\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50](g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

Then we show the diagram

\[
\begin{tikzcd}
\arrow[dr, "i_x"]
\arrow[ddr, swap, "f", bend right]
\arrow[dl, swap, "i_y"]
\arrow[ddl, "g", bend left]
\\&
x \otimes y
\arrow[d, "h"]
\\&q
\end{tikzcd}
\]

commutes by the following calculations.

\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=blackdot] (y) at (0.5, 0.5) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.2, -0.25) {=};
\node [style=none] () at (1.7, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50] (g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=blackdot] (y) at (0.5, 0) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.2, -0.25) {=};
\node [style=none] () at (1.7, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 45] (y.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (0, 0.75) {};
\node [style=construct] (f) at (0, -0.25) {$f$};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (f.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (x) at (-0.5, 0.5) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.2, -0.25) {=};
\node [style=none] () at (1.7, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50] (g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (x) at (-0.5, 0) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.3, -0.25) {=};
\node [style=none] () at (1.8, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right = 45] (x.center) to (m.center);
\draw (y.center) to (g.center);
\draw [bend left = 50] (g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (0, 0.75) {};
\node [style=construct] (g) at (0, -0.25) {$g$};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (g.center);
\draw (g.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

Let $k \maps x \otimes y \to q$ be a map which makes that diagram commute. Then

\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (k) at (0, -0.125) {$h$};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.1, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (x.center) to (k.center);
\draw [bend left] (y.center) to (k.center);
\draw (k.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (f) at (-0.5, 0) {$f$};
\node [style=construct] (g) at (0.5, 0) {$g$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.25, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (x.center) to (f.center);
\draw (y.center) to (g.center);
\draw [bend right = 50] (f.center) to (m.center);
\draw [bend left = 50](g.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (-0.875, 0.75) {};
\node [style=blackdot] (2) at (-0.14, 0.5) {};
\node [style=blackdot] (3) at (0.14, 0.5) {};
\node [style=none] (4) at (0.9, 0.75) {};
\node [style=construct] (k1) at (-0.5, 0) {$k$};
\node [style=construct] (k2) at (0.5, 0) {$k$};
\node [style=blackdot] (m) at (0, -0.75) {};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.3, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right = 20] (1.center) to (k1);
\draw [bend left = 20] (2.center) to (k1);
\draw [bend right = 30] (3.center) to (k2.center);
\draw [bend left = 27] (4.center) to (k2.center);
\draw [bend right = 50] (k1.center) to (m.center);
\draw [bend left = 50] (k2.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (-0.875, 0.75) {};
\node [style=blackdot] (2) at (-0.14, 0.5) {};
\node [style=blackdot] (3) at (0.14, 0.5) {};
\node [style=none] (4) at (0.9, 0.75) {};
\node [style=blackdot] (k1) at (-0.5, 0) {};
\node [style=blackdot] (k2) at (0.5, 0) {};
\node [style=construct] (m) at (0, -0.65) {$k$};
\node [style=none] (q) at (0, -1.25) {};
\node [style=none] () at (1.3, 0) {=};
\node [style=none] () at (1.7, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right = 20] (1.center) to (k1);
\draw [bend left = 20] (2.center) to (k1);
\draw [bend right = 30] (3.center) to (k2.center);
\draw [bend left = 27] (4.center) to (k2.center);
\draw [bend right = 35] (k1.center) to (m.center);
\draw [bend left = 35] (k2.center) to (m.center);
\draw (m.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (x) at (-0.5, 0.75) {};
\node [style=none] (y) at (0.5, 0.75) {};
\node [style=construct] (k) at (0, -0.125) {$k$};
\node [style=none] (q) at (0, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (x.center) to (k.center);
\draw [bend left] (y.center) to (k.center);
\draw (k.center) to (q.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

Thus $h$ is the unique such map. This demonstrates $x \otimes y$ as the coproduct of $x$ and $y$.

There is a dual statement which characterizes cartesian monoidal categories, but in order to state it, we must first define comonoid.

A comonoid object in a monoidal category $\C$ is monoid in $\C\op$. Equivalently, a comonoid is an object $x \in \C$ equipped with a comultiplication map $\mu \maps x \to x \otimes x$ and a counit map $\varepsilon \maps x \to I$, which we express as

\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, 0.5) {};
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.5, -0.5) {};
\node [style=none] (2) at (0.5, -0.5) {};
\node [style=none] () at (1, 0) {,};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (1.center) to (r.center);
\draw [bend right] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.5, -0.5) {};
\node [style=none] (2) at (0.5, -0.5) {};
\node [style=none] (3) at (0, 0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

satisfying the following equations.

\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -1) {};
\node [style=none] (2) at (0.5, -1) {};
\node [style=none] (3) at (1, -1) {};
\node [style=none] (4) at (1, -0.5) {};
\node [style=none] (5) at (0.625, 0.5) {};
\node [style=blackdot] (b1) at (0.25, -0.5) {};
\node [style=blackdot] (b2) at (0.625, 0) {};
\node [style=none] () at (1.5, -0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (1.center) to (b1.center);
\draw [bend right] (2.center) to (b1.center);
\draw (3.center) to (4.center);
\draw [bend right] (4.center) to (b2.center);
\draw [bend left] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -1) {};
\node [style=none] (2) at (0.5, -1) {};
\node [style=none] (3) at (1, -1) {};
\node [style=none] (4) at (0, -0.5) {};
\node [style=none] (5) at (0.375, 0.5) {};
\node [style=blackdot] (b1) at (0.75, -0.5) {};
\node [style=blackdot] (b2) at (0.375, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw [bend left] (2.center) to (b1.center);
\draw [bend right] (3.center) to (b1.center);
\draw [bend left] (4.center) to (b2.center);
\draw [bend right] (b1.center) to (b2.center);
\draw (b2.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0.25, -0.75) {};
\node [style=blackdot] (2) at (1, -0.5) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, 0.5) {};
\node [style=none] () at (1.5, -0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (3.center);
\draw [bend left] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\node [style=none] () at (4, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=blackdot] (1) at (0.25, -0.5) {};
\node [style=none] (2) at (1, -0.75) {};
\node [style=blackdot] (3) at (0.625, 0) {};
\node [style=none] (4) at (0.625, 0.5) {};
\node [style=none] () at (1.5, -0.25) {=};
\node [style=none] () at (2, 0.5) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (3.center);
\draw [bend left] (1.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}\]

Let $\Comon(\C, \otimes)$ denote the category of comonoid objects in $\C$ and their homomorphisms.

A cocommutative comonoid is a comonoid for which the following equation holds.

\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, 0.5) {};
\node [style=blackdot] (r) at (0, 0) {};
\node [style=none] (1) at (-0.25, -0.5) {};
\node [style=none] (2) at (0.25, -0.5) {};
\node [style=none] () at (1, 0) {=};
\node [style=none] () at (1.5, 0) {}; % spacing
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left] (1.center) to (r.center);
\draw [bend right] (2.center) to (r.center);
\draw (r.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, 0.5) {};
\node [style=blackdot] (m) at (0, 0) {};
\node [style=none] (2) at (0, -0.55) {};
\node [style=none] (3) at (-0.25, -1) {};
\node [style=none] (4) at (0.25, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (m.center);
\draw [bend left = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend right = 45] (2.center) to (3.center);
\draw [bend right = 90, white, line width = 3, looseness = 1.5] (m.center) to (2.center);
\draw [bend left = 45, white, line width = 3] (2.center) to (4.center);
\draw [bend right = 90, looseness = 1.5] (m.center) to (2.center);
\draw [bend left = 45] (2.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

Let $\CoComon(\C, \otimes)$ denote the category of cocommutative comonoids in $\C$ and their homomorphisms.

A symmetric monoidal category is cartesian if and only if each object has a natural cocommutative comonoid structure.

This is dual to <ref>.

Let $(\C, \otimes, I)$ be a symmetric monoidal category. Then $\CMon(\C, \otimes)$ has a cocartesian monoidal structure given by $\otimes$, and $\CoComon(\C, \otimes)$ has a cartesian monoidal structure given by $\otimes$.

CHAPTER: MONOIDAL 2-CATEGORIES AND PSEUDOMONOIDS

There are many sources for the basic theory of 2-categories and bicategories <cit.>. Below we sketch some basic definitions and constructions regarding monoidal 2-categories, necessary for what follows; relevant references where explicit axioms can be found are <cit.>.

§ MONOIDAL 2-CATEGORIES

A monoidal 2-category $\K$ is a 2-category equipped with a pseudofunctor $\otimes \maps \K \times \K \to \K$ and a unit object $I \maps 1 \to \K$ which are associative and unital up to coherent equivalence. A lax monoidal pseudofunctor $\F \maps \K \to \L$ between monoidal 2-categories is a pseudofunctor equipped with pseudonatural transformations

\begin{equation}\label{eq:weakmonpseudo}
\begin{tikzcd}
\K \times \K
\arrow[r, "\F \times \F"]
\arrow[d, "\otimes_\K"']
& |[alias=doma]|
\L \times \L
\arrow[d, "\otimes_\L"]
\\
\K
\arrow[r, "\F"']
& \L
\arrow[Rightarrow, from=doma, to=coda, "\mu"', shorten >=.25in, shorten <=.25in]
\end{tikzcd}
\quad\quad
\begin{tikzcd}
\1\arrow[d, "I_{\K}"']
\arrow[dr, bend left, "I_{\L}"{name=doma}]
\\
\K
\arrow[r, "\F"']
& \L
\arrow[Rightarrow, from=doma, to=coda, "\mu_0"', shorten >=.1in, shorten <=.1in]
\end{tikzcd}
\end{equation}

with components $\mu_{a, b} \maps \F a \otimes \F b \to \F (a \otimes b)$, $\mu_0 \maps I \to \F I$, and invertible modifications

\begin{equation}\label{eq:omega}
\begin{tikzcd}
\L^3
\arrow[d, phantom, "\Downarrow{\scriptstyle\mu \times 1}"]
\arrow[r, "\otimes _\L \times 1"]
\L^2
\arrow[dd, phantom, "\Downarrow{\scriptstyle\mu}"]
\arrow[dr, "\otimes _\L"]
\L^3
\arrow[dd, phantom, "\Downarrow{\scriptstyle 1 \times \mu}"]
\arrow[r, "\otimes _\L \times 1"]
\arrow[dr, "1 \times \otimes _\L"description]
\L^2
\arrow[d, phantom, "{\scriptstyle\cong}"]
\arrow[dr, "\otimes _\L"]
\\
\K^3
\arrow[ur, "\F \times \F \times \F"]
\arrow[r, "\otimes _\K \times 1"]
\arrow[dr, "1 \times \otimes _\K", swap]
\K^2
\arrow[ur, "\F \times \F"description]
\arrow[dr, "\otimes _\K"description]
\L
\arrow[r, phantom, "\stackrel{\omega}{\Rrightarrow}"]
\K^3
\arrow[ur, "\F \times \F \times \F"]
\arrow[dr, "1 \times \otimes _\K", swap]
\L^2
\arrow[d, phantom, "\Downarrow{\scriptstyle\mu}"]
\arrow[r, "\otimes _\L"]
\L
\\&
\K^2
\arrow[u, phantom, "{\scriptstyle\cong}"]
\arrow[r, "\otimes _\K", swap]
\K
\arrow[ur, "\F", swap]
\K^2
\arrow[ur, "\F \times \F"description]
\arrow[r, "\otimes _\K", swap]
\K
\arrow[ur, "\F", swap]
\end{tikzcd}
\end{equation}

[column sep=.25in]
[dr, "1×I"']
[rr, "×I"name=doma]
[rrd, bend right=80, "1"', "≅"]
[rrr, bend left=30, ""]
[rrr, phantom, bend left=15, "≅"description]

