non-
symmetric categorical operad. For any unital categorical operad $\mathscr{O}$
we have the forgetful functors

${\it{IColl}(\mathscr{O})}$${\it{IOp}_{\it{ns}}(\mathscr{O})}$${\it{IOp}(\mathscr{O})}$${\it{IOp}_{u}(\mathscr{O}).}$${U_{1}}$${U_{2}}$${U_{3}}$

In Section 5 we study the left adjoint to the composite $U=U_{1}\circ
U_{2}\circ U_{3}.$

A categorical operad can be presented as a collection of categories
$\mathscr{O}(n)$, for each $n\geq 0$, together with

  * •

functors

${\mathscr{O}(n)}$${\mathscr{O}(n),}$${\sigma(-)}$

for any $n\geq 1$ and permutation $\sigma\in\Sigma_{n}$, making each
$\mathscr{O}(n)$ a non-strict left $\Sigma_{n}$-module, i.e. there are
coherent natural isomorphism $\tau(\sigma(-))\cong\tau\sigma(-)$ and
$\mathbb{1}_{n}(-)\cong\mathbb{1}_{\mathscr{O}(n)}$,

  * •

partial compositions functors

${\mathscr{O}(n)\times\mathscr{O}(m)}$${\mathscr{O}(m+n-1)}$${\odot_{i}}$

for $n>0,m\geq 0,$ and $i\in\\{1,\ldots,n\\}$,

satisfying classical associativity and equivariance axioms (see [Markl08,
Definition 11]), only up to coherent isomorphisms.

Consequently, an internal operad $A$ in $\mathscr{O}$ is a collection of
objects $A_{n}\in\mathscr{O}(n)$, for each $n\geq 0$, with the following
additional structure.

  * •

Each $A_{n}$ is equipped with an internal $\Sigma_{n}$-action: For every
$\sigma\in\Sigma_{n}$ there is a map

${\sigma(A_{n})}$${A_{n}}$${\overline{\sigma}}$

in the category $\mathscr{O}(n)$, such that

${\sigma(\tau(A_{n}))}$${\sigma(A_{n})}$${\sigma\tau(A_{n})}$${A_{n}}$${\sigma(\overline{\tau})}$${\overline{\sigma}}$${\overline{\sigma\tau}}$${\cong}$

commutes and

${\mathbb{1}_{n}(A_{n})}$${A_{n}}$${\overline{\mathbb{1}_{n}}}$

is the natural isomorphisms of the non-strict $\Sigma_{n}$-action on
$\mathscr{O}(n)$.

  * •

For every $n>0,m\geq 0$ and $1\leq i\leq n$, there are internal composition
maps

${A_{n}\odot_{i}A_{m}}$${A_{m+n-1}}$${\bullet_{i}}$

in the category $\mathscr{O}(m+n-1)$.

The structure maps further satisfy internal versions of classical operad
axioms:

  * •

Associativity. For any $1\leq j\leq n,0\leq m$ and $0\leq k$,

$\bullet_{i}\cdot(\bullet_{j}\odot_{i}\mathbb{1})=\begin{cases*}\bullet_{j+k-1}\cdot(\bullet_{i}\odot_{j+k-1}\mathbb{1})&if
$1\leq i<j\leq n$,\\\ \bullet_{j}\cdot(\mathbb{1}\odot_{j}\bullet_{i-j+1})&if
$j\leq i<j+m$,\\\ \bullet_{j}\cdot(\bullet_{i-m+1}\odot_{j}\mathbb{1})&if
$j+m\leq i\leq n+m-1$.\end{cases*}$ (17)

In formulas (17) we ignored the structure isomorphisms of the operad
$\mathscr{O}$. For example, the middle case thus means the commutativity of
the diagram

${{(A_{n}\odot_{j}A_{m})\odot_{i}A_{k}}}$${{A_{n+m-1}\odot_{i}A_{k}}}$${{A_{n}\odot_{j}(A_{m}\odot_{i-j+1}A_{k})}}$${{A_{n}\odot_{j}A_{m+k-1}}}$${{A_{n+m+k-2}}.}$${\cong}$${\mathbb{1}\odot_{j}\bullet_{i-j+1}}$${\bullet_{j}\odot_{i}\mathbb{1}}$${\bullet_{j}}$${\bullet_{i}}$

  * •

Equivariance. For $\sigma\in\Sigma_{n}$, $\tau\in\Sigma_{m}$, $1\leq i\leq n$
and $\pi=\sigma\circ_{i}\tau$, the following diagram commutes.

${{\pi(A_{n}\odot_{\sigma(i)}A_{m})}}$${{\pi(A_{n+m-1})}}$${{\sigma(A_{n})\odot_{i}\tau(A_{m})}}$${{A_{n}\odot_{i}A_{m}}}$${{A_{n+m-1}}}$${\overline{\sigma}\odot_{i}\overline{\tau}}$${\pi(\bullet_{\sigma(i)})}$${\cong}$${\bullet_{i}}$${\overline{\pi}}$
(18)

The isomorphisms in the diagrams are the structure isomorphisms of the operad
$\mathscr{O}$. Non-symmetric internal operads admit a similar presentations
without the structure maps and axioms involving internal $\Sigma$-actions. In
the unital case, a unital categorical operad $\mathscr{O}$ further has a unit
$U\in\mathscr{O}(1)$, satisfying the unit axioms up to coherent natural
isomorphisms. A unital internal operad then has an internal unit, i.e. a
morphism

${U}$${A_{1}}$${u}$

in the category $\mathscr{O}(1)$, such that for $1\leq i\leq n$, the following
diagrams commute.

${{A_{n}\odot_{i}U}}$${{A_{n}\odot_{i}A_{1}}}$${{A_{n}}}$${\cong}$${\mathbb{1}\odot_{i}u}$${\bullet_{i}}$
${{U\odot_{1}A_{m}}}$${{A_{1}\odot_{1}A_{m}}}$${{A_{m}}}$${\bullet_{1}}$${u\odot_{1}\mathbb{1}}$${\cong}$
(19)

The following example is an analogue of Lemma 9.1 of [Batanin08].

######  Example 3.5.

Let $(\mathscr{V},\otimes,I)$ be a symmetric monoidal category. Define
$\mathscr{O}(n):=\mathscr{V}$, with the trivial $\Sigma$-actions,
$\odot_{i}:=\otimes$ and $U:=I$. The non-strict associativity of $\otimes$
makes $\mathscr{O}$ a unital categorical operad. The category of internal
operads in $\mathscr{O}$ is isomorphic to the category of classical operads in
$\mathscr{V}$.

######  Remark 3.6.

In the context of Example 3.5, one can think of the structure operations
$\odot_{i}$ of $\mathscr{O}$ as a generalisation of the product $\otimes$ of
$\mathscr{V}$. When composing three (or more) operations of a classical operad
$P$, we loose the shape of the source $P_{n}\otimes P_{m}\otimes P_{k}$. On
the other hand, when defining operads in a categorical operad $\mathscr{O}$,
the source $(P_{n}\odot_{j}P_{m})\odot_{i}P_{k}$ of multiple compositions
retains its geometrical shape, which makes the compositions easier to handle,
for example when describing free operads.

##  4 Categorical operad $C_{\mathscr{V}}$ for $C$-operads

In this section, we describe a categorical operad $C_{\mathscr{V}}$, for a
small category $C$ and cocomplete symmetric monoidal closed category
$\mathscr{V}$, and prove that internal operads in $C_{\mathscr{V}}$ are
$C$-operads. Recall the notation $C_{\mathscr{V}}(n)$, for the category of
functors

${\underbrace{C^{\it{op}}\times\cdots\times
C^{\it{op}}}_{n\textit{-times}}\times C}$${\mathscr{V},}$${X}$

and the operations

${C_{\mathscr{V}}(n)\times
C_{\mathscr{V}}(m)}$${C_{\mathscr{V}}(n+m-1)}$${\otimes_{i}}$

of Definition 1.5. Let us describe the left $\Sigma_{n}$-module structure on
$C_{\mathscr{V}}(n)$. Let $\underline{n}$ denote the set $\\{1,\ldots,n\\}$,
regarded as a discrete category. There is a canonical right
$\Sigma_{n}$-module structure on the functor category
$\it{Cat}(\underline{n},C)$ given by precomposition:

${\underline{n}}$${C.}$${\underline{n}}$${\sigma}$${f}$${f\circ\sigma}$

We transfer this right $\Sigma_{n}$-module structure from
$\it{Cat}(\underline{n},C)$ to $C^{\times n}$ by

${C^{\times n}}$${C^{\times
n}}$${{\it{Cat}(\underline{n},C)}}$${{\it{Cat}(\underline{n},C)}}$${\cong}$${(-)\circ\sigma}$${\cong}$${(-)\sigma}$
${{\underline{c}=(c_{1},\ldots,c_{n})}}$${{(c_{\sigma(1)},\ldots,c_{\sigma(n)})=\underline{c}\sigma}}$${(-)\sigma}$

using the obvious isomorphism

$C^{\times n}\cong\it{Cat}(\underline{n},C).$

The left $\Sigma_{n}$-module structure on $C_{\mathscr{V}}(n)$ is given by

${{(C^{\it{op}})^{\times n}\times C}}$${\mathscr{V}.}$${{(C^{\it{op}})^{\times
n}\times C}}$${X}$${(-)\sigma\times\mathbb{1}}$${\sigma(X)}$

######  Lemma 4.1.

The operations $\otimes_{i}$ are equivariant.