Ł×Ł[r, "⊗_Ł"']

Ł[Rightarrow, from=doma, to=coda, "1×μ_0  "', shorten <=.5em, shorten >=.5em]

×[r, "⊗_"']
[ur, "×"description]
[urr, phantom, "⇓μ"description]

[ur, ""']

[column sep=.25in]

[rr, bend left, ""][rr, bend right, ""'][rr, phantom, "⇓1"description]        Ł

[column sep=.25in]
[dr, "I×1"']
[rr, "I×"name=doma]
[rrd, bend right=80, "1"', "≅"]
[rrr, bend left=30, ""]
[rrr, phantom, bend left=15, "≅"description]

Ł×Ł[r, "⊗_Ł"']

Ł[Rightarrow, from=doma, to=coda, "μ_0×1  "', shorten <=.5em, shorten >=.5em]

×[r, "⊗_"']
[ur, "×"description]
[urr, phantom, "⇓μ"description]

[ur, ""']

subject to coherence conditions which can be found in Definition 2 in <cit.>. A monoidal pseudonatural transformation $\tau \maps \F \Rightarrow \G$ between two lax monoidal pseudofunctors $(\F, \mu, \mu_0)$ and $(\G, \nu, \nu_0)$ is a pseudonatural transformation equipped with two invertible modifications

\begin{equation}\label{eq:monpseudonat}
\begin{tikzcd}[column sep = .2in, row sep = .1in]
\K \times \K
\arrow[rr, bend left, "\F \times \F"]
\arrow[rr, bend right, "\G \times \G"']
\arrow[dd, "\otimes "']
\arrow[ddrr, phantom, bend right = 10, "\Downarrow{\scriptstyle \nu}"']
\arrow[rr, phantom, description, "\Downarrow{\scriptstyle \tau \times \tau}"]
\L \times \L
\arrow[dd, "\otimes "]
\K \times \K
\arrow[rr, "\F \times \F"]
\arrow[dd, "\otimes "']
\arrow[ddrr, phantom, bend left=10, "\Downarrow{\scriptstyle \mu}"]
\L \times \L
\arrow[dd, "\otimes "]
\\&&&
\stackrel{u}{\Rrightarrow}
\K
\arrow[rr, "\G"']
\L
\K
\arrow[rr, bend left, "\F"]
\arrow[rr, bend right, "\G"']
\arrow[rr, phantom, description, "\Downarrow{\scriptstyle\tau}"]
\L
\end{tikzcd}
\end{equation}

[row sep = .1in, column sep = .3in]

[ddrr, phantom, "⇓ν_0" description]
[rr, "I_Ł"]
[dd, "1_"']

[ddrr, phantom, near start, "⇓μ_0" description]
[rr, "I_Ł"]
[dd, "I_"']

[uurr, bend right = 40,

[uurr, "" description]
[uurr, bend right=60, ""']
[uurr, phantom, bend right, "⇓τ" description]

that consist of natural isomorphisms with components

\begin{equation}\label{eq:monpseudocomponents}
u_{a, b}\maps \nu_{a, b} \circ (\tau_a \otimes \tau_b) \xrightarrow{\sim} \tau_{a \otimes b} \circ \mu_{a, b},
\quad
u_0\maps \nu_0 \xrightarrow{\sim}\tau_I \circ \mu_0
\end{equation}

satisfying coherence conditions which can be found in <cit.>.

The above notions of course generalize those of an ordinary monoidal category, lax monoidal functor and monoidal natural transformation. However, in our higher dimensional setting, there is now room for a structure not present for monoidal 1-categories.

A monoidal modification
between two monoidal pseudonatural transformations $(\tau, u, u_0)$ and $(\sigma, v, v_0)$ is a modification

[rr, bend left=40, "", ""'name = F]
[rr, bend right=40, ""', ""name = G]
[rr, phantom, "m⇛"description]

Ł[from = F, to = G, Rightarrow, "τ"', bend right=50]
[from = F, to = G, Rightarrow, "σ", bend left=50]

which consists of pseudonatural transformations $m_a\maps \tau_a\Rightarrow\sigma_a$ compatible with the monoidal structures, in the sense that

\begin{equation}\label{eq:monoidalmodaxioms}
\begin{tikzcd}[column sep=.27in]
\G a \otimes \G b
\arrow[dr, bend left, "\nu_{a, b}"]
\G a \otimes \G b
\arrow[dr, bend left, "\nu_{a, b}"]
\\
\F a\otimes \F b
\arrow[ur, bend left, "\sigma_a\otimes \sigma_b"]
\arrow[dr, bend right, "\mu_{a, b}"']
\arrow[rr, phantom, near start, "\Downarrow{\scriptstyle v_{a, b}}"description]
\G (x\otimes y)
\arrow[r, phantom, "="description]
\F a\otimes \F b
\arrow[dr, bend right, "\mu_{a, b}"']
\arrow[ur, bend right, "\sigma_a\otimes \sigma_b"']
\arrow[rr, phantom, near end, "\Downarrow{\scriptstyle u_{a, b}}"description]
\arrow[ur, bend left, "\tau_a\otimes\tau_b"]
\arrow[ur, phantom, "\Downarrow{\scriptstyle m_x\otimes m_y}"description]
\G (a\otimes b)
\\
\F(a\otimes b)
\arrow[ur, bend left, "\tau_{a\otimes b}"]
\arrow[ur, bend right, "\sigma_{a\otimes b}"']
\arrow[ur, phantom, "\Downarrow{\scriptstyle m_{a\otimes b}}"description]
\F(a\otimes b)
\arrow[ur, bend right, "\tau_{a\otimes b}"']
\end{tikzcd}
\end{equation}

[row sep=.2in]
[dr, bend right, "μ_0"']
[rr, bend left, "ν_0"]
[rr, phantom, near start, "⇓v_0"description]

[r, phantom, "="description]

[rr, phantom, "⇓u_0"description]
[dr, bend right, "μ_0"']
[rr, bend left, "ν_0"]

[ur, bend left, "τ_I"]
[ur, bend right, "σ_I"']
[ur, phantom, "⇓m_I"description]

[ur, bend right, "τ_I"']

For any monoidal 2-categories $\K, \L$ there are 2-categories $\MonTCat_\pse(\K, \L)$ denoted by $\namedcat{WMonHom}(\K, \L)$ in <cit.> for bicategories. If we take lax monoidal 2-functors and monoidal 2-transformations, the corresponding sub-2-category is denoted by $\MonTCat(\K, \L)$.

§ PSEUDOMONOIDS

A pseudomonoid in a monoidal 2-category $(\K, \otimes , I)$ is an object $a$ equipped with multiplication $m \maps a\otimes a\to a$, unit $j \maps I \to a$, and invertible 2-cells

\begin{equation}
\label{alphalambdarho}
\begin{tikzcd}
a \otimes a \otimes a
\arrow[r, "1 \otimes m"]
\arrow[d, "m \otimes 1"']
\arrow[dr, phantom, "\scriptstyle \stackrel{\assoc}{\cong}"description]
a \otimes a \arrow[d, "m"] \arrow[d, "m"]
& a \otimes I \arrow[r, "1 \otimes j"] \arrow[dr, "\sim"']
& a \otimes a \arrow[d, "m", "\stackrel{\lambda}{\cong}\quad"'near start, "\quad\;\stackrel{\rho}{\cong}"near start]
& I \otimes a \arrow[l, "j \otimes 1"']
\arrow[dl, "\sim"]  \\
a \otimes a\arrow[r, "m"']
& a && a &
\end{tikzcd}
\end{equation}

expressing associativity and unitality up to isomorphism, that satisfy appropriate coherence conditions. A lax morphism between pseudomonoids $a, b$ is a 1-cell $f\maps a\to b$ equipped with 2-cells

\begin{equation}\label{eq:laxmorphism}
\begin{tikzcd}[row sep=.5in, column sep=.5in]
a \otimes a\arrow[d, "m"']\arrow[r, "f\otimes f"] & |[alias=doma]| b\otimes b\arrow[d, "m"] \\
|[alias=coda]| a\arrow[r, "f"'] & b
\arrow[Rightarrow, from=doma, to=coda, "\phi"', shorten >=.35in, shorten <=.35in]
\end{tikzcd}
\qquad
\begin{tikzcd}[row sep=.5in, column sep=.5in]
I\arrow[d, "j"']\arrow[dr, bend left, "j"{name=doma}] \\
|[alias=coda]| a\arrow[r, "f"'] & b
\arrow[Rightarrow, from=doma, to=coda, "\phi_0"', shorten >=.15in, shorten <=.15in]
\end{tikzcd}
\end{equation}

such that the following conditions hold:

\begin{equation}\label{eq:axiomslaxmorphism}
\adjustbox{scale=.9, center}{
\begin{tikzcd}[column sep=.3in]
& b\otimes b\otimes b\arrow[r, "m\otimes1"]\arrow[d, phantom, "\Downarrow{\scriptstyle\phi\otimes 1_f}"description] & b\otimes b\otimes a\arrow[dr, "m"] & \\
a\otimes a\otimes a\arrow[r, "m\otimes 1"]\arrow[dr, "1\otimes m"']\arrow[ur, "f\otimes f\otimes f"] &
a\otimes a\arrow[dr, "m"]\arrow[ur, "f\otimes f"']\arrow[rr, phantom, "\Downarrow{\scriptstyle\phi}"description]
\arrow[d, phantom, "\stackrel{\assoc}{\cong}"description] && b \\
& a\otimes a\arrow[r, "m"'] & a\arrow[ur, "f"']
\end{tikzcd}
\begin{tikzcd}[column sep=.3in]
& b\otimes b\otimes b\arrow[r, "m\otimes1"]\arrow[dr, "1\otimes m"']\arrow[dd, phantom, "\Downarrow{\scriptstyle1_f\otimes \phi}"description] &
b\otimes b\arrow[dr, "m"]\arrow[d, phantom, "\stackrel{\alpha}{\cong}"description] & \\
a\otimes a\otimes a\arrow[ur, "f\otimes f\otimes f"]\arrow[dr, "1\otimes m"'] & & b\otimes b\arrow[r, "m"]
\arrow[d, phantom, "\Downarrow{\scriptstyle\phi}"description] & b \\
& a\otimes a\arrow[ur, "f\otimes f"]\arrow[r, "m"'] & b\otimes b\arrow[ur, "f"']
\end{tikzcd}
\end{equation}

scale=.9, center
[column sep=.25in]
a≅a⊗I[dr, "1⊗j"'][rr, "f⊗j"name=doma][rrd, bend right=60, "1_a"', "λ≅"]
[rrr, bend left=30, "f"][rrr, phantom, bend left=15, "1_f⊗λ≅"description]        b⊗b[r, "m"']     b
[Rightarrow, from=doma, to=coda, "1_f⊗ϕ_0  "', shorten <=.5em, shorten >=.5em]

|[alias=coda]|a⊗a[r, "m"'][ur, "f⊗f"description][urr, phantom, "⇓ϕ"description]
a[ur, "f"']

[column sep=.25in]

a[rr, bend left, "f"][rr, bend right, "f"'][rr, phantom, "⇓1_f"description]        b

[column sep=.25in]
a≅I⊗a[dr, "j⊗1"'][rr, "j⊗1"name=doma][rrd, bend right=60, "1_a"', "ρ≅"]
[rrr, bend left=30, "f"][rrr, phantom, bend left=15, "ρ⊗1_f≅"description]        b⊗b[r, "m"']     b
[Rightarrow, from=doma, to=coda, "ϕ_0⊗1_f  "', shorten <=.5em, shorten >=.5em]

|[alias=coda]|a⊗a[r, "m"'][ur, "f⊗f"description][urr, phantom, "⇓ϕ"description]
a[ur, "f"']

If $(f, \phi, \phi_0)$ and $(g, \psi, \psi_0)$ are two lax morphisms between pseudomonoids $a$ and $b$, a 2-cell between them $\sigma \maps f \Rightarrow g$ in $\K$ which is compatible with multiplications and units, in the sense that

\begin{equation}\label{monoidal2cell}
\begin{tikzcd}[row sep=.15in]
& b\otimes b\arrow[dr, bend left=20, "m"] & \\
a\otimes a\arrow[ur, bend left, "f\otimes f"]\arrow[ur, bend right, "g\otimes g"']\arrow[dr, bend right=20, "m"']
\arrow[ur, phantom, "\Downarrow{\scriptstyle\sigma\otimes\sigma}"description]
\arrow[rr, phantom, "{\scriptstyle\psi}\Downarrow"{description, near end}] && b \\
& a\arrow[ur, bend right=20, "g"'] &
\end{tikzcd}
\quad=\quad
\begin{tikzcd}[row sep=.15in]
& b\otimes b\arrow[rd, bend left=20, "m"] & \\
a\otimes a\arrow[ur, bend left=20, "f\otimes f"]\arrow[dr, bend right=20, "m"']
\arrow[rr, phantom, "{\scriptstyle\phi}\Downarrow"{description, near start}] && b \\
& a \arrow[ur, phantom, "\Downarrow{\scriptstyle\sigma}"description]
\arrow[ur, bend left, "f"]\arrow[ur, bend right, "g"'] \\
\end{tikzcd}
\end{equation}

[row sep=.15in, column sep=.6in]
I[rr, bend left, "j"][dr, bend right=20, "j"'][rr, phantom, "ϕ_0⇓"description, near start]        b

a[ur, bend left, "f"][ur, bend right, "g"'][ur, phantom, "⇓σ"description]

=  [row sep=.15in, column sep=.6in]
I[dr, bend right=20, "j"'][rr, bend left, "j"][rr, phantom, "⇓ψ_0"description]        b

a[ur, bend right=20, "g"']

We obtain a 2-category $\PsMon_\lax(\K)$ for any monoidal 2-category $\K$, which is sometimes
denoted by $\Mon(\K)$ <cit.>. By changing the direction of the 2-cells in <ref> and the rest of the axioms appropriately, or asking for them to be invertible, we have 2-categories $\PsMon_\opl(\K)$ and $\PsMon(\K)$ of pseudomonoids with oplax or (strong) morphisms between them.

The prototypical example is that of the monoidal 2-category $\K = (\Cat, \times, \1)$ of categories, functors, and natural transformations with the cartesian product of categories and the unit category with a unique object and arrow. A pseudomonoid in $(\Cat, \times, \1)$ is a monoidal category, a lax (resp. oplax, strong) morphism between two of these is precisely a lax (resp. oplax, strong) monoidal functor, and a 2-cell is a monoidal natural transformation. Therefore we obtain the well-known 2-categories $\Mon\Cat_\lax$, $\Mon\Cat_\opl$ and $\Mon\Cat$.

There is an evident similarity between the structures defined above, e.g. <ref> and <ref>, or <ref> and <ref>. This is due to the fact that monoidal 2-categories, lax monoidal pseudofunctors and monoidal pseudonatural transformations are themselves appropriate pseudomonoid-related notions in a higher level; we do not get into such details, as they are not pertinent to the present work.

For our purposes, we are interested in a different observation: any pseudomonoid $a$ in a monoidal 2-category $\K$ can in fact be expressed as a lax monoidal normal pseudofunctor $A \maps \1 \to \K$ with $A(*) = a$, namely one where $A(1_*)$ is equal to $1_a$. Moreover, a monoidal pseudonatural transformation $\tau\maps A\Rightarrow B\maps \1\to\K$ bijectively corresponds to a strong morphism between the pseudomonoids $a$ and $b$, and similarly for monoidal modifications and 2-cells. Since every pseudofunctor is equivalent to a normal one, the 2-category of pseudomonoids $\PsMon(\K)$ can be equivalently viewed as $\MonTCat_{\pse}(\1, \K)$, the 2-category of lax monoidal pseudofunctors $\1\to\K$, monoidal pseudonatural transformations and monoidal modifications.

As was already shown in <cit.>, any lax monoidal 2-functor $\F\maps \K\to\L$ takes pseudomonoids to pseudomonoids, and in fact
<cit.> there is a functor $\PsMon(\F)$ that commutes with the respective forgetful functors

[column sep=.6in]
()[r, "()"][d]     (Ł)[d]

[r, ""']     Ł.

Based on the above, and since every pseudofunctor from $\1$ into a 2-category trivially preserves composition on the nose and every pseudonatural transformation is really 2-natural, we can define a hom-2-functor that clarifies these assignments.

There is a 2-functor

\begin{equation}
\label{eq:PsMon}
\PsMon(-) \simeq \MonTCat_{\pse}(\1, -) \maps \MonTCat \to \TCat
\end{equation}

which maps a monoidal 2-category to its 2-category of pseudomonoids, strong morphisms and 2-cells between them.

The theory in <cit.> extends the above definitions to the case of braided and symmetric pseudomonoids in braided and symmetric monoidal 2-categories. Briefly recall that a braiding for $(\K, \otimes, I)$ is a pseudonatural equivalence with components $\braid_{a, b} \maps a \otimes b \to b \otimes a$ and invertible modifications, whereas a syllepsis is an invertible modification

\[
a \otimes b \xrightarrow{1} a \otimes b \Rrightarrow a \otimes b \xrightarrow{\braid_{a, b}} b \otimes a \xrightarrow{\braid_{b, a}} a \otimes b
\]

which is called symmetry if it satisfies extra axioms. With the appropriate notions of braided and symmetric lax monoidal pseudofunctors and monoidal pseudonatural transformations (and usual monoidal modifications), we have 3-categories $\BrMonTCat_{\pse}$ and $\SymMonTCat_{\pse}$. Indicatively, a lax monoidal pseudofunctor comes equipped an invertible modification with components

\begin{equation}\label{eq:brweakmonpseudo}
\begin{tikzcd}
\F a \otimes \F b
\arrow[r, "\mu_{a, b}"]
\arrow[d, "\braid_{\F a, \F b}"']
\arrow[dr, phantom, "\Downarrow{\scriptstyle v_{a, b}}" description]
\F b \otimes \F a
\arrow[d, "\F(\braid_{a, b})"]
\\
\F b \times \F a
\arrow[r, "\mu_{b, a}"']
\F(b\otimes a)
\end{tikzcd}
\end{equation}

As earlier, there exist 2-categories of braided and symmetric pseudomonoids with strong morphisms between them, expressed as
\[
\BrPsMon(\K)=\BrMonTCat_{(\pse)}(\1, \K)
\]

\[
\SymPsMon(\K)=\SymMonTCat_{(\pse)}(\1, \K).
\]

There are 2-functors

→,   →
which map a braided or symmetric monoidal 2-category to its 2-category of braided or symmetric pseudomonoids.

Finally, recall the notion of a monoidal 2-equivalence arising as the equivalence internal to the 2-category $\MonTCat$.

A monoidal 2-equivalence is a 2-equivalence $\F\maps \K \simeq \L \maps \G$ where both 2-functors
are lax monoidal, and the 2-natural isomorphisms $1_\K \cong \F \G$, $\G \F \cong 1_\L$ are monoidal. Similarly for braided and symmetric monoidal 2-equivalences.

As is the case for any 2-functor between 2-categories, $\PsMon$ as well as $\BrPsMon$ and $\SymPsMon$ map equivalences
to equivalences.

Any monoidal 2-equivalence $\K\simeq\L$ induces a 2-equivalence between the respective 2-categories
of pseudomonoids $\PsMon(\K)\simeq\PsMon(\L)$. Similarly any braided or symmetric monoidal 2-equivalence induces $\BrPsMon(\K)\simeq\BrPsMon(\L)$ or $\SymPsMon(\K)\simeq\SymPsMon(\L)$.

CHAPTER: FIBRATIONS AND INDEXED CATEGORIES

We recall some basic facts and constructions from the theory of fibrations and indexed categories, as well as the equivalence between them via the Grothendieck construction. Several indicative references for the general theory are <cit.>.

§ FIBRATIONS

Consider a functor $P \maps \A \to \X$. A morphism $\phi \maps a \to b$ in $\A$ over a morphism $f = P(\phi) \maps x \to y$ in $\X$ is called cartesian if and only if, for all $g \maps x' \to x$ in $\X$ and $\theta \maps a' \to b$ in $\A$ with $P \theta = f \circ g$, there exists a unique arrow $\psi \maps a' \to a$ such that $P \psi = g$ and $\theta = \phi \circ \psi$:

\begin{equation}
\begin{tikzcd}[column sep = huge]
\arrow[drr, "\theta"]
\arrow[dr, dashed, swap, "\exists!\psi"]
\arrow[dd, dotted, bend right]
\\&
\arrow[r, swap, "\phi"]
\arrow[dd, dotted, bend right]
\arrow[dd, dotted, bend right]
\text{in }\A
\\
\arrow[drr, "f \circ g = P \theta"]
\arrow[dr, swap, "g"]
\\&
\arrow[r, swap, "f = P \phi"]
\text{in }\X
\end{tikzcd}
\end{equation}

For $x \in \Ob\X$, the fibre of $P$ over $x$ written $\A_x$, is the subcategory of $\A$ which consists of objects $a$ such that $P(a) = x$ and morphisms $\phi$ with $P(\phi) = 1_x$, called vertical morphisms. The functor $P \maps \A \to \X$ is called a fibration if and only if, for all $f \maps x \to y$ in $\X$ and $b\in\A_Y$, there is a cartesian morphism $\phi$ with codomain $b$ above $f$; it is called a cartesian lifting of $f$ to $b$. The category $\X$ is then called the base of the fibration, and $\A $ its total category.

Dually, the functor $U \maps \C \to \X$ is an opfibration if $U^\mathrm{op}$ is a fibration, i.e. for every $c \in \C _x$ and $h \maps x \to y$ in $\X$, there is a cocartesian morphism with domain $c$ above $h$, the cocartesian lifting of $h$ to $c$ with the dual universal property:

[column sep = huge]

[dd, dotted, bend right]

[urr, "γ"]
[r, swap, "β"]
[dd, dotted, bend right]

[ur, dashed, swap, "∃! δ"]
[dd, dotted, bend right]

[urr, "k ∘h = U γ"]
[r, swap, "h = U β"]

[ur, swap, "k"]

A bifibration is a functor which is both a fibration and opfibration.

If $P\maps \A \to \X$ is a fibration, assuming the axiom of choice we may select a cartesian arrow over each $f\maps x \to y$ in $\X$ and $b \in \A_y$, denoted by $\Cart(f, b) \maps f^*(b) \to b$. Such a choice of cartesian liftings is called a cleavage for $P$, which is then called a cloven fibration; any fibration is henceforth assumed to be cloven. Dually, if $U$ is an opfibration, for any $c \in \C_x$ and $h \maps x \to y$ in $\X$ we can choose a cocartesian lifting of $h$ to $c$, $\Cocart(h,c)\maps c \to h_!(c)$. The choice of (co)cartesian liftings in an (op)fibration induces a so-called reindexing functor between the fibre categories

\begin{equation}\label{reindexing}
f^*\maps \A _y \to \A _x\quad\textrm{ and }\quad h_! \maps \C _x \to \C _y
\end{equation}

respectively, for each morphism $f\maps x \to y$ and $h \maps x \to y$ in the base category.
It can be verified by the (co)cartesian universal property that $1_{\A _x}\cong(1_x)^*$
and that for composable morphism in the base category, $g^*\circ f^*\cong(g\circ f)^* $, as well as $(1_x)_!\cong1_{\C_x}$ and $(k\circ h)_!\cong k_!\circ h_!$. If these isomorphisms are equalities, we have the notion of a split (op)fibration.