For any $X\in C_{\mathscr{V}}(n)$, $Y\in C_{\mathscr{V}}(m)$,
$\sigma\in\Sigma_{n}$, $\tau\in\Sigma_{m}$, $f\colon c\rightarrow d\in C$, and
$i\in\\{1,\ldots,n\\}$, there is a strict equality

$\sigma(X)^{i}_{c}\otimes\prescript{d}{}{\tau(Y)}=(\sigma\circ_{i}\tau)(X^{\sigma(i)}_{c}\otimes\prescript{d}{}{Y}).$

Further, the map

${C_{\mathscr{V}}(n+m-1)}$${C_{\mathscr{V}}(n+m-1)}$${(\sigma\circ_{i}\tau)(-)}$

is an isomorphism, so it commutes with colimits, and thus
$(\sigma\circ_{i}\tau)(X\otimes_{\sigma(i)}Y)$ is the colimit of the diagram

${{\displaystyle\bigoplus_{f\colon d\rightarrow
c}(\sigma\circ_{i}\tau)(X^{\sigma(i)}_{c}\otimes\prescript{d}{}{Y})}}$${{\displaystyle\bigoplus_{b\in
C}(\sigma\circ_{i}\tau)(X^{i}_{b}\otimes\prescript{b}{}{Y}).}}$${\displaystyle\bigoplus_{f\colon
d\rightarrow
c}(\sigma\circ_{i}\tau)(X^{\sigma(i)}_{f}\otimes\mathbb{1})}$${\displaystyle\bigoplus_{f\colon
d\rightarrow c}(\sigma\circ_{i}\tau)(\mathbb{1}\otimes\prescript{f}{}{Y})}$

One obtains the canonical natural equivariance isomorphism

$\sigma(X)\otimes_{i}\tau(Y)\cong(\sigma\circ_{i}\tau)(X\otimes_{\sigma(i)}Y).$

The associativity of $\otimes_{i}$ was shown in Lemma 1.2. Since the
isomorphisms constructed in Lemma 1.6 and Lemma 4.1 are canonical, we have
proved:

######  Proposition 4.2.

The categorical left $\Sigma_{n}$-modules $C_{\mathscr{V}}(n)$ together with
operations $\otimes_{i}$ form a categorical operad.

The operad $C_{\mathscr{V}}$ is in fact unital. The unit for $\otimes_{i}$ is
the functor $C(-,-)\in C_{\mathscr{V}}(1)$, defined by (16). An internal
operad $A$ in the categorical operad $C_{\mathscr{V}}$ consists of a
collection $A_{n}\in C_{\mathscr{V}}(n)$ for each $n\geq 0$, i.e. a collection
of functors

${\underbrace{C^{\it{op}}\times\cdots\times
C^{\it{op}}}_{n\textit{-times}}\times C}$${\mathscr{V},}$${A_{n}}$

together with natural transformations

${\sigma(A_{n})}$${A_{n}.}$${\overline{\sigma}}$

By definition,

$\sigma(A_{n}){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)={A_{n}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{\sigma(1)}\,\cdots\,x_{\sigma(n)}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$

and so the components of $\overline{\sigma}$ are

${{A_{n}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{\sigma(1)}\,\cdots\,x_{\sigma(n)}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{A_{n}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).}$${\overline{\sigma}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$

Further, there are natural transformations

${A_{n}\otimes_{i}A_{m}}$${A_{n+m-1},}$${\bullet_{i}}$

the unit is a natural transformation

${C(x,y)}$${{A_{1}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}y\\\
x\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),}$${u_{xy}}$

and they satisfy the internal operad axioms (17), (18), and (19).

###### Theorem A 1

The category of internal operads (resp. unital internal operads) in the
categorical operad $C_{\mathscr{V}}$ (resp. unital categorical operad
$C_{\mathscr{V}}$), constructed above, is isomorphic to the category of
$C$-operads (resp. unital $C$-operads) in $\mathscr{V}$:

$\it{IOp}(C_{\mathscr{V}})\cong\it{Op}^{C}(\mathscr{V})\text{ and
}\it{IOp}_{u}(C_{\mathscr{V}})\cong\it{Op}_{u}^{C}(\mathscr{V}).$

The data of the underlying collection of a $C$-operad and of an internal
operad agree. The internal actions of an internal operad

${\sigma{A_{n}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${A_{n}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${\overline{\sigma}}$

translate as right $\Sigma_{n}$-actions of a $C$-operad

${P_{n}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{\sigma(1)}\,\cdots\,x_{\sigma(n)}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${P_{n}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}x\\\
x_{1}\,\cdots\,x_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).}$${(-)\sigma^{-1}}$

By definition, the operations $\circ_{i}$ of a $C$-operad and $\bullet_{i}$ of
an internal operad are the same. The units for both structures agree and the
equivariance and associativity axioms can be compared directly. When $C$ is a
small $\mathscr{V}$-enriched category, the categories
$\hat{C_{\mathscr{V}}}(n)$ of $\mathscr{V}$-functors and their
$\mathscr{V}$-natural transformations again form a categorical operad. The
composition operations $\otimes_{i}$ are defined using $\mathscr{V}$-enriched
colimits. Theorem A suggest the following definition of $C$-operads for a
$\mathscr{V}$-enriched category $C$.

######  Definition 4.3.

Let $C$ be a small $\mathscr{V}$-enriched category. A $C$-operad $P$ is an
internal operad in the categorical operad $\hat{C_{\mathscr{V}}}$.

An example of a $\it{Vect_{\mathbb{k}}}$-enriched $C$-operad is given in
Example 6.9.

##  5 Free internal operads

We recall some combinatorial material and terminology of [BM23a], which we
will use to describe free internal operads.

### The operadic category $\Delta$

Let $\Delta$ be the category of finite ordered sets
$\underline{n}:=\\{1,\ldots,n\\}$ and order preserving maps. It is an operadic
category in the sense of [BM23a, Section 1]. Recall that a map
$\phi\colon\underline{m}\rightarrow\underline{n}$ of $\Delta$ is elementary if
it has precisely one non-trivial fiber, i.e. there is a unique index
$i\in\underline{n}$, such that $|\phi^{-1}(i)|\neq\underline{1}$. We write
$(\phi,i)$ for an elementary morphism $\phi$ with a non-trivial fiber over
$i$. Two composable elementary morphisms

$\underline{n}\xrightarrow{(\psi,j)}\underline{m}\xrightarrow{(\phi,i)}\underline{k}$

have disjoint fibers if $\phi(j)\neq i$.

Consider the following commutative diagram of elementary morphisms with
disjoint fibers.

${{\underline{m}^{\prime}}}$${{\underline{n}}}$${{\underline{k}}.}$${{\underline{m}^{\prime\prime}}}$${(\psi^{\prime},p)}$${(\phi^{\prime},q)}$${(\psi^{\prime\prime},r)}$${(\phi^{\prime\prime},s)}$
(20)

In this situation the non-trivial fibers of $\psi^{\prime}$ and
$\phi^{\prime\prime}$ are the same (cf. Figure 6):

$\psi^{\prime-1}(p)=\phi^{\prime\prime-1}(s),$

and similarly

$\psi^{\prime\prime-1}(r)=\phi^{\prime-1}(q).$

Figure 6: Pair of morphisms with disjoint fibers.