A fibred 1-cell $(H,F) \maps P \to Q$ between fibrations $P \maps \A \to \X$ and $Q \maps \B \to \Y$ is given by a commutative square of functors and categories

\begin{equation}
\label{commutativefibredcell}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, "H"]
\arrow[d, swap, "P"]
\B
\arrow[d, "Q"]
\\
\X
\arrow[rr , "F", swap]
\Y
\end{tikzcd}
\end{equation}

where the top $H$ preserves cartesian liftings, meaning that if $\phi$ is $P$-cartesian, then $H\phi$ is $Q$-cartesian. In particular, when $P$ and $Q$ are fibrations over the same base category, we may consider fibred 1-cells of the form $(H,1_{\X})$ displayed as

\begin{equation}
\label{eq:fibredfunctor}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, "H"]
\arrow[dr, swap, "P"]
\B
\arrow[dl, "Q"]
\\&
\X
\end{tikzcd}
\end{equation}

and $H$ is then called a fibred functor. Dually, we have the notion of an opfibred 1-cell and opfibred functor. Notice that any such (op)fibred 1-cell induces functors between the fibres, by commutativity of <ref>:

\begin{equation}\label{eq:functorbetweenfibres}
H_{x} \maps \A_x\longrightarrow\B_{Fx}
\end{equation}

A fibred 2-cell between fibred 1-cells $(H,F)$ and $(K,G)$ is a pair of natural transformations ($\braid\maps H\Rightarrow K,\alpha\maps F\Rightarrow G$) with $\braid$ above $\alpha$, i.e. $Q(\braid_a)=\alpha_{Pa}$ for all $a\in\A $, displayed as

\begin{equation}
\label{eq:fibred2cell}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, bend left, "H"]
\arrow[rr, phantom, "\Downarrow \beta"]
\arrow[rr, bend right, swap, "K"]
\arrow[d, swap, "P"]
\B
\arrow[d, "Q"]
\\
\X
\arrow[rr, bend left, "F"]
\arrow[rr, phantom, "\Downarrow \alpha"]
\arrow[rr, bend right, swap, "G"]
\Y
\end{tikzcd}
\end{equation}

A fibred natural transformation is of the form $(\braid,1_{1_{\X}})\maps(H,1_{\X})\Rightarrow(K,1_\X)$

\begin{equation}\label{eq:fibrednaturaltrans}
\begin{tikzcd}[row sep = huge]
\A
\arrow[rr, bend left, "H"]
\arrow[rr, phantom, "\Downarrow \beta"]
\arrow[rr, bend right, swap, "K"]
\arrow[dr, swap, "P"]
\B
\arrow[dl, "Q"]
\\&
\X
\end{tikzcd}
\end{equation}

Dually, we have the notion of an opfibred 2-cell and opfibred natural transformation between opfibred 1-cells and functors respectively.

We thus obtain a 2-category $\Fib$ of fibrations over arbitrary base categories, fibred 1-cells and fibred 2-cells. There is also a 2-category $\Fib(\X)$ of fibrations over a fixed base category $\X$, fibred functors and fibred natural transformations. Dually, we have the 2-categories $\OpFib$ and $\OpFib(\X)$. Moreover, we also have 2-categories $\Fib_\mathrm{sp}$ and $\OpFib_\mathrm{sp}$ of split (op)fibrations, and (op)fibred 1-cells that preserve the cartesian liftings `on the nose'.

Notice that $\Fib$ and $\OpFib$ are both sub-2-categories of $\Cat^\2 = [\2, \Cat]$, the arrow 2-category of $\Cat$. Similarly, $\Fib(\X)$ and $\OpFib(\X)$ are sub-2-categories of $\Cat/\X$, the slice 2-category of functors into $\X$. Due to that, both these (1-)categories form fibrations themselves. Explicitly, the functor $\cod \maps \Fib \to \Cat$ which maps a fibration to its base is a fibration, with fibres $\Fib(\X)$ and cartesian liftings pullbacks along fibrations. In fact, it is a 2-fibration <cit.>.

§ INDEXED CATEGORIES

We now turn to the world of indexed categories. Given an ordinary category $\X$, an $\X$-indexed category is a pseudofunctor
\[\M \maps \X\op \to \Cat\]
where $\X$ is viewed as a 2-category with trivial 2-cells; it comes with natural isomorphisms $\delta_{g,f} \maps (\M g) \circ (\M f) \xrightarrow\sim  \M(g \circ f)$ and $\gamma_x \maps 1_{\M x} \xrightarrow\sim \M (1_x)$ for every $x\in\X$ and composable morphisms $f$ and $g$, satisfying coherence axioms.
Dually, an $\X$-opindexed category is an $\X\op$-indexed category, i.e. a pseudofunctor $\X \to \Cat$.
If an (op)indexed category strictly preserves composition, i.e. is a (2-)functor, then it is called strict.

An indexed $1$-cell $(F, \tau) \maps \M \to \psN$ between indexed categories $\M \maps \X\op \to \Cat$ and $\psN \maps \Y\op \to \Cat$ consists of an ordinary functor $F \maps \X \to \Y$ along with a pseudonatural transformation $\tau \maps \M \Rightarrow \psN \circ F\op$

\begin{equation}\label{eq:indexed1cell}
\begin{tikzcd}[column sep=.7in,row sep=.2in]
\X\op
\arrow[dr, "\M"]
\arrow[dd, "F\op"']
\\
\arrow[r, phantom, "\Downarrow{\scriptstyle\tau}"description]
\Cat
\\
\Y\op \arrow[ur, "\psN"']
\end{tikzcd}
\end{equation}

with components functors $\tau_x \maps \M x \to \psN Fx$, equipped with coherent natural isomorphisms $\tau_f \maps (\psN Ff) \circ \tau_x \xrightarrow{\sim} \tau_y \circ (\M f)$ for any $f \maps x \to y$ in $\X$.
For indexed categories with the same base, we may consider indexed 1-cells of the form $(1_\X, \tau)$

\begin{equation}\label{eq:ifun}
\begin{tikzcd}[column sep=.7in]
\X\op
\arrow[r, bend left, "\M"]
\arrow[r, bend right, swap, "\psN"]
\arrow[r, phantom, "\Downarrow \scriptstyle \tau"]
\Cat
\end{tikzcd}
\end{equation}

which are called indexed functors. Dually, we have the notion of an opindexed 1-cell and opindexed functor.

An indexed 2-cell $(\alpha,m)$ between indexed 1-cells $(F, \tau)$ and $(G, \sigma)$, pictured as

[column sep=.6in,row sep=.4in]
[drr, ""]
[drr, ""name = M, swap, pos = 0.5]
[dd,bend right=40,"F" description]
[dd, "G"description, bend left=40]

[urr,""name = H, pos = 0.4]
[from = M, to = H, Rightarrow, "σ", pos = 0.4, bend left=45]
[from = M, to = H, Rightarrow, "τ", pos = 0.56, bend right=45,swap]
[from = M, to = H, phantom, "m⇛"]

consists of an ordinary natural transformation $\alpha \maps F \Rightarrow G$ and a modification $m$

\begin{equation}\label{eq:indexed2cell}
\begin{tikzcd}[column sep=.5in,row sep=.2in]
\X\op
\arrow[rr,bend left,"\M"]
\arrow[dr,bend right=10,"F\op"']
\arrow[rr,phantom,"\Downarrow{\scriptstyle \tau}"description]
\Cat \arrow[r,phantom,"\stackrel{m}{\Rrightarrow}"description]
\X\op
\arrow[rr,bend left=30,"\M"]
\arrow[dr,bend left=35,"G\op"]
\arrow[dr,bend right=35,"F\op"']
\arrow[rr, phantom, near end, "\Downarrow{\scriptstyle \sigma}"description]
\arrow[dr,phantom, "\Downarrow{\scriptstyle \alpha\op}"description]
\Cat
\\
\Y\op
\arrow[ur,bend right=10,"\psN"']
\Y\op
\arrow[ur,bend right,"\psN"'] &
\end{tikzcd}
\end{equation}

given by a family of natural transformations $m_x \maps \tau_x \Rightarrow\psN \alpha_x \circ\sigma_x$. Notice that taking opposites is a 2-functor $(-)\op \maps \Cat \to \Cat^{co}$, on which the above diagrams rely.
An indexed natural transformation between two indexed functors is an indexed 2-cell of the form $(1_{1_\X},m)$.
Dually, we have the notion of an opindexed 2-cell and opindexed natural transformation between opindexed 1-cells and functors respectively.

Notice that an indexed 2-cell $(\alpha,m)$ is invertible if and only if both $\alpha$ is a natural isomorphism and the modification $m$ is invertible, due to the way vertical composition is formed.

We obtain a 2-category $\ICat$ of indexed categories over arbitrary bases, indexed 1-cells and indexed 2-cells. In particular, there is a 2-category $\ICat(\X)$ of indexed categories with fixed domain $\X$, indexed functors and indexed natural transformations, which coincides with the functor 2-category $\TCat_\pse(\X\op,\Cat)$.

Dually, we have the 2-categories $\OpICat$ and $\OpICat(\X)=\TCat_\pse(\X,\Cat)$. Notice that due to the absence of opposites in the world of opindexed categories, opindexed 2-cells have a different form than <ref>, namely

[column sep=.5in, row sep=.15in]
[rr, bend left=30, ""]
[dr, bend left=30, "F"]
[dr, bend right=30, "G"']
[rr, phantom, near end, "⇓τ"description]
[dr, phantom, "⇓α"description]
[r, phantom, "m⇛"description]

[rr, bend left, ""]
[dr, bend right=10, "G"']
[rr, phantom, "⇓σ"description]

[ur, bend right, ""']

[ur, bend right=10, ""']

Moreover, we have 2-categories of strict (op)indexed categories and (op)indexed 1-cells that consist of strict natural transformations $\tau$ <ref>, i.e.$\ICat(\X)=[\X\op,\Cat]$ and $\OpICat_\mathrm{sp}(\X)=[\X,\Cat]$ the usual functor 2-categories.

Notice that these categories also form fibrations over $\Cat$, this time essentially using the family fibration also seen in <ref>. The functor $\ICat \to \Cat$ sends an indexed category to its domain and an indexed 1-cell to its first component. It is a split fibration, with fibres $\ICat(\X)$ and cartesian liftings pre-composition with functors. In fact, it is also a 2-fibration as explained in <cit.>.

§ THE GROTHENDIECK CONSTRUCTION

In the first volume of the Séminaire de Géométrie Algébrique du Bois Marie  <cit.>, Grothendieck introduced a construction for a fibration $P_\M \maps \inta \M \to \X$ from a given indexed category $\M \maps \X\op \to \Cat$ as follows. If $\delta$ and $\gamma$ are the structure pseudonatural transformations of the pseudofunctor $\M$,
the total category $\inta \M$ has

* objects $(x,a)$ with $x \in \X$ and $a \in \M x$;

* morphisms $(f,k) \maps (x,a) \to (y,b)$ with $f \maps x \to y$ a morphism in $\X$, and $k \maps a \to (\M f)(b)$ a morphism in $\M x$;

* composition $(g, \ell) \circ (f, k) \maps (x,a) \to (y,b) \to (z,c)$ is given by $g \circ f \maps a \to b \to c$ in $\X$ and

\begin{equation}\label{eq:comp_intM}
a\xrightarrow{k}(\M f)(b)\xrightarrow{(\M g)(\ell)}(\M g\circ\M f)(c)\xrightarrow{(\delta_{f,g})_c}\M(g\circ f)(c)\quad\textrm{in }\M x;
\end{equation}

* unit $1_{(x,a)} \maps (x,a) \to (x,a)$ is given by $1_x \maps x \to x$ in $\X$ and
\[a=1_{\M x}a\xrightarrow{(\gamma_x)_a}(\M 1_x)(a)\quad\textrm{in }\M x.\]

The fibration $P_\M \maps \inta \M \to \X$ is given by $(x,a) \mapsto x$ on objects and $(f,k) \mapsto f$ on morphisms, and the cartesian lifting of
any $(y,b)$ in $\inta \M$ along $f \maps x \to y$ in $\X$ is precisely $(f,1_{(\M f)b})$. Its fibres are precisely $\M x$ and the reindexing functors between them are $\M f$.