### Free construction

Fix a unital categorical operad $\mathscr{O}$, for which every
$\mathscr{O}(n)$ is cocomplete and the operations $\odot_{i}$ commute with
colimits in both variables. Recall the restriction functors

${\it{IColl}(\mathscr{O})}$${\it{IOp}_{\it{ns}}(\mathscr{O})}$${\it{IOp}(\mathscr{O})}$${\it{IOp}_{u}(\mathscr{O}).}$${U_{1}}$${U_{2}}$${U_{3}}$
(21)

of Section 3 and denote their composite by $U$. A left adjoint $F\dashv U$
exists and can be computed by the methods of [BB17], using the relative
internal algebra classifiers. We describe $F$ by constructing the left
adjoints $F_{i}\dashv U_{i}$ for each functor of the sequence (21).

For the construction of $F_{1}\dashv U_{1}$ we use a modification of the
construction of [BM23b] for the operadic category $\Delta$, which produces the
free operad as a colimit of a functor from the groupoid of leveled planar
rooted trees. We replace the monoidal products $\otimes$ in the construction
with the operations $\odot_{i}$.

######  Definition 5.1.

A leveled planar rooted tree $T$ is a sequence of composable elementary
morphisms

$\underline{n}_{1}\xrightarrow{(\tau_{1},i_{1})}\cdots\xrightarrow{(\tau_{k-1},i_{k-1})}\underline{n}_{k}\xrightarrow{!}\underline{1}$

in $\Delta$. Its fiber sequence $f_{1},\ldots,f_{k}$ is a sequence of non-
trivial fibers of the morphisms. The height of a tree is the number of
morphisms in the sequence.

Let us define a groupoid lT of leveled planar rooted trees. The objects are
leveled planar rooted trees and two trees

$\begin{split}\mathscr{T}^{\prime}=\underline{n_{1}}\xrightarrow{}\cdots\xrightarrow{}\underline{n}\xrightarrow{}\underline{m}^{\prime}\xrightarrow{}\underline{k}\xrightarrow{}\cdots\xrightarrow{}\underline{1}\\\
\mathscr{T}^{\prime\prime}=\underline{n_{1}}\xrightarrow{}\cdots\xrightarrow{}\underline{n}\xrightarrow{}\underline{m}^{\prime\prime}\xrightarrow{}\underline{k}\xrightarrow{}\cdots\xrightarrow{}\underline{1}\end{split}$
(22)

are isomorphic if their sequences differ by a commutative square of elementary
morphisms with disjoint fibers as in (20). Three isomorphic leveled planar
rooted trees are depicted in Figure 7.

Figure 7: The groupoid of leveled trees.

Let $\texttt{lT}(n)$ denote the full subcategory of trees with
$\underline{n}_{1}=\underline{n}$, so the collection of $\texttt{lT}(n)$ is an
object in $\it{Cat}/\mathbb{N}$. The maps

${\texttt{lT}(n)\times\texttt{lT}(m)}$${\texttt{lT}(m+n-1)}$${\odot_{i}}$

are given by grafting of leveled trees, producing a (strict) non-unital non-
symmetric categorical operad lT. Let

$\mathscr{T}=\underline{n}_{1}\xrightarrow{(\tau_{1},i_{1})}\cdots\xrightarrow{(\tau_{k},i_{k})}\underline{1}$

be a tree with fiber sequence $t_{1},\ldots,t_{k}$. For an internal collection
$X$ in $\mathscr{O}$ we define a functor

lT${\mathscr{O}}$${\hat{X}}$

by the formula

$\hat{X}(\mathscr{T})=(\cdots\,(X_{t_{k}}\odot_{i_{k-1}}X_{t_{k-1}})\odot_{i_{k-2}}\cdots)\odot_{i_{1}}X_{t_{1}}.$

The level-interchange isomorphisms of lT are mapped by $\hat{X}$ to the
parallel associativity isomorphisms of $\mathscr{O}$. Further, we have

$\hat{X}(\mathscr{T}\odot_{i}\mathscr{S})\cong\hat{X}(\mathscr{T})\odot_{i}\hat{X}(\mathscr{S}).$

The isomorphism is given by the repeated use of the sequential associativity
isomorphisms of $\mathscr{O}$.

######  Definition 5.2.

Let $X$ be an internal collection in $\mathscr{O}$. We define the functor
$F_{1}$ by

${F_{1}(X):=\it{colim}\big{(}\texttt{lT}}$${\mathscr{O}\big{)}.}$${\hat{X}}$

By assumption the operations $\odot_{i}$ of the operad $\mathscr{O}$ commute
with colimits, and thus

$F_{1}(X)_{n}\odot_{i}F_{1}(X)_{m}=(\underset{\texttt{lT}(n)}{\it{colim}}\hskip
5.0pt\hat{X}_{n})\odot_{i}(\underset{\texttt{lT}(m)}{\it{colim}}\hskip
5.0pt\hat{X}_{m})\cong\underset{\texttt{lT}(n)\times\texttt{lT}(m)}{\it{colim}}\,(\hat{X}_{n}\odot_{i}\hat{X}_{m}).$
(23)

Next, every pair of tree isomorphisms
$\mathscr{T}^{\prime}\cong\mathscr{T}^{\prime\prime}$,
$\mathscr{S}^{\prime}\cong\mathscr{S}^{\prime\prime}$, leads to an isomorphism

$\mathscr{T}^{\prime}\odot_{i}\mathscr{S}^{\prime}\cong\mathscr{T}^{\prime\prime}\odot_{i}\mathscr{S}^{\prime\prime},$

i.e. the image

${\it{im}\big{(}\texttt{lT}(n)\times\texttt{lT}(m)}$${\texttt{lT}(m+n-1)\big{)}}$${\odot_{i}}$

is a full subgroupoid of $\texttt{lT}(m+n-1)$, and hence there is a map

${\underset{\texttt{lT}(n)\times\texttt{lT}(m)}{\it{colim}}(\hat{X}_{n}\odot_{i}\hat{X}_{m})}$${\underset{\texttt{lT}(m+n-1)}{\it{colim}}\hskip
5.0pt\hat{X}_{m+n-1}=F_{1}(X)_{m+n-1}.}$${\odot_{i}}$ (24)

The composite of (23) and (24) provides the composition maps $\circ_{i}$ for
the operad $F_{1}(X)$. It is straightforward to check that the associativity
axioms hold and we have:

######  Lemma 5.3.

The operad $F_{1}(X)$ with $\circ_{i}$ defined above is the free non-symmetric
non-unital internal operad generated by $X$ in $\mathscr{O}$, i.e. $F_{1}$ is
the left adjoint of $U_{1}$.

The proof is standard. Let $A$ be a non-unital non-symmetric internal operad
in $\mathscr{O}$ and $y\colon X\rightarrow A$ a map of internal collections in
$\mathscr{O}$. We have to construct a map of operads $Y\colon
F_{1}(X)\rightarrow A$, such that

${F_{1}(X)}$${X}$${A}$${i}$${y}$${Y}$ (25)

commutes in the category of internal collections. We describe the map $Y$ by
constructing a map of collections
$\hat{X}(\mathscr{T})\xrightarrow{Y(\mathscr{T})}A,$ for each tree
$\mathscr{T}$, which will respect the level-interchange isomorphism, hence
inducing a map from the colimit $F_{1}(X)$.