In the other direction, given a (cloven) fibration $P \maps \A \to \X$, we can define an indexed category $\M_P \maps \X\op \to \Cat$ that sends each object $x$ of $\X$ to its fibre category $\A_x$, and each morphism $f \maps x \to y$ to the corresponding reindexing functor $f^* \maps \A_y \to \A_x$ as in <ref>. The isomorphisms of cartesian liftings $f^* \circ g^* \cong (g \circ f)^*$ and $1_{\A_x} \cong 1_x^*$  render this assignment pseudofunctorial.

Details of the above, as well as the correspondence between 1-cells and 2-cells, can be found in the provided references. Briefly, given a pseudonatural transformation $\tau \maps \M \to \psN\circ F\op$ <ref> with components $\tau_x \maps \M x \to \psN Fx$, define a functor $P_\tau \maps \inta\M \to \inta\psN$ mapping $(x\in\X,a\in\M x)$ to the pair $(Fx\in\Y,\tau_x(a)\in\psN Fx)$ and accordingly for arrows. This makes the square

\begin{equation}
\label{eq:inducedfibred1cell}
\begin{tikzcd}
\inta\M
\arrow[r,"P_\tau"]
\arrow[d,"P_\M"']
\inta\psN
\arrow[d,"P_\psN"]
\\
\X\arrow[r,"F"']
\Y
\end{tikzcd}
\end{equation}

commute, and moreover $P_\tau$ preserves cartesian liftings due to pseudonaturality of $\tau$. Moreover, given an indexed 2-cell $(\alpha,m) \maps (F,\tau)\Rightarrow(G,\sigma)$ as in <ref>, we can form a fibred 2-cell

\begin{equation}\label{eq:inducedfibred2cell}
\begin{tikzcd}[column sep=.8in,row sep=.6in]
\inta\M
\arrow[r,bend left,"P_\tau"]
\arrow[r,bend right,"P_\sigma"']
\arrow[r,phantom,"\Downarrow{\scriptstyle P_m}"description]
\arrow[d,"P_\M"']
\inta\psN
\arrow[d,"P_\psN"]
\\
\X
\arrow[r,bend left,"F"]
\arrow[r,bend right,"G"']
\arrow[r,phantom,"\Downarrow{\scriptstyle\alpha}"description]
\Y
\end{tikzcd}
\end{equation}

where $\alpha \maps F\Rightarrow G$ is piece of the given structure, whereas $P_m$ is given by components

\[
(P_m)_{(x,a)} \maps P_\tau(x,a)=(Fx,\tau_xa) \to P_\sigma(x,a)=(Gx,\sigma_xa)\quad\textrm{in } \inta \psN
\]

explicitly formed by $\alpha_x \maps Fx \to Gx$ in $\Y$ and $(m_x)_a \maps \tau_xa \to (\psN\alpha_x)\sigma_xa$ in $\psN Fx$.

The following theorem summarizes these standard results.

* Every fibration $P \maps \A \to \X$ gives rise to a pseudofunctor $\M_P \maps \X\op { \to }\Cat$.

* Every indexed category $\M \maps \X\op \to \Cat$ gives rise to  a fibration $P_\M \maps \inta\M \to \X$.

* The above correspondences yield an equivalence of 2-categories

() ≃()

so that $\M_{P_\M} \simeq \M$ and $P_{\M_P} \simeq P$.

* The above 2-equivalence extends to one between 2-categories of arbitrary-base fibrations and arbitrary-domain indexed categories

\begin{equation}\label{eq:Gr_equiv}
\ICat\simeq\Fib
\end{equation}

If we combine the above with the fact that the 2-categories $\Fib$ and $\ICat$ are fibred over $\Cat$ with fibres $\Fib(\X)$ and $\ICat(\X)$ respectively, we obtain the following $\Cat$-fibred equivalence

\begin{equation}\label{FibICat}
\begin{tikzcd}
\ICat
\arrow[rr, "\simeq"]
\arrow[dr]
\Fib
\arrow[dl]
\\&
\Cat
\end{tikzcd}
\end{equation}

There is an analogous story for opindexed categories and opfibrations that results into a 2-equivalences $\OpICat(\X)\simeq\OpFib(\X)$ and $\OpICat\simeq\OpFib$, as well as for the split versions of (op)indexed and (op)fibred categories.

§ EXAMPLES

§.§ Fundamental Fibration

Let $2$ denote the category with two objects, and one non-identity morphism $\star \to \bullet$. For a category $\X$, the functor category $\X^2$ then consists of the arrows of $\X$ as objects, and commuting squares between them as the morphisms.

For any category $\X$, the codomain or fundamental opfibration is the usual functor from its arrow category

\[\cod \maps \X^2 \longrightarrow \X\]
mapping every morphism to its codomain and every commutative square to its right-hand side leg. It uniquely corresponds to the strict opindexed category, i.e. functor

\begin{equation}
\begin{tikzcd}[row sep=.05in]
\X\arrow[r] & \Cat \\
x\arrow[r,mapsto]\arrow[dd,"f"'] &     \X/x\arrow[dd,"f_!"] \\
\\
y\arrow[mapsto, r] & \X/y
\end{tikzcd}
\end{equation}

that maps an object to the slice category over it and a morphism to the post-composition functor $f_!=f\circ-$ induced by it.

§.§ Graphs

Consider (directed, multi) graphs, i.e. presheaves on the category $G = \begin{tikzcd} V \arrow[r, shift right, swap, "t"] \arrow[r, shift left, "s"] & E\end{tikzcd}$. For a presheaf $g \maps G\op \to \Set$, the set $g_V$ is the set of vertices of the graph, the set $g_E$ is the set of edges of the graph, and the maps $g_s, g_t \maps g_E \to g_V$ assign to an edge its starting and terminating vertex respectively. Let $\Grph$ denoted the category of graphs, $\Set^{G\op}$. Sometimes it is helpful to think of a graph as a single map of the form $(g_s, g_T) \maps g_E \to g_V \times g_V$. When convenient, we will abuse notation by simply referring to this map as $g \maps g_E \to g_V^2$.

Consider the inclusion of a terminal category $1$ into $G$ which selects the object $V$. This induces a functor $\mathsf V \maps \Grph \to \Set$ by precomposing, which sends a graph $g$ to its vertex set $g_V$. As we show below, this functor is in fact a bifibration. The idea here is that if you have a function $f \maps x \to y$, you can pull a graph on $y$ back to a graph on $x$, and you can also push a graph on $x$ forward to a graph on $y$.

A morphism $\phi\maps g \to h$ in $\Grph$ is $\mathsf V$-cartesian if and only if the square

\[
\begin{tikzcd}
\arrow[d, swap, "{(g_s, g_t)}"]
\arrow[r, "\phi_E"]
\arrow[d, "{(h_s, h_t)}"]
\\
\arrow[r, swap, "\phi_V^2"]
\end{tikzcd}
\]

is a pullback in $\Set$.

A simple manipulation shows that the universal property of $\phi$ forming a pullback square is the same as the universal property for it to be $\mathsf V$-cartesian.

The functor $\mathsf V \maps \Grph \to \Set$ is a fibration.

Let $f \maps x \to y$ be a function, and $g \in \Grph$ with $g_V=y$. Then we can take a pullback of the following diagram.

\[x^2 \xrightarrow{f^2} y^2 = h_V^2 \xleftarrow{(h_s,h_t)} h_E\]

By <ref>, this map is a cartesian lift of $f$.

By the Grothendieck correspondence, there is a indexed category $\Set\op \to \Cat$.This pseudofunctor assigns to a set $X$ the category $\Grph_X$ of graphs which have vertex set $X$, and graph morphisms which fix the vertices. Given a function $f \maps X \to Y$, this pseudofunctor gives a functor $f^* \maps \Grph_Y \to \Grph_X$ which sends a graph $g$ over $Y$ to the pullback, as in the proof of <ref>. Since there is also an opindexed category with the same fibres, we denote this by $\Grph^*$, referring to the action on morphisms.

To show that $\mathsf V$ is also an opfibration, it is actually easier to construct an explicit splitting. We can derive a characterization of the cocartesian maps from there.

The functor $\mathsf V \maps \Grph \to \Set$ is an opfibration.

Let $g \in \Grph$, $y \in \Set$, and $f \maps g_V \to y$ a function. Then we can obtain a graph with vertex set $y$ by taking the following composite.

\[g_E \xrightarrow{g} g_V^2 = g_V^2 \xrightarrow{f^2} y^2\]

We claim the induced map of graphs displayed below is in fact a cocartesian lift of $f$.

\[
\begin{tikzcd}
\arrow[r, "1"]
\arrow[d, swap, "g"]
\arrow[d, "f^2 \circ g"]
\\
\arrow[r, swap, "f^2"]
\end{tikzcd}
\]

Let $h$ be a graph, $\phi \maps g \to h$ a map of graphs, and $\phi \maps y \to h_V$.

\[
\begin{tikzcd}[column sep = huge]
\arrow[d, "h"]
\\&&
\arrow[ddll, bend right = 15, leftarrow, swap, "\phi_V^2"]
\\
\arrow[uurr, bend left = 15, "\phi_E"]
\arrow[r, -, white, line width = 5]
\arrow[r, "1", pos = 0.4]
\arrow[d, swap, "g"]
\arrow[d, swap, "f^2 \circ g"]
\arrow[uur, -, white, line width = 5]
\arrow[uur, dashed, ""]
\\
\arrow[r, swap, "f^2"]
\arrow[uur, swap, "1"]
\end{tikzcd}
\]

The only map which may take the place of the dashed arrow is $\phi_E$.

A morphism $\phi\maps g \to h$ in $\Grph$ is $\mathsf V$-cocartesian if and only if it is bijective on edges.

By the Grothendieck correspondence, there is a corresponding opindexed category $\Grph_* \maps \Set \to \Cat$, again referring to the action on morphisms. This must have the same fibres $\Grph_X$ as the indexed category $\Grph^*$ above. Given a function $f \maps X \to Y$, this pseudofunctor gives a functor $f_* \maps \Grph_X \to \Grph_Y$ which sends a graph $g$ over $X$ to the composite, as in the proof of <ref>.

The functor $\mathsf V \maps \Grph \to \Set$ is a bifibration.

§.§ Ring Modules

For a ring $R$, denote by $\Mod_R$ the category of $R$-modules and their homomorphisms. Given a ring homomorphism $f \maps R \to S$, and an $S$-module $N$, we can give the underlying abelian group of $N$ the structure of an $R$-module, denoted $f^\ast N$, by the formula

\[r.x := f(r).x\]

where $r \in R$ and $x \in N$.
This pullback construction is functorial:

\[f^\ast \maps \Mod_S \to \Mod_R\]

and preserves ring homomorphism composition.

\[(f\circ g)^\ast = g^\ast f^\ast \]

Indeed, the above defines a functor $\Mod_- \maps \Ring\op \to \Cat$, a (strict) indexed category. Note, one could choose to be persnickety about size here, but we do not. We can then apply the Grothendieck construction, resulting in a category $\Mod := \int \Mod_-$ where

* an object is a pair $(R \in \Ring, M \in \Mod_R)$

* a morphism is a pair $(f, \phi)$ where $f \maps R \to S$ is a ring homomorphism, and $\phi \maps M \to f^\ast N$ is an $S$-module homomorphism.