Let $\mathscr{T}\in\texttt{lT}_{n}$,
$\mathscr{T}=\underline{n}\xrightarrow{(\tau_{1},i_{1})}\cdots\xrightarrow{(\tau_{k},i_{k})}\underline{1}$
with fiber sequence $t_{1},\ldots,t_{k}$, and recall that

$\hat{X}(\mathscr{T})=(\cdots\,(X_{t_{k}}\odot_{i_{k-1}}X_{t_{k-1}})\odot_{i_{k-2}}\cdots)\odot_{i_{1}}X_{t_{1}}.$

Let us similarly denote

$\hat{A}(\mathscr{T})=(\cdots\,(A_{t_{k}}\odot_{i_{k-1}}A_{t_{k-1}})\odot_{i_{k-2}}\cdots)\odot_{i_{1}}A_{t_{1}}$

and

$\hat{y}(\mathscr{T})\colon\hat{X}(\mathscr{T})\rightarrow\hat{A}(\mathscr{T}),$
$\hat{y}(\mathscr{T})=(\cdots\,(y_{t_{k}}\odot_{i_{k-1}}y_{t_{k-1}})\odot_{i_{k-2}}\cdots)\odot_{i_{1}}y_{t_{1}}.$

We can compose the sequence $\hat{A}(\mathscr{T})$ in the operad $A$ by

$\hat{A}(\mathscr{T})\xrightarrow{\circ_{i_{k-1}}\,\cdots\,\circ_{i_{1}}}A_{n}.$

Thus, we define $Y(\mathscr{T})$ as the composite

$\hat{X}(\mathscr{T})\xrightarrow{\hat{y}(\mathscr{T})}\hat{A}(\mathscr{T})\xrightarrow{\circ_{i_{k-1}}\cdots\circ_{i_{1}}}A_{n}.$

The parallel associativity of the operad $A$ ensures that the diagram

${{\hat{X}(\mathscr{T}^{\prime})}}$${{A_{n}}}$${{\hat{X}(\mathscr{T}^{\prime\prime})}}$${Y(\mathscr{T}^{\prime})}$${Y(\mathscr{T}^{\prime\prime})}$${\cong}$

commutes for any isomorphism
$\mathscr{T}^{\prime}\cong{\mathscr{T}^{\prime\prime}}$ of $\texttt{lT}_{n}$,
and hence the maps $Y(\mathscr{T})$ induce a unique map $Y\colon
F_{1}(X)\rightarrow A$. By construction, the diagram (25) commutes. It remains
to show that $Y$ preserves composition. This is ensured by the sequential
associativity of the operad $A$. The symmetrisation and adding the unit to a
non-unital non-symmetric internal operad is analogous to the classical case.
Let $A$ be an internal non-symmetric operad in $\mathscr{O}$ and define

$F_{2}(A)_{n}=\displaystyle\bigoplus_{\pi\in\Sigma_{n}}\pi(A_{n}).$

The internal actions

${\sigma(F_{2}(A)_{n})}$${F_{2}(A)_{n}}$${\overline{\sigma}}$

are induced by the component maps

${\sigma(\pi(A_{n}))}$${\sigma\pi(A_{n})}$${F_{2}(A)_{n},}$${\cong}$

and the composition

${F_{2}(A)_{n}\odot_{i}F_{2}(A)_{m}}$${F_{2}(A)_{m+n-1}}$${\circ_{i}}$

is induced by the maps

${\sigma(A_{n})\odot_{i}\tau(A_{m})}$${(\sigma\circ_{i}\tau)(A_{n}\odot_{\sigma(i)}A_{m})}$${(\sigma\circ_{i}\tau)(A_{m+n-1})}$${F_{2}(A)_{m+n-1}.}$${\cong}$${(\sigma\circ_{i}\tau)(\circ_{\sigma(i)})}$

######  Lemma 5.4.

The functor $F_{2}$ is left adjoint to the forgetful functor $U_{2}$.

The verification is straightforward. For the categorical operad
$C_{\mathscr{V}}$, Lemma 5.4 recovers the symmetrisation of [DS03]. Let $U$
denote the unit of a unital categorical operad $\mathscr{O}$. We define an
internal unital operad $I$ in $\mathscr{O}$ by

$I_{1}:=U,$

and for $n\neq 1$,

$I_{n}:=\emptyset,$

the initial object of $\mathscr{O}(n)$.

######  Lemma 5.5.

The coproduct $F_{3}(A)=A\oplus I$ defines left adjoint to the forgetful
functor $U_{3}$.

The verification is straightforward.

Putting together Lemmas 5.3, 5.4, and 5.5 we obtain:

######  Proposition 5.6.

The composite $F=F_{3}\circ F_{2}\circ F_{1}$ is left adjoint to the functor

$U\colon\it{IOp}(\mathscr{O})\rightarrow\it{IColl}(\mathscr{O}).$

######  Example 5.7.

We analyze the free construction for $\mathscr{O}=C_{\mathscr{V}}$, the operad
defined in Section 4, and $\mathscr{V}=\it{Vect_{\mathbb{k}}}$, the category
of vector spaces over a field $\mathbb{k}$. This gives a description of the
free $C$-operad over a non-symmetric $C$-collection $X$. Fix a leveled tree

$\mathscr{T}=\underline{n}\xrightarrow{(\tau_{1},i_{1})}\cdots\xrightarrow{(\tau_{k},i_{k})}\underline{1}$

with fiber sequence $t_{1},\ldots,t_{k}.$ The tree $\mathscr{T}$ can be
regarded as a planar graph with an order of vertices given by the levels of
$\mathscr{T}$. We label each edge of $\mathscr{T}$ by an object of $C$, i.e.
we fix a labeling function

$\lambda\colon\it{Edges}(\mathscr{T})\rightarrow\it{ob}(C).$

Then each vertex $v_{j}$ has a type

$\overline{v}_{j}={}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\cdots
c_{t_{j}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$

given by the labels of adjacent edges. Let $X$ be a non-symmetric
$C$-collection of functors

${\underbrace{C^{\it{op}}\times\cdots\times C^{\it{op}}}_{n-times}\times
C}$${\it{Vect}_{\mathbb{k}}.}$${X_{n}}$

We define

$\hat{X}(\mathscr{T},\lambda):=X_{t_{k}}(\overline{v}_{k})\otimes\cdots\otimes
X_{t_{1}}(\overline{v}_{1}).$

The value
$F_{1}(X){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}r\\\
l_{1}\,\cdots\,l_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
is the coproduct

$F_{1}(X){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}r\\\
l_{1}\,\cdots\,l_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)=\displaystyle\bigoplus_{(\mathscr{T},\lambda)}\hat{X}(\mathscr{T},\lambda),$

taken over all leveled planar trees with $n$ leaves, such that $\lambda$
labels the leaves of $\mathscr{T}$ by the objects $l_{1},\ldots,l_{n}\in C$
and the root edge by $r\in C$, with the following identifications:

  * •

Let $\mathscr{T}^{\prime}\cong\mathscr{T}^{\prime\prime}$, such that the two
trees differ only in the $i$-th position by a square of elementary morphisms
with disjoint fibers as in (22). Then

$a_{k}\otimes\cdots\otimes a_{i}\otimes a_{i-1}\otimes\cdots\otimes
a_{1}\in\hat{X}(\mathscr{T}^{\prime},\lambda)$

is identified with

$a_{k}\otimes\cdots\otimes a_{i-1}\otimes a_{i}\otimes\cdots\otimes
a_{1}\in\hat{X}(\mathscr{T}^{\prime\prime},\lambda).$

  * •

Let an edge $e$ of a labeled tree $(\mathscr{T},\lambda)$ connect vertices
$v_{i}$ and $v_{j}$ (with $v_{j}$ closer to the root) of types

$\overline{v}_{j}={}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}y\\\
y_{1}\,\cdots\,d\,\cdots\,y_{t_{j}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),\overline{v}_{i}={}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
x_{1}\,\cdots\,x_{t_{i}}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$

and let $f\colon c\rightarrow d$ in $C$. Then

$a_{k}\otimes\cdots\otimes a_{j}\cdot f\otimes\cdots\otimes
a_{i}\otimes\cdots\otimes a_{1}$

is identified with

$a_{k}\otimes\cdots\otimes a_{j}\otimes\cdots\otimes f\cdot
a_{i}\otimes\cdots\otimes a_{1}.$

The notation $a_{j}\cdot f$ and $f\cdot a_{i}$ was introduced in Example 1.7.
Note, that in the graded context signs may be involved. The free symmetric
$C$-operad $F(X)$ has

$F(X){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}r\\\
l_{1}\,\cdots\,l_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)=\displaystyle\bigoplus_{\sigma\in\Sigma_{n}}F_{1}(X){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}r\\\
l_{\sigma(1)}\,\cdots\,l_{\sigma(n)}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

Elements of $F(X)$ are thus sums of trees as in Figure 1, with specified
orders on their leaves.

##  6 Algebras

The first part of this section modifies some standard notions to the category-
colored setting. In the second part we express algebras of classical operads
with values in a functor category $\mathscr{V}^{C}$ as algebras of $C$-operads
in $\mathscr{V}$. For this we need $\mathscr{V}$ to be a cocomplete symmetric
closed monoidal category and $C$ a small symmetric monoidal
$\mathscr{V}$-category, so that the category $\mathscr{V}^{C}$ is symmetric
monoidal closed via the Day convolution [Day70].

From this point on we work only with the monoidal category
$\mathscr{V}=\it{Vect}_{\mathbb{k}}$, the category of vector spaces over a
field ${\mathbb{k}}$. This allows us to work with elements, which simplifies
the presentation. We believe that the reader can easily translate the
definitions and constructions to other relevant monoidal categories.

The category of algebras of a $C$-operad $P$ lives over the functor category
$\it{Vect}_{\mathbb{k}}^{C}.$ For a functor $A\colon
C\rightarrow\it{Vect_{\mathbb{k}}}$, we define its endomorphism $C$-operad
$\it{End}_{A}$. Let $c_{1},\ldots,c_{n},c$ be objects of $C$ and let
$\it{Hom}(U,V)$ denote the space of linear maps from $U$ to $V$. We define

${\it{End}_{A}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right):=\it{Hom}(A(c_{1})\otimes\cdot\cdot\cdot\otimes
A(c_{n}),A(c)).$

For a map

${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
\sigma\,f_{1}\,\cdots\,f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\colon{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
d_{1}\,\cdots\,d_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$

in $(\mathbb{S}C)^{\it{op}}\times C$ and
$\varphi\in{\it{End}_{A}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$,
we define
${\it{End}_{A}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
\sigma\,f_{1}\cdots
f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\varphi)$ to
be the composite

${{A(c_{1})\otimes\cdots\otimes
A(c_{n})}}$${{A(c)}}$${{A(d_{\sigma(1)})\otimes\cdots\otimes
A(d_{\sigma(n)})}}$${{A(d_{1})\otimes\cdots\otimes
A(d_{n})}}$${{A(d)}.}$${{\it{End(A)}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}f\\\
\sigma\,f_{1}\cdots
f_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\varphi)}$${\varphi}$${A(f)}$${(-)\sigma}$${A(f_{1})\otimes\cdots\otimes
A(f_{n})}$

For $n=0$, let
${\it{End}_{A}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
\emptyset\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right):=A(c)$.
Let
$\varphi\in{\it{End}_{A}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\cdots c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
and
$\psi\in{\it{End}_{A}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c_{i}\\\
d_{1}\cdots
d_{m}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$. The operad
composition is defined by

$\varphi\circ_{i}\psi:=\varphi\circ(\mathbb{1}_{A(c_{1})}\otimes\cdots\otimes\psi\otimes\cdots\otimes\mathbb{1}_{A(c_{n})}).$

The operad $\it{End}_{A}$ is unital with units
$A(f)\in{\it{End}_{A}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
c\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$, for $f\colon
c\to d$ in $C$.

######  Definition 6.1.

Let $P$ be a $C$-operad. A $P$-algebra is a functor $A\colon
C\rightarrow\it{Vect_{\mathbb{k}}}$ together with a morphism of $C$-operads

${P}$${\it{End}_{A}.}$${\alpha}$

A $P$-algebra is thus a functor $A\colon C\rightarrow\it{Vect_{\mathbb{k}}}$,
together with maps

$\displaystyle\int^{c_{1}\cdots
c_{n}\in\mathbb{S}C^{\it{op}}}{P}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes
A(c_{1})\otimes\cdots\otimes A(c_{n})\xrightarrow{{\alpha}}A(c),$

for any $n\geq 0$ and object $c\in C$, which satisfy some obvious axioms. Due
to our Proposition 2.9, Proposition 5.1.6 of [BW22] offers other
characterisations of algebras of unital $C$-operads.

######  Definition 6.2.

Let $F\colon D\rightarrow\it{Vect_{\mathbb{k}}}$ be a functor. An equivalence
relation $E$ on the functor $F$ is a subfunctor

$E(d)\subseteq F(d)\times F(d),$

such that, for every $d\in D$, $E(d)$ is an equivalence relation. The quotient
$F/E$ is the functor

$F/E(d):=F(d)/E(d).$

######  Definition 6.3.

Let $R$ be a set of pairs

$R=\\{(a_{s},b_{s})\,|\,a_{s},b_{s}\in F(c_{s})\\}_{s\in S},$

for some set $S$. An equivalence relation generated by $R$ is the smallest
equivalence relation on $F$ which contains $R$. The generating pairs will be
denoted $a_{s}\sim b_{s}$.

For a non-symmetric $C$-collection $X$, the notation $\alpha\in X$ will mean
$\alpha\in{X_{n}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$,
for some $n\in\mathbb{N}$ and objects $c_{1},\ldots,c_{n},c$ of $C$.

######  Definition 6.4.

An operadic ideal $I$ of a $C$-operad $P$ is a subfunctor $I\subseteq P$, such
that every composition $\gamma\circ_{i}\beta$ of two elements of $P$, where at
least one of the elements is in $I$, is again in $I$.

######  Definition 6.5.

An operadic ideal of a $C$-operad $P$ generated by a set

$A=\\{\alpha_{s}\in P\\}_{s\in S},$

for some set $S$, is the smallest operadic ideal of $P$, which contains $A$.
We will denote it by $(A)$.

For an operadic ideal $I$ of a $C$-operad $P$, the $C$-collection $P/I$
acquires an obvious $C$-operad structure. For a non-symmetric $C$-collection
$X$ in $\it{Vect_{\mathbb{k}}}$, every operation $\alpha\in F(X)$ is a finite
sum $\alpha=\sum_{s\in S}\alpha_{s}$ with each
$\alpha_{s}\in\hat{X}(\mathscr{T}_{s},\lambda_{s})$ for some tree
$\mathscr{T}_{s}$ with labeling $\lambda_{s}$, cf. Example 5.7.

######  Definition 6.6.

We say that $\alpha\in F(X)$ is homogeneous of weight $k$, if every tree
$\mathscr{T}_{s}$ appearing in the sum $\alpha=\sum_{s\in S}\alpha_{s}$, with
$\alpha_{s}\in\hat{X}(\mathscr{T}_{s},\lambda_{s}),$ has $k$ vertices. We
denote by $F(X)^{(k)}$ the subfunctor of $F(X)$ of operations of weight $k$.

######  Definition 6.7.

A $C$-operad $P$ is quadratic binary if it is of the form $P=F(X)/(A)$, such
that $X_{n}$ is trivial unless $n=2$, and $A\subseteq F(X)^{(2)}$.

Let $(C,\oplus,0)$ be a small symmetric monoidal linear category (i.e. a
$\it{Vect_{\mathbb{k}}}$-enriched category), and $P$ an operad with values in
the category $\it{Vect}_{\mathbb{k}}^{C}$ of linear functors and their linear
natural transformations. With the Day convolution,
$\it{Vect}_{\mathbb{k}}^{C}$ is a cocomplete symmetric monoidal closed
category. Let $P(n,r)$ denote the value of $P(n)$ on an object $r$ of $C$. Our
goal is to define a $C$-operad $P^{C}$ with values in $\it{Vect_{\mathbb{k}}}$
which has the same algebras as the operad $P$. The object
${P^{C}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
is the linear coend

$\displaystyle\int^{r\in C}C(c_{1}\oplus\cdots\oplus c_{n}\oplus r,c)\otimes
P(n,r),$

and the action of a morphism
${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0pt\alpha\\\
\sigma\,\alpha_{1}\,\cdots\,\alpha_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
of $\mathbb{S}C^{\it{op}}\times C$ is induced by precomposition with
$((\alpha_{1}\oplus\cdots\oplus\alpha_{n})\circ\overline{\sigma})\oplus 1_{r}$
and postcomposition with $\alpha$ on $C(c_{1}\oplus\cdots\oplus c_{n}\oplus
r,c)$. The composition maps $\circ_{i}$ of $P^{C}$ are induced by the
composition maps $\circ_{i}^{P}$ of $P$, the composition $\circ^{C}$ of $C$,
and the symmetry isomorphism $s$ of $\mathscr{V}$:

${{C(c_{1}\oplus\cdots\oplus c_{n}\oplus p,c)\otimes P(n,p)\otimes
C(d_{1}\oplus\cdots\oplus d_{m}\oplus q,c_{i})\otimes
P(m,q)}}$${{C(c_{1}\oplus\cdots\oplus c_{n}\oplus p,c)\otimes
C(d_{1}\oplus\cdots\oplus d_{m}\oplus q,c_{i})\otimes P(n,p)\otimes
P(m,q)}}$${{C(c_{1}\oplus\cdots\oplus d_{1}\oplus\cdots\oplus
d_{m}\oplus\cdots\oplus c_{n}\oplus p\oplus q,c)\otimes P(n+m-1,p\oplus
q)}.}$${\mathbb{1}\otimes
s\otimes\mathbb{1}}$${\circ^{C}\otimes\circ_{i}^{P}}$

######  Proposition 6.8.

The categories of $P$-algebras in $\it{Vect}_{\mathbb{k}}^{C}$ and
$P^{C}$-algebras in $\it{Vect_{\mathbb{k}}}$ are isomorphic.