The category $\Mod$ admits a fibration $\Mod \to \Ring$ which forgets the module in a ring-module pair.

CHAPTER: SPECIES AND OPERADS

§ COMBINATORIAL SPECIES

Combinatorial species were introduced by Joyal <cit.>. A standard reference for the combinatorial perspective is <cit.>. In the previous section, we noted that the category $\Set$ can be given both a cartesian and cocartesian monoidal structure. Moreover, these satisfy a distributive law reminiscent of rings.

\[A \times (B + C) \cong A \times B + A \times C\]

Consider the subcategory $\FinBij$ consisting of finite sets and bijections. This subcategory is closed under both finite sums and products, and thus inherits both monoidal structures. However, $\FinBij$ lacks the maps that would be the projections and inclusions necessary for these structures to be cartesian or cocartesian themselves. By abuse of notation, we will continue to denote them by $+$ and $\times$.

A combinatorial species or simply species is a functor $F \maps \FinBij \to \Set$. The category of species is the functor category $\Set^\FinBij$.

There are several operations which have been defined on species. These operations make up the building blocks of a calculus for counting families of combinatorial gadgets.

Being a presheaf category, $\Set^\FinBij$ has colimits given pointwise, thus giving it a cocartesian structure. We refer to this operation simply as addition. On objects, this operation is given by $(F+G)(U) = F(U) + G(U)$.

If we apply Day convolution as in <ref> to the $+$ monoidal structure on $\FinBij$, we get the operation which we refer to as multiplication of species.
On objects, this operation is given by the following formula.

\[(F \cdot G)(U) = \sum_{V \subseteq U} F(V) \times G(U \setminus V)\]

Being a presheaf category, $\Set^\FinBij$ has products which are given pointwise, thus giving a cartesian monoidal structure. This is called the Hadamard product. On objects, it is given by $(F \times G) (U) = F(U) \times G(U)$. This tends to be less useful than multiplication, but it certainly has its purposes.

We define the Dirichlet product on species to be the Day convolution (as in <ref>) of the $\times$ monoidal structure on $\FinBij$.

To define differentiation of species, we will need to make use of the shift operator, denoted $+1$, on $\FinBij$, which is defined as the composite

\[
\begin{tikzcd}[column sep = huge]
\FinBij
\arrow[r, dashed, "+1"]
\arrow[d, swap, "\sim"]
\FinBij
\\
\FinBij \times 1
\arrow[r, swap, "1_{\FinBij} \times \Delta1"]
\FinBij \times \FinBij
\arrow[u, swap, "+"]
\end{tikzcd}
\]

where the map $\Delta1 \maps 1 \to \FinBij$ is the monoidal unit with respect to $\times$.

The differentiation operator on species is given by $D = +1^\ast \maps \Set^\FinBij \to \Set^\FinBij$. In other words, for a given species $F$, the derivative of $F$ is given by $F' = F \circ +1$, or $F'(U) = F(U+1)$ on objects. The motivation for calling this operation differentiation is that it actually corresponds to taking the formal derivative of its generation series.

The composition product or substitution product is given by the following formula.

\[(F \circ G)(U) = \sum_{\pi \text{ partition of } U} \left( F(\pi) \times \prod_{p \in \pi} G(p) \right)\]

The species which acts as unit for this product is the singleton indicator functor, i.e. $I_\circ (U)$ is a singleton is $U$ is, and is empty otherwise. This monoidal structure is not symmetric.

§ OPERADS

An operad is a generalization of category which incorporates the notion of an arrow having multiple inputs. A category is an arrow-like compositional system, consisting of a collection of directed arrows, a collection of labels for the endpoints called objects, and a rule for turning a path of such arrows into a single arrow which is associative and unital. An operad is also a compositional system, but now tree-like. An operad consists of a collection of directed “short” trees

\begin{equation*}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (3) at (0, 0.7) {};
\node [style=none] () at (0.5, 0.7) {\dots};
\node [style=none] (1) at (1, 0.7) {};
\node [style=none] (2) at (0, 0.5) {};
\node [style=none] (6) at (0.33, 0.5) {};
\node [style=none] (7) at (0.66, 0.5) {};
\node [style=none] (4) at (1, 0.5) {};
\node [style=downtri] (mu) at (0.5, 0) {};
\node [style=none] (5) at (0.5, -0.7) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (4.center);
\draw (3.center) to (2.center);
\draw [bend right] (2.center) to (mu.center);
\draw [bend left] (4.center) to (mu.center);
\draw (6.center) to (mu.center);
\draw (7.center) to (mu.center);
\draw (mu.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}

a collection of labels for the endpoints called objects, and a rule for turning a big tree of short trees

\begin{equation*}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=downtri] (a) at (0.5, 2.5) {};
\node [style=downtri] (b) at (1, 1.5) {};
\node [style=downtri] (c) at (0, 0.5) {};
\node [style=none] (1) at (0.25, 3) {};
\node [style=none] (2) at (0.5, 3) {};
\node [style=none] (3) at (0.75, 3) {};
\node [style=none] (5) at (1.25, 2) {};
\node [style=none] (6) at (-0.5, 1) {};
\node [style=none] (7) at (-0.125, 1) {};
\node [style=none] (8) at (0.125, 1) {};
\node [style=none] (9) at (0, -0.3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (a.center);
\draw (2.center) to (a.center);
\draw [bend left] (3.center) to (a.center);
\draw [bend right] (a.center) to (b.center);
\draw [bend left] (5.center) to (b.center);
\draw [bend left] (b.center) to (c.center);
\draw [bend right] (6.center) to (c.center);
\draw (7.center) to (c.center);
\draw (8.center) to (c.center);
\draw (c.center) to (9.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}

into a single short tree

\begin{equation*}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=downtri] (a) at (0, 0.5) {};
\node [style=none] (1) at (-0.375, 1) {};
\node [style=none] (2) at (-0.25, 1) {};
\node [style=none] (3) at (-0.125, 1) {};
\node [style=none] (4) at (0, 1) {};
\node [style=none] (5) at (0.125, 1) {};
\node [style=none] (6) at (0.25, 1) {};
\node [style=none] (7) at (0.375, 1) {};
\node [style=none] (8) at (0, -0.3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (1.center) to (a.center);
\draw [bend right = 15] (2.center) to (a.center);
\draw (3.center) to (a.center);
\draw (4.center) to (a.center);
\draw (5.center) to (a.center);
\draw [bend left = 15] (6.center) to (a.center);
\draw [bend left] (7.center) to (a.center);
\draw (8.center) to (a.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}

which is associative and unital. There are several good references for the theory of operads <cit.>. Here, we follow the treatment given by Yau in <cit.>.

§.§ Definition of Operad

Let $C$ be a non-empty set, whose elements we call colors. Recall that we denote the free symmetric monoidal category on $C$ by $\S(C)$. Below, we define operads to be monoids in the presheaf category $\Set^{\S(C) \times C}$ with respect to a certain monoidal structure. First we must define this monoidal structure, which is somewhat involved.

For this section, we denote objects of $\S(C)$ by either $\underline c$ or $(c_1, \dots, c_n)$ depending on context. We denote the monoidal structure on $\S(C)$ by $+$. For an object $X \in \Set^{\S(C)\op \times C}$, we denote the set assigned to $(\underline c, d) \in \S(C)\op \times C$ by $X(\underline c; d)$.

Let $X, Y \in \Set^{\S(C) \times C}$. For each $\underline c \in \S(C)$, define $Y^{\underline c}$ by the following coend formula.

\[Y^{\underline c} (\underline b) = \int^{\{\underline a_j\} \in \prod_{j=1}^m \S(C)\op} \S(C)\op (\underline a_1 + \dots + \underline a_m, \underline b) \times \left[\prod_{j=1}^m Y(\underline a_j; c_j) \right]\]

The $C$-colored circle product of $X$ and $Y$ is given by

\[X \circ Y(\underline b; d) = \int^{\underline c \in \S(C)} X(\underline c; d) \otimes Y^{\underline c}(\underline b)\]

Define the unit object $I$ as follows.

\[I(\underline c; d) = \begin{cases}
1 & \text{if } \underline c = d\\
\emptyset & \text{otherwise}
\end{cases}\]

This reduces to the composition monoidal product of species in the case where $C \cong 1$.

$(\Set^{\S(C) \times C}, \circ, I)$ is a monoidal category.

Let $C$ be a set. Define the category of operads by
\[\Opd_C = \Mon(\Set^{\S(C) \times C}, \circ, I).\]
We refer to an object of $\Opd_C$ as a $C$-colored operad, and a morphism as a color-fixing $C$-operad functor. Let $f \maps C \to D$ be a function. Let $f(\underline c)$ denote $(f(c_1), \dots, f(c_n))$ for $\underline c \in \S(C)$.
For a $D$-operad $P$, we can pullback along $f$ to get a $C$-operad given by $f^*P(\underline c; d) = P(f(\underline c); f(d))$. For a color-fixing $D$-operad functor $\phi \maps P \to Q$, we get a color-fixing $C$-operad functor $f^*\phi \maps f^*P \to f^*Q$ which sends an operation $\theta \in f^*P(\underline c; d) = P(f(\underline c); d)$ to $\phi\theta \in Q(f(\underline c); d)$. These assignments give a functor $f^* \maps \Opd_D \to \Opd_D$, and we get an indexed category $\Opd_- \maps \Set\op \to \Cat$. Define $\Opd = \int \Opd_-$. We refer to an object of $\Opd$ as an operad, and to a morphism as an operad functor.

§.§ Operads from symmetric monoidal categories

There is a standard method of constructing an single-colored operad from an object $x$ in a strict symmetric monoidal category $\C$. Namely, we define the set of $n$-ary operations to be $\hom_{\C}(x^{\otimes n}, x)$, and compose these operations using composition in $\C$. This gives the so-called endomorphism operad of $x$. Here we give the generalization of this idea to the multi-color case, using all the objects of $\C$ as the objects of the operad. In what follows we let $\Ob(\C)$ be the set of objects of a small category $\C$.

If $\C$ is a small strict symmetric monoidal category then there is an $\Ob(\C)$-colored operad $\Op(\C)$ for which:

* the set of operations $\Op(\C)(c_1, \dots, c_k; c)$ is defined to be $\hom_\C(c_1 \otimes \dots \otimes c_k, c)$,

* given operations

\[f\in \hom_\C(c_1 \otimes \cdots \otimes c_k; c) \]

\[g_i \in \hom_\C(c_{ij_1} \otimes \cdots \otimes c_{ij_i},c_i)
\]

for $1 \le i \le k$, their composite is defined by

\begin{equation}
\label{eq:composition_of_operations}
f \circ (g_1, \dots, g_k) = f \circ (g_1 \otimes \cdots \otimes g_k) .
\end{equation}

* identity operations are identity morphisms in $\C$, and

* the action of $S_k$ on $k$-ary operations is defined using the braiding in $\C$.

The various axioms of a colored operad can be checked for $\Op(\C)$ using the corresponding laws in the definition of a strict symmetric monoidal category.  The associativity axiom for $\Op(\C)$ follows from associativity of composition and the functoriality of the tensor product in $\C$.  The left and right unit axioms for $\Op(\C)$ follow from the unit laws for composition and the functoriality of the tensor product in $\C$. The two equivariance axioms for $\Op(\C)$ follow from the laws governing the braiding in $\C$.