By definition, a $P$-algebra is a linear functor $X$ together with a linear
natural transformation

$\displaystyle\int^{r\in C,c_{1}\cdots
c_{n}\in\mathbb{S}C^{\it{op}}}C(c_{1}\oplus\cdots\oplus c_{n}\oplus
r,c)\otimes P(n,r)\otimes X(c_{1})\otimes\cdots\otimes
X(c_{n})\xrightarrow{\alpha_{c}}X(c),$

natural in $c$, which preserves the operad structure of $P$. This is the same
as

$\displaystyle\int^{c_{1}\cdots
c_{n}\in\mathbb{S}C^{\it{op}}}{P^{C}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\,\cdots\,c_{n}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\otimes
X(c_{1})\otimes\cdots\otimes X(c_{n})\xrightarrow{{\alpha_{c}}}X(c).$

The map ${\alpha_{c}}$ respects the $C$-operad structure, since the
compositions of $P^{C}$ are induced by the composition of $P$, and hence
$(X,\alpha)$ is a $P^{C}$-algebra. This correspondence induces isomorphism of
the categories of algebras.

######  Example 6.9.

As an instance of Proposition 6.8, we show that differential graded
associative algebras are algebras of a non-unital
$\it{Vect_{\mathbb{k}}}$-enriched category-colored operad. The coloring
category in this case is not a groupoid, which is related to the question of
Remark 3.12 of [Petersen13]. Let us define the coloring linear category $D$.
Objects of $D$ are integers, the spaces $D(n,n-1)$, for any integer $n$, are
one dimensional with generator $\partial_{n}$, and the other Hom-spaces of $D$
are trivial. This forces $\partial\partial=0$, hence linear functors
$D\rightarrow\it{Vect_{\mathbb{k}}}$ are exactly chain complexes of vector
spaces. We define a linear functor

$\displaystyle\bigoplus_{n\in\mathbb{N}}(D^{\it{op}})^{n}\times D\xrightarrow{X}\it{Vect_{\mathbb{k}}},$ ${X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}m+n\\\ m\hskip 8.0ptn\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right):=\mathbb{k}[\mu_{m,n}]$, | ${X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}m+n-1\\\ m\hskip 8.0ptn\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right):=\mathbb{k}[\mu_{0},\mu_{1},\mu_{2}]/\langle\mu_{1}+(-1)^{n}\mu_{2}-\mu_{0}\rangle,$
---|---

for any $m,n\in D$, and $X$ is trivial otherwise. The only non-trivial
$D$-actions are

${X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\ \partial\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\mu_{m-1,n}):=\mu_{1},$ | ${X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\ \mathbb{1}\,\partial\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\mu_{m,n-1}):=\mu_{2},$ | ${X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\partial\\\ \mathbb{1}\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\mu_{m,n}):=\mu_{0}.$
---|---|---

The functor $X$ generates the free non-unital $D$-operad $F(X)$ which we
further divide by an operadic ideal $I$ generated by the elements

$\mu_{m+n,k}\circ_{1}\mu_{m,n}-\mu_{m,n+k}\circ_{2}\mu_{n,k},$

for all $m,n,k\in D$. One can check, that algebras of the $D$-operad $F(X)/I$
are precisely differential graded associative algebras.

##  7 Hyperoperad for operads

We construct a quadratic operad $\mathbb{H}$, whose algebras are classical
symmetric non-unital Markl operads. This serves as a toy example. By the same
procedure, we describe a quadratic operad $\mathbb{H}_{\bO}$, whose algebras
are non-unital Markl operads in the operadic category $\bO$, and further a
quadratic operad $\mathbb{H}_{C}$, for a fixed small category $C$, whose
algebras are (non-unital) $C$-operads. We will work with the category
$\mathscr{V}=\it{Vect_{\mathbb{k}}}$, believing that the reader can easily
reformulate the construction into other monoidal categories.

### Classical non-unital Markl operads

We denote by $\Sigma$ the free symmetric monoidal category on one generator.
Its objects are natural numbers and morphisms are permutations. This will be
the category of colors for the operad $\mathbb{H}$. Let us begin with a
generating non-symmetric $\Sigma$-collection

${(\Sigma^{\it{op}})^{k}\times\Sigma}$${\it{Vect_{\mathbb{k}}},}$${X_{k}}$

defined by:

  * •

For any $m\geq 0,n\geq 1$, the space
${X_{2}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}n+m-1\\\
n\hskip
14.22636ptm\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$ is
freely generated by the set

$\\{*_{i}\,|\,1\leq i\leq
n\\}\times\Sigma_{n}\times\Sigma_{m}\times\Sigma_{n+m-1}.$

The generators will be written as schemes
${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi\\\
{}_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
and the actions of $\Sigma$ on individual inputs and output are given by

${X_{2}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\pi^{\prime}\\\
\sigma^{\prime}\tau^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi\\\
{}_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right):={}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi^{\prime}\pi\\\
{}_{i},\sigma\sigma^{\prime}\,\tau\tau^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

  * •

$X_{k}$ is trivial for $k\neq 2$.

Each generator
${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi\\\
{}_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
represents the composite

${P(n)\otimes P(m)}$${P(n+m-1)}$${P(n)\otimes
P(m)}$${P(n+m-1)}$${(-)\sigma\otimes(-)\tau}$${\circ_{i}}$${(-)\pi}$

of symmetric actions and the operation $\circ_{i}$ of a classical Markl operad
$P$. The generators are depicted in Figure 8.

Figure 8: Generators of the operad $\mathbb{H}$.

Next, we form the quotient $Q=X/\textit{Eq}$, where Eq is the equivalence
relation, generated by pairs

${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\mathbb{1}\\\
{}_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\sim{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
28.0pt\sigma\circ_{i}\tau\\\ {}_{\sigma(i)},\mathbb{1}\hskip
10.0pt\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$

for each $m\geq 0,1\geq i\geq n,\sigma\in\Sigma_{n}$ and $\tau\in\Sigma_{m}$
(cf. Figure 9). The permutation $\sigma\circ_{i}\tau$ is given by inserting
the permutation $\tau$ at the $i$-th place in $\sigma$.

Figure 9: Equivariance of the symbols $*_{i}$.

Each space
${Q_{2}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}n+m-1\\\
n\hskip
14.22636ptm\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$ is
thus generated by equivalence classes
${}\left[\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi\\\
{}_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right]$.
Let us denote $[*_{i}]$ $:=$
${}\left[\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\mathbb{1}\\\
{}_{i},\mathbb{1}\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right].$
Consider the free symmetric $\Sigma$-operad $F(Q)$ (cf. Example 5.7) and form
an operadic ideal As generated by the elements

$\displaystyle\circ_{1}[*_{j}]$
$\displaystyle-\tau([*_{j+k-1}]\circ_{1}[*_{i}]),$ (26)
$\displaystyle[*_{i}]\circ_{1}[*_{j}]$
$\displaystyle-[*_{j}]\circ_{2}[*_{i-j+1}],$
$\displaystyle[*_{i}]\circ_{1}[*_{j}]$
$\displaystyle-\tau([*_{j}]\circ_{1}[*_{i-m+1}]),$

for all $i,j,k,m,n$ as in (9). The symbol $\tau$ stands for the map

${{F(Q)}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}n+m+k-2\\\
n\hskip 14.22636ptm\hskip
14.22636ptk\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$${{F(Q)}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}n+m+k-2\\\
n\hskip 14.22636ptk\hskip
14.22636ptm\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),}$${{F(Q)}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
8.0pt\mathbb{1}\\\
\tau\,\mathbb{1}\,\mathbb{1}\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}$

which swaps the second and third input. An element of the second type is
depicted in Figure 10.

Figure 10: Parallel associativity identification.

###### Theorem B1 1

The category of algebras of the binary quadratic $\Sigma$-operad

$\mathbb{H}:=F(Q)/\textit{As}$

is isomorphic to the category of classical symmetric non-unital Markl operads.

The theorem follows directly from definitions and the construction of
$\mathbb{H}$. The data of an $\mathbb{H}$-algebra $(M,\alpha)$ consists of a
symmetric collection of vector spaces

${\Sigma}$${\it{Vect_{\mathbb{k}}}}$${M}$

and of an operad map

${\mathbb{H}}$${\it{End}_{M}.}$${\alpha}$

The map $\alpha$ interprets each class
$[*_{i}]\in{\mathbb{H}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}n+m-1\\\
n\hskip
14.22636ptm\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$ as a
map $\circ_{i}:=\alpha([*_{i}])$,

${M(n)\otimes M(m)}$${M(n+m-1).}$${\circ_{i}}$

Since $\alpha$ is a map of $\Sigma$-collections, we have

$\displaystyle\circ_{i}\cdot\big{(}(-)\sigma\otimes(-)\tau\big{)}$
$\displaystyle=\it{End}_{M}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\
\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\circ_{i})=\big{(}\it{End}_{M}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\
\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\cdot\alpha\big{)}([*_{i}])$
$\displaystyle=\big{(}\alpha\cdot{\mathbb{H}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\mathbb{1}\\\
\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\big{)}\big{(}{}\left[\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\mathbb{1}\\\
_{i},\mathbb{1}\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right]\big{)}=\alpha\big{(}{}\left[\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\mathbb{1}\\\
_{i},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right]\big{)}=\alpha\big{(}{}\left[\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
28.0pt\sigma\circ_{i}\tau\\\ _{\sigma(i)},\mathbb{1}\hskip
10.0pt\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right]\big{)}$
$\displaystyle=\big{(}\alpha\cdot{\mathbb{H}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\sigma\circ_{i}\tau\\\
\mathbb{1}\hskip
10.0pt\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\big{)}\big{(}{}\left[\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
28.0pt\mathbb{1}\\\
_{\sigma(i)},\mathbb{1}\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right]\big{)}=\it{End}_{M}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\sigma\circ_{i}\tau\\\
\mathbb{1}\hskip
10.0pt\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)(\circ_{\sigma(i)})$
$\displaystyle=\big{(}(-)(\sigma\circ_{i}\tau)\big{)}\cdot\circ_{\sigma(i)}.$

This shows that the operations $\circ_{i}$ are equivariant. The associativity
of the operations $\circ_{i}$ is ensured by the identifications (26) and the
fact, that $\alpha$ is an operad morphism. On the other hand, one can
reconstruct the map $\alpha$ from a non-unital operad structure on $M$.
Further, the morphisms of non-unital operads and of $\mathbb{H}$-algebras
agree.

### Non-unital Markl $\bO$-operads

The construction in the previous section generalises to the context of non-
unital Markl operads in operadic categories. We use freely the terminology of
[BM23a, BM23b].

Let $\bO$ denote an operadic category, which satisfies the axioms required for
the definition of Markl $\bO$-operads [BM23a, Section 6]. That is, we assume
that $\bO$ is a graded, factorizable operadic category in which all
quasibijections are invertible, the strong blow-up and unique ﬁber axioms are
fulfilled, and a morphism $f$ is an isomorphism if and only it is of grade 0.
We also assume that $\bO$ is small, but this is the case for the examples of
[BM23a, BM23b]. Let $\bO_{\it{iso}}$ denote the full subcategory of
isomorphisms of $\bO$. An elementary morphism $\phi\colon T\rightarrow S$ with
the non-trivial fiber $F$ over $i\in|S|$ will be written as $\phi\colon
F\triangleright_{i}T\rightarrow S$ or simply $\phi\colon F\triangleright
T\rightarrow S$, when the concrete $i$ is not important. We construct an
$\bO_{\it{iso}}^{\it{op}}$-operad $\mathbb{H}_{\bO}$ whose algebras are (non-
unital) Markl $\bO$-operads.

Let $Bq(\bO_{\it{iso}}^{\it{op}})$ denote the category $\bigoplus_{n\geq
0}(\bO_{\it{iso}})^{\times n}\times\bO_{\it{iso}}^{\it{op}}$ and
$Bq^{\textit{Disc}}(\bO_{\it{iso}}^{\it{op}})$ the discrete category of
objects of $Bq(\bO_{\it{iso}}^{\it{op}})$. We define a collection
$X^{\textit{Disc}}\colon
Bq^{\textit{Disc}}(\bO_{\it{iso}}^{\it{op}})\rightarrow\it{Vect_{\mathbb{k}}}$
by

  * •

${X^{\textit{Disc}}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\
S\,F\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$ is generated
by the set $\\{*_{\phi}\,|\,\phi\colon F\triangleright T\rightarrow S\text{ is
elementary in $\bO$}\\}$,

  * •

$X^{\textit{Disc}}$ is trivial otherwise.

We equip the generating collection with free
$\bO_{\it{iso}}^{\it{op}}$-actions on inputs and output, which is done using
Kan extensions. The left Kan extension $X=\textit{Lan}(X^{\textit{Disc}})$ of
the functor $X^{\textit{Disc}}$ along the inclusion functor

$Bq^{\textit{Disc}}(\bO_{\it{iso}}^{\it{op}})\subseteq
Bq(\bO_{\it{iso}}^{\it{op}})$

is then a non-symmetric $\bO_{\it{iso}}^{\it{op}}$-collection. Explicitly, $X$
is given by

$\displaystyle{X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T^{\prime}\\\
S^{\prime}\,F^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)=\big{\\{}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi\\\
_{\phi},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\,|\,$
$\displaystyle\tau\colon F^{\prime}\rightarrow F^{\prime\prime},\sigma\colon
S^{\prime}\rightarrow S^{\prime\prime},\pi\colon T^{\prime\prime}\rightarrow
T^{\prime}\text{ are isomorphisms of $\bO$,}$ (31) $\displaystyle\phi\colon
F^{\prime\prime}\triangleright T^{\prime\prime}\rightarrow
S^{\prime\prime}\text{ is elementary in $\bO$}\big{\\}},$

and the $\bO_{\it{iso}}$-actions are

${X}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\pi^{\prime}\\\
\sigma^{\prime}\,\tau^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi\\\
{}_{\phi},\sigma\,\tau\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)={}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\pi^{\prime}\pi\\\
{}_{\phi},\sigma\sigma^{\prime}\,\tau\tau^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

Recall from [BM23a, Definition 6.1] that for any square

${T^{\prime}}$${{T^{\prime\prime}}}$${S^{\prime}}$${{S^{\prime\prime}}}$${\phi^{\prime}}$${\phi^{\prime\prime}}$${\cong}$${\omega}$${\sigma}$${\sim}$

in $\bO$ with $\omega$ an isomorphism, $\sigma$ a quasibijection, and
$\phi^{\prime},\phi^{\prime\prime}$ elementary, there is an induced
isomorphism $\overline{\omega}$ between the unique non-trivial fibers
$F^{\prime}$ and $F^{\prime\prime}$ of $\phi^{\prime}$ and
$\phi^{\prime\prime}$. We form the quotient $Q=X/\textit{Eq}$, where Eq is an
equivalence relation generated by

${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
18.0pt\omega\\\
{}_{\phi^{\prime}},\mathbb{1}\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\sim{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\mathbb{1}\\\
{}_{\phi^{\prime\prime}},\sigma\,\,\overline{\omega}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$
(32)

for any $\sigma,\omega,\phi^{\prime},\phi^{\prime\prime}$ as above. Denote the
class of
${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}\hskip
15.0pt\mathbb{1}\\\
{}_{\phi},\mathbb{1}\,\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$
by $[*_{\phi}]$. We consider the free symmetric $\bO_{\it{iso}}$-operad
$F(Q)$, which we divide by the operadic ideal generated by two types of
associators:

Sequential associators. Let $\phi\colon F\triangleright_{j}T\rightarrow S$ and
$\psi\colon G\triangleright_{i}S\rightarrow P$ be two elementary maps in
$\bO$, such that $|\psi|(j)=i$. Their composite is an elementary map

$\xi=\psi\phi\colon H\triangleright_{i}T\rightarrow P$

and there is an induced elementary map $\phi_{i}\colon F\triangleright
H\rightarrow G$:

${F}$${H}$${G}$${F}$${T}$${S}$${P.}$${\phi}$${\psi}$${\psi\phi}$${\triangleright}$${\triangleright}$${\triangleright}$${\phi_{i}}$${\triangleright}$

We have

$[*_{\phi}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\ S\,F\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$ | $[*_{\psi}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}S\\\ P\,G\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$ | $[*_{\phi_{i}}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}H\\\ G\,F\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$ | $[*_{\xi}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\ P\,H\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$
---|---|---|---

and the composites $[*_{\phi}]\circ_{2}[*_{\psi}]$ and
$[*_{\xi}]\circ_{1}[*_{\phi_{i}}]$ are both elements of
${F(Q)}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\
P\,G\,F\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$ The
sequential associators are

$[*_{\phi}]\circ_{2}[*_{\psi}]-[*_{\xi}]\circ_{1}[*_{\phi_{i}}],$ (33)

for any $\phi$ and $\psi$ as above.