The assignment $\Op \maps \Sym\Mon\Cat_\spl \to \Opd$ defined on objects as in <ref> and sending any strict symmetric monoidal functor $F \maps \C \to \C'$ to the operad morphism $\Op(F) \maps \Op(\C) \to \Op(\C')$ that acts by $F$ on types and also on operations:

\[
\Op(F) = F \maps \hom_C(c_1 \otimes \cdots \otimes c_n, c) \to \hom_{\C'}(F(c_1) \otimes \cdots \otimes F(c_n), F(c))
\]
is a functor.

This is a straightforward verification.

§.§ Operad Algebras

As a sort of monoid, operads exist to act. The elements of $\O(\underline c;d)$ for some operad $\O$ are meant to be thought of as “abstract operations” with $\underline c$ as the input types, and $d$ as the output type. When $\O$ acts on something, it is meant to be thought of as realizing these abstract operations as real operations on some family of sets indexed by the elements of $C$.

Let $C$ be a set. A $C$-colored set is a functor $C \to \Set$, where $C$ is thought of as a discrete category. This is of course the same as a function $C \to \ob\Set$. For $\underline c = (c_1, \dots, c_n) \in \S(C)$, let $X_{\underline c}$ denote the set $\prod_{j=1}^n X_{c_j}$. A map of $C$-colored sets $f \maps X \to Y$ is a natural transformation $f \maps X \To Y$. This is the same as a family of functions $\{f_c \maps X_c \to Y_c\}_{c \in C}$ with no further conditions. For $\underline c = (c_1, \dots, c_n) \in \S(C)$, let $f_{\underline c} \maps X_{\underline c} \to Y_{\underline c}$ denote the function $\prod_{j=1}^n f_{c_j} \maps \prod_{j=1}^n X_{c_j} \to \prod_{j=1}^n Y_{c_j}$.

Let $\O$ be a $C$-colored operad, with operad composition denoted by $\gamma$, and unit operation denoted by $I$. An $\O$-algebra consists of

* a $C$-colored set $X \maps C \to \Set$, also denoted $\{X_c\}_{c \in C}$

* for $\underline c \in \S(C)$ and $d \in C$, a map $\theta \maps \O(\underline c;d) \times X_{\underline c} \to X_d$

which makes the following diagrams commute for $c,d, c_j \in C$, $\underline c, \underline b_j \in \S(C)$, $\underline b = \sum^n \underline b_j$, and $\sigma \in S_n$.

* associativity:

\[
\begin{tikzcd}
\O(\underline c, d) \times \prod_{j=1}^n \O(\underline b_j, c_j) \times X_{\underline b}
\arrow[d, "\cong", "\text{permute}"']
\arrow[r, "\gamma \times 1"]
\O(\underline b; d) \times X_{\underline b}
\arrow[dd, "\theta"]
\\
\O(\underline c; d) \times \prod_{j=1}^n [\O(\underline b_j, c_j) \times X_{\underline b_j}]
\arrow[d, swap, "1 \times \prod_j \theta"]
\\
\O(\underline c; d) \times X_{\underline c}
\arrow[r, swap, "\theta"]
\end{tikzcd}\]

* unity:

\[
\begin{tikzcd}
1 \times X_c
\arrow[dl, swap, "I \times 1"]
\arrow[dr, "\cong"]
\\
\O(c;c) \times X_c
\arrow[rr, swap, "\theta"]
\end{tikzcd}\]

* equivariance:

\[
\begin{tikzcd}
\O(\underline c; d) \times X_{\underline c}
\arrow[dr, swap, "\theta"]
\arrow[rr, "\sigma \times \sigma\inv"]
\O(\underline c \sigma; d) \times X_{\underline c \sigma}
\arrow[dl, "\theta"]
\\&
\end{tikzcd}\]

A map of $\O$-algebras $(X, \theta) \to (Y, \xi)$ consists of a map of $C$-colored sets $\alpha \maps X \to Y$ such that the following diagram commutes.

\[
\begin{tikzcd}
\O(\underline c; d) \times X_{\underline c}
\arrow[r, "1 \times f_{\underline c}"]
\arrow[d, swap, "\theta"]
\O(\underline c; d) \times Y_{\underline c}
\arrow[d, "\xi"]
\\
\arrow[r, swap, "f"]
\end{tikzcd}\]

Let $\Alg(\O)$ denote the category of $\O$-algebras and maps of $\O$-algebras.


$\langle$acception date$\rangle$ $\langle$publication date$\rangle$ Kurahara
et al.Diffuse radio source in Abell 1060

††affiliationtext:  1Mizusawa VLBI Observatory, National Astronomical
Observatory Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
2Department of Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
3Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi
464-8601, Japan
4Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan

galaxies: clusters: individual (Abell 1060) — galaxies: individual (NGC 3309,
NGC 3311, NGC 3312)

# Discovery of Diffuse Radio Source in Abell 1060

Kohei Kurahara11affiliationmark:  Takuya Akahori11affiliationmark:  Aika
Oki11affiliationmark:  22affiliationmark:  Yuki Omiya33affiliationmark:  and
Kazuhiro Nakazawa33affiliationmark:  44affiliationmark:
<EMAIL_ADDRESS>
###### Abstract

Non-thermal components in the intra-cluster medium (ICM) such as turbulence,
magnetic field, and cosmic rays imprint the past and current energetic
activities of jets from active galactic neuclie (AGN) of member galaxies as
well as disturbance caused by galaxy cluster mergers. Meter- and centimeter-
radio observations of synchrotron radiation allow us to diagnose the
nonthermal component. Here we report on our discovery of an unidentified
diffuse radio source, named the Flying Fox, near the center of the Abell 1060
field. The Flying Fox has an elongated ring-like structure and a central bar
shape, but there is no obvious host galaxy. The average spectral index of the
Flying Fox is $-1.4$, which is steeper than those for radio sources seen at
meter wavelength. We discussed the possibilities of radio lobes, phoenixes,
radio halos and relics, and Odd Radio Circle (ORC). In conclusion, the Flying
Fox is not clearly explained by known radio sources.

##  1 Introduction

Galaxy clusters are formed by multiple mergers of groups and clusters of
galaxies. While major mergers, where the mass of the two systems are
comparable to each other, are dramatic and spectacular phenomena, minor
mergers, in which a sub system falls by the gravity of the main system, are
more frequent and play a role in disturbing the thermal and dynamical state of
the intra-cluster medium (ICM) and facilitating chemical enrichment in galaxy
clusters. Merger events impact on the galaxy evolution through galaxy mergers
and ram-pressure stripping as head-tail galaxies as well as on the energy
feedback of radio jets from AGN such as wide-angled-tailed (WAT) radio
galaxies.

Hydrodynamic simulations of minor merger of galaxy clusters have suggested
that the co-existence and mixing of multi-temperature ICM around the
gravitational center of the main cluster (Churazov et al. 2003; Fujita et al.
2004; ZuHone et al. 2016). This phenomenon is called “sloshing”, where a
relatively cool ICM is moving into the hotter ICM. The sloshing exhibits the
temperature gap between the two ICM without raising shock front, because the
sloshing motion is sub-sonic in the hot ICM. Actually, such a temperature gap
has been observed in X-rays and the gap is called “cold front” (see,
Markevitch & Vikhlinin 2007, for a review).

The sloshing is thought to be one of the key mechanisms that link non-thermal
components of the ICM such as turbulence, magnetic field, and cosmic rays. The
sloshing motion results in forming turbulence, amplifying magnetic field, and
driving turbulent re-acceleration (i.e. second-order Fermi acceleration) of
cosmic rays. The turbulent acceleration models predict a steep
($\alpha\lesssim-1.5$ of $S\propto\nu^{\alpha}$) spectrum radio halos (e.g.,
Brunetti et al. 2001; Petrosian 2001; Brunetti et al. 2008; Cassano et al.
2023), which have been found in some galaxy clusters (e.g., Bruno et al. 2021;
Duchesne et al. 2021; Di Gennaro et al. 2021; Riseley et al. 2022; Di Gennaro
et al. 2023).

Centimeter and meter wavelength radio observations have provided fruitful
information for understanding the non-thermal aspects of galaxies, AGNs, and
galaxy clusters (Akahori et al. 2018, for a review). Cosmic-ray
(re)-acceleration and magnetic-field amplification have been examined to
explain classical radio halos, mini-halos, and relics, as well as faint radio
fossil and phoenix which are thought to be aged radio sources but be
brightening due to disturbance caused by subcluster mergers and/or injection
of cosmic-rays from jets (see van Weeren et al. 2019, for a review). The
fossils often exhibit an ultra-steep radio spectrum ($\alpha\lesssim-2$; e.g.,
de Gasperin et al. 2017; Mandal et al. 2020; Ignesti et al. 2020), and are
more paid into attention, thanks to modern high sensitivity radio telescopes.

More recently, using the Australian Square Kilometre Array Pathfinder (ASKAP)
(Hotan et al. 2021) and the Giant Metrewave Radio Telescope (GMRT), Norris et
al. (2021) reported unidentified diffuse radio sources in the extragalactic
space. They are called Odd Radio Circle (ORC). Only several ORCs have been
reported and their spectral indice range from $-0.7$ to $-1.1$ (Norris et al.
2022). Several origins of ORCs have been discussed, including the shock wave
from the central galaxy, the termination shock of galactic wind, the end-on-
view double radio lobes, and the interactions between galaxies (Koribalski et
al. 2021; Norris et al. 2022) as well as the throats of wormholes (e.g.,
Kirillov et al. 2021). Cross-identification of host galaxies is essential to
understand their origins.

In this letter, we report on our discovery of an unidentified diffuse radio
source in the field of Abell 1060, using the GMRT. In Section 2, we briefly
summarize the observation and our data reduction. The results and our
discussions are described in Sections 3 and 4, respectively. Throughout the
letter, we adopt the $\lambda$CDM cosmology giving $\sim 0.27$ kpc arcsec-1 or
$\sim 16$ kpc arcmin-1 for the redshift to Abell 1060, $z=0.013$.

##  2 Observation and Data Reduction

We mainly used the archival data taken with GMRT in December 2010. The project
code is ddtB015. The data were recorded by the GMRT Hardware Backend (GHB) and
the GMRT software correlator backend (GSB) in parallel. We analyzed the GSB
data, where its center frequency and the bandwidth are 338 MHz and 33 MHz,
respectively. The radio source 3C147 was used as the flux density and bandpass
calibrator, and J1033-343 as the phase calibrator. The TGSS point-source
catalog (Intema et al. 2017) was used for self-calibration and peeling of
compact sources. Moreover, the SPAM (Source Peeling and Atmospheric Modeling;
Intema 2014) pipeline was applied to achieve the direction-dependent
calibration (DDC; Intema et al. 2017). See Kurahara et al. (2023) for details
of the data reduction and the analysis.

To calculate the radio spectral index, we referred to GLEAM (Hurley-Walker et
al. 2017), TGSS (Intema et al. 2017), NVSS (Condon et al. 1998) and RACS
(McConnell et al. 2020) data. Those data were taken from the NASA’s SkyView
facility (http://skyview.gsfc.nasa.gov) or the archival system. We also took
WISE (Wright et al. 2010) and DSS2-IR (Lasker et al. 1990; McLean et al. 2000)
data from SkyView and XMM-Newton (Jansen et al. 2001) data from the archival
system, for the cross-identification purpose. The X-ray data for a 64 ks
exposure was analyzed following the previous method (Omiya et al. 2023). The
redshift was fixed at 0.013 and the metal abundance table of Lodders et al.
(2009) was used.

##  3 Results

The top panel of figure 1 shows the 338 MHz total intensity (Stokes I) map
near the center of Abell 1060. The brightest source possesses a bi-polar shape
and its midpoint well-aligns with the center of the elliptical galaxy NGC 3309
(figure 2 left), suggesting that the source is radio jets ejected from AGN of
NGC 3309. The peak brightness is 23 mJy beam-1 or 245$\sigma$, where $\sigma$
= 0.094 mJy beam-1 is the image rms noise level. The integrated flux density
at 338 MHz is 334 mJy, which is in agreement with 288 mJy derived from the
spectral index of $-1.12$ that is obtained from the power-law fit of the TGSS
(874 mJy at 150 MHz), GLEAM (413 mJy at 200 MHz), and NVSS (61 mJy at 1400
MHz) data.

(80mm,80mm)fig1_uGMRT338MHz.eps (80mm,80mm)fig1_GLEAM200MHz.eps

Figure 1: The GMRT 338 MHz (top) and GLEAM 200 MHz (bottom) images of the
Abell 1060 center. The white contours are the XMM-Newton X-ray surface
brightness profile. Also drawn at the lower left are the synthesized beam
sizes of $13^{\prime\prime}.1\times 7^{\prime\prime}.2$ with the position
angle of 17.5 degrees for uGMRT and $2^{\prime}.2\times 2^{\prime}.1$ with
$-$6.8 degrees.

(55mm,55mm)fig2_ngc3309_wise12.eps (55mm,55mm)fig2_ngc3312_wise12.eps
(55mm,55mm)fig2_ngc3311_wise12.eps (55mm,55mm)fig2_ngc3309_DSS2IR.eps
(55mm,55mm)fig2_ngc3312_DSS2IR.eps (55mm,55mm)fig2_ngc3311_DSS2IR.eps

Figure 2: The WISE 12 micron images (top) and DSS2-IR optical images (bottom)
overlaid with the GMRT radio contours. The left, middle, and right panels show
the zoom-in views around NGC 3309, NGC 3312, and NGC 3311, respectively.

The second brightest radio source is located at the south-east part of the
image. The peak brightness is 4.3 mJy beam-1 or 45$\sigma$. The brightest
center is associated with the center of the spiral galaxy NGC 3312 and an
elongated ring shape matches with its arms (figure 2 middle), suggesting that
the source is the diffuse synchrotron emission of NGC 3312. The integrated
flux density at 338 MHz is 106 mJy, consistent with 94 mJy or the spectral
index of $-0.48$ obtained from the TGSS (173 mJy at 150 MHz), GLEAM (95 mJy at
200 MHz), and NVSS (49 mJy at 1400 MHz) data.

With the above confirmation of the validity of our synthesized image and the
flux scale, we find with GMRT that there is a spectacular diffuse radio source
near the center of the image. This source shows an elongated ring-like shape
with the middle part brightening strongly in a straight bar. We hereafter call
it the Flying Fox based on its silhouette. The head of the Flying Fox is
aiming at south-west. Its major axis (or the wingspan) is $\sim 235$ arcsec
($\sim 66$ kpc) and its minor axis (head-and-body length) is $\sim 100$ arcsec
($\sim 28$ kpc). The brightest bar (the inner right-wing to the chest) has a
brightness of 3.2 mJy beam-1 (34$\sigma$) and the outer ring has $\sim 0.65$
mJy beam-1 (7$\sigma$), both confirming solid detection. The elliptical galaxy
NGC 3311 overlaps at the north-western edge of the ring structure (or the
left-wing tip), where a point-like excess is associated with the center of NGC
3311 (figure 2 right).

We confirmed that the Flying Fox was detected in MWA GLEAM as an unresolved
source with the flux density of $431\pm 23$ mJy at 200 MHz (figure 1 bottom).
With our detection of the integrated flux density at 338 MHz, $208\pm 21$ mJy,
the average spectral index $\alpha$ at the meter-wavelength is $-1.4$, which
is notably steep compared to the sources in the GLEAM catalog (typically
$-0.77$). We confirmed that the extrapolated brightness of the radio peak is
below the detection levels of RACS at 900 MHz and NVSS at 1400 MHz. There is a
lack of the sensitivity with GMRT to split the data and obtain the in-band
spectral index. Thus, the spectral index map was not obtained.

##  4 Discussion

We discuss the possible origin of the Flying Fox. In conclusion, the Flying
Fox is not clearly explained by known radio sources such as radio lobes and
their fossils, diffuse cluster emissions, and ORCs. A future exploration of
host galaxies with optical/IR telescopes will help understanding the origin of
the Flying Fox. A future study of the spectrum-index map will provide the
information of the age and evolution of the Flying Fox, while the polarization
map will reveal the magnetic-field structure. Those information are also
crucial to understanding the origin and nature of the Flying Fox.

###  4.1 Remnant of Radio Lobes of Member Galaxies?

The overall morphology of the ring-like structure and the central bar shape
implies that the Flying Fox resembles radio lobes of AGN jets. However, we
confirmed no clear host AGN and galaxy at the midpoint of the Flying Fox in
radio/optical/IR/X-ray images. NGC 3311 connects to the ring-like structure
and possesses radio-bright AGN core, so that one possibility is that the
Flying Fox is a WAT-like radio jets from NGC 3311 and the bright bar is the
hot spots. This scenario suggests that NGC 3311 is moving to the north-west
direction to form a stripped tail or a remnant of radio lobes, where the
cosmic-ray aging from the north-west to the south-east direction is expected
and is needed to be consistent with the observed steep spectrum of the
integrated flux density at meter wavelength. This scenario would be supported
by X-ray (figure 3); the temperature and metal abundance maps implies a
sloshing structure and the Flying Fox is aligned with the high-temperature and
metal-rich region, that seems to be a tail-like structure from NGC 3311.

The steep spectrum implies that the Flying Fox is an aged source. One may
consider that the Flying Fox is an old radio lobe made by the past jet
activity of NGC 3311, and NGC 3311 was escaped from the radio lobe with a
relative velocity to the ICM. Such a relative velocity can be seen in merger
simulations including sloshing; dark matter and galaxies are collision-less
components and are moving forward from the ICM. Assuming a relative velocity
of 100 km s-1 on the sky-plane, the distance between NGC 3311 and the center
of the ring, 24 kpc, gives the travel time of 240 Myr. This time scale is
broadly consistent with radio fossils discussed in the literature (e.g., van
Weeren et al. 2019), although it is difficult to precisely estimate such a
relative velocity between the ICM and the galaxy.

(80mm,80mm)fig3_1.eps (80mm,80mm)fig3_2.eps

Figure 3: The XMM-Newton temperature (top) and metal abundance (bottom) maps
overlaid with the GMRT radio contours.

The motion of NGC 3311 toward the north-west direction is, however, a
controversy. NGC 3311 is located around the center of the line-of-sight
velocity distribution of member galaxies (see Hess et al. 2022) and around the
center of the X-ray surface brightness (figure 1). Interestingly, the MeerKAT
HI observation revealed that NGC 3312 and NGC 3314, which is located at 222
kpc south of NGC 3312 (see figure 3), have HI tails to the south-west
direction, suggesting that they are stripped gas from the galaxies both moving
to the north-east direction. This motion of NGC 3312 does not align with the
hypothetical motion of NGC 3311 toward the north-west direction. Note that it
is difficult to examine the motion of NGC 3311 from HI observation, because
NGC 3311 is a HI-poor elliptical galaxy and no HI emission has been detected
(Hess et al. 2022).

Leaving from NGC 3311, another potential origin of the Flying Fox is NGC 3309.
If a relative motion between the ICM and NGC 3309, likely caused by a
sloshing, is a clockwise or counter-clockwise circular motion around the
gravitational center close to NGC 3311, NGC 3309 is expected to be located
near the current position of the Flying Fox in the past. Thus, the Flying Fox
may be a remnant of the radio lobe made by past jets from NGC 3309. However,
the current jets and radio lobes associated with NGC 3309 do not indicate
clear stripped tails, suggesting that NGC 3311 does not have such a circular
motion. Note that Abell 1060 is classified into the Bautz-Morgan type III.
Neither NGC 3311 nor NGC 3309 is the cluster galaxy which dominates the
cluster center.

Finally, let us also discuss the scenario that the Flying Fox was formed by
NGC 3312. Hess et al. (2022) revealed that NGC 3312 is located at the nearside
in Abell 1060 and it is moving toward us according to its blue-shift with
respect to the redshift distribution of Abell 1060. They suggest that NGC 3312
already experienced the passage of the pericenter of the Abell 1060
gravitational potential, so that the HI tails of NGC 3312 were formed when NGC
3312 passed through the dense ICM on its orbit. The Flying Fox may be a
remnant of this disturbance at the passage. A weakness of this scenario is no
clear enhancement of X-ray brightness and temperature. It is also possible
that NGC 3312 is before the pericenter passage and is located at the backside
in Abell 1060. In this case, the Flying Fox is not related to NGC 3312.

###  4.2 Radio Relic or Halo?

No detection of host galaxy suggests that the Flying Fox is classical cluster
diffuse emission such as radio relic and halo. Radio mini-halo is also
classified into the cluster diffuse radio source, but a mini halo is often
associated with the central brightest cluster galaxy (BCG). There is no galaxy
around the center of the Flying Fox and no dominant galaxy in Abell 1060.
Those natures disfavor that the Flying Fox is a radio mini-halo.

There is a well-known group of galaxies, HGC 48, located at about 500 kpc away
to the north-east from NGC 3311. The redshift of HGC 48 is $z=0.094$, close to
the redshift of Abell 1060. If HGC 48 is a post-merging group and it disturbed
the ICM of Abell 1060 in the past, the Flying Fox may be a remnant of this
event. There are both scenarios of radio relic and radio halo described below.
Note that there is no data of the polarization fraction, which is often used
to distinguish between these two.

Based on the X-ray observations as shown in figure 3, there is no notable edge
of the temperature distribution associated with the Flying Fox. Therefore, it
is difficult to explain the Flying Fox as an edge-on view of a radio relic.
The Flying Fox is located at only $\sim 24$ kpc away from the cluster center.
This location is very different from the nature that radio relics are often
seen in the outskirts of clusters. Meanwhile, if HGC 48 is orbiting in the
Abell 1060 gravitational potential, and it formed the Flying Fox when the
projected position was near the Abell 1060 center on the sky plane, the Flying
Fox should be located at the nearside or farside in Abell 1060. This orbital
motion does not significantly disturb the central part of Abell 1060, showing
no clear temperature jump in the projected X-ray map.

Another scenario which links to radio halo is that HGC 48 collided through the
central part of Abell 1060. The collision could form the turbulence that can
stretch and amplify magnetic fields and accelerate cosmic-rays by turbulent
(re)-acceleration, resulting in radio halo. Although the spectral index of the
Flying Fox is steeper than those of classical halos, recent observations have
reported the steep-spectrum ones (see Introduction). However, the elongated
morphology of the Flying Fox is different from those of radio halos. The ring-
like structure is not expected in general for radio halos made by turbulent
(re)-acceleration.

###  4.3 Odd Radio Circle?

It is interesting to argue whether the Flying Fox belongs to a category of
ORC. A clear difference between the Flying Fox and ORCs is that the Flying Fox
is very elongated while ORCs exhibit circular morphology. Another discrepancy,
but which is not yet statistically significant, is that the Flying Fox is a
steep spectrum source while ORCs are not.

One possibility is that the Flying Fox is the aged ORC that was stretched
toward the east-west direction by the gas sloshing with NGC 3311 toward the
north-west direction, or it was compressed along the north-south direction by
the ram pressure with aligning the motions of NGC 3312 and NGC 3314. Whichever
the case is, no host galaxy is present and thus the origin is unclear.

This work was supported in part by JSPS KAKENHI Grant Numbers, JP21H01135(TA).
AIPS is produced and maintained by the National Radio Astronomy Observatory, a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc.

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