Parallel associators. Let $\psi^{\prime},\psi^{\prime\prime},\phi^{\prime},$
and $\phi^{\prime\prime}$ form a square of elementary morphisms with disjoint
fibers:

${{F^{\prime\prime}}}$${{G^{\prime}}}$${{F^{\prime}}}$${T}$${{P^{\prime}}}$${{G^{\prime\prime}}}$${{P^{\prime\prime}}}$${S.}$${\phi^{\prime}}$${\psi^{\prime}}$${\psi^{\prime\prime}}$${\phi^{\prime\prime}}$${\triangleright}$${\triangleright}$${\triangleright}$${\triangleright}$

It follows from the assumptions that $F^{\prime}=G^{\prime\prime}$ and
$F^{\prime\prime}=G^{\prime}$ (cf. Corollary 5.7 of [BM23a]), and we have

$[*_{\psi^{\prime}}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}P^{\prime}\\\ S\,F^{\prime\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$, | $[*_{\psi^{\prime\prime}}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}P^{\prime\prime}\\\ S\,F^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$ | $[*_{\phi^{\prime}}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\ P^{\prime}\,F^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$ | $[*_{\phi^{\prime\prime}}]\in{Q}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\ P^{\prime\prime}\,F^{\prime\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$
---|---|---|---

The composite $[*_{\phi^{\prime}}]\circ_{2}[*_{\psi^{\prime}}]$ is an element
of
$F(Q){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\
S\,F^{\prime\prime}\,F^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$
$[*_{\phi^{\prime\prime}}]\circ_{2}[*_{\psi^{\prime\prime}}]$ is an element of
$F(Q){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\
S\,F^{\prime}\,F^{\prime\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right),$
and hence $\tau([*_{\phi^{\prime\prime}}]\circ_{2}[*_{\psi^{\prime\prime}}])$
is an element of
$F(Q){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\
S\,F^{\prime\prime}\,F^{\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$
The parallel associators are

$[*_{\phi^{\prime}}]\circ_{2}[*_{\psi^{\prime}}]-\tau([*_{\phi^{\prime\prime}}]\circ_{2}[*_{\psi^{\prime\prime}}]),$
(34)

for any $\phi^{\prime},\phi^{\prime\prime},\psi^{\prime}$ and
$\psi^{\prime\prime}$ as above. The symbol $\tau$ stands for the isomorphism
which swaps the second and third input of
$F(Q){}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}T\\\
S\,F^{\prime}\,F^{\prime\prime}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$.

Finally, we define the binary quadratic $\bO_{\it{iso}}^{\it{op}}$-operad

$\mathbb{H}_{\bO}:=F(Q)/\textit{As},$

where As is the operadic ideal generated by elements (33) and (34).

###### Theorem B2 1

The category of algebras of the binary quadratic
$\bO_{\it{iso}}^{\it{op}}$-operad $\mathbb{H}_{\bO}$ is isomorphic to the
category of (non-unital) Markl $\bO$-operads in $\it{Vect_{\mathbb{k}}}$.

The proof is analogous to the proof of Theorem B1. An algebra of $\mathbb{H}$
is a functor

${\bO_{\it{iso}}^{\it{op}}}$${\it{Vect_{\mathbb{k}}},}$${M}$

together with structure maps

${M(S)\otimes M(F)}$${M(T)}$${\circ_{\phi}}$

for any elementary morphism $\phi\colon F\triangleright T\rightarrow S$. Due
to the identifications (32), (33), and (34), the maps $\circ_{\phi}$ are
equivariant and associative in the sense of (44),(46), and (47) of [BM23a].
Unital Markl $\bO$-operads can be described by the operad $\mathbb{H}_{\bO}$,
extended by generators

$\eta_{U}\in{\mathbb{H}}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}U\\\
\emptyset\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)$

for every chosen local terminal object $U$ of $\bO$, with suitable relations
corresponding to (48) of [BM23a].

######  Remark 7.1.

The Theorem B1 is a direct consequence of the Theorem B2 for
$\bO=\textit{Fin}$, the skeletal category of finite sets. In [BM23a], it is
shown that operadic categories of various kinds of graphs, such as rooted
trees or genus graded connected graphs, satisfy the assumptions given in the
beginning of this subsection. In this case, Markl $\bO$-operads are graph-
indexed operads. All sorts of operad-like structures and their strongly
homotopy versions can be encoded as algebras of Markl $\bO$-operads, cf.
[BMO23]. The Theorem B2 says, that the encoding hyperoperads are themselves
algebras of a binary quadratic $\bO_{\it{iso}}^{\it{op}}$-operad
$\mathbb{H}_{\bO}$. The Theorem B3 below is a separate result, since there is
no operadic category in the classical sense (i.e. with chosen local terminal
objects for each connected component), whose operads are $C$-operads.

###  $C$-operads

The key ingredient in the construction of the hyperoperad $\mathbb{H}_{C}$ is
the description of $C$-operads in terms of the operations $\circ_{i}^{f}$ of
Proposition 2.7. The category of colors for the hyperoperad $\mathbb{H}_{C}$
is

$Bq^{\Sigma}(C):=\bigoplus_{n\in\mathbb{N}}(C^{\it{op}})^{\times n}\times
C\times\Sigma_{n}.$

The objects of $Bq^{\Sigma}(C)$ are ’bouquets’

${}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\ \cdots\ c_{n}\
\sigma\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right).$

The hyperoperad $\mathbb{H}_{C}$ will be generated by symbols

$*^{f}_{i}\in{\mathbb{H}_{C}}\begin{pmatrix}{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\cdots d_{1}\cdots d_{m}\cdots c_{n}\
\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)\\\
{{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}c\\\
c_{1}\ \cdots\ c_{n}\
\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}\
{{}\left(\rule{0.0pt}{15.0pt}\right.\hskip-8.53581pt\begin{array}[]{c}d\\\
d_{1}\ \cdots\ d_{m}\
\mathbb{1}\end{array}\hskip-8.53581pt\left.\rule{0.0pt}{15.0pt}\right)}\end{pmatrix},$

with $f\colon d\to c_{i}$, equipped with free action of $(C^{\it{op}})^{\times
k}\times C\times\Sigma_{k}$ on inputs and output, for $k=n,m$ and $n+m-1$,
respectively. By the free actions we mean the left Kan extension along the
discrete subcategory inclusion functor, as in (31). We make the symbols
$C$-equivariant by dividing the generating collection by the relation (14) of
Proposition 2.7. The hyperoperad $\mathbb{H}_{C}$ is obtained as a free
operad, divided by the operadic ideal of $C$-associative relations (15).

###### Theorem B3 1

For a fixed small category $C$, the category of algebras of the binary
quadratic $Bq^{\Sigma}(C)$-operad $\mathbb{H}_{C}$, described above, is
isomorphic to the category of (non-unital) $C$-operads.

## References

  * [Batanin08] M.A. Batanin.  The Eckmann–Hilton argument and higher operads.  Advances in Mathematics, 217(1):334–385, 2008.
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  * [BM15] M.A. Batanin and M. Markl.  Operadic categories and duoidal Deligne’s conjecture.  Advances in Mathematics, 285:1630–1687, 2015.
  * [BM23a] M.A. Batanin and M. Markl.  Operadic categories as a natural environment for Koszul duality,  Compositionality, 5(3), 2023.
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