text:
\section{Introduction}

The $n$th Catalan number $C_n$, given explicitly by $\frac{1}{n+1}\binom{2n}{n}$, is well-known to be the answer
to many different counting problems.
For example, it is the number of bracketings of an $(n+1)$-fold product.
Thus there are many families of sets, $\mathbb C_n$, indexed by the natural numbers, whose cardinalities are the
Catalan numbers; such families might then be called ``Catalan sets''. Stanley~\cite{StanleyEC2, StanleyAddendum} describes at least 205 such families.
We shall show how to define functions between these sets, in such a way as to produce a simplicial set $\mathbb C$, which is the ``Catalan simplicial set'' of the title.
This is done using what seems to be a new description of the Catalan sets, which relies heavily on the Boolean algebra $\mathbf 2$.

Simplicial sets are abstract, combinatorial models of spaces, most often used in homotopy theory. They also arise, however,
as models of higher-dimensional categories, and that is their main role in this paper: our primary goal is to show that the simplicial
set $\mathbb C$ encodes, in a precise sense, a particular categorical structure called a {\em skew-monoidal category}. We shall show
that a skew-monoidal category is the same thing as a simplicial map from $\mathbb C$ to another simplicial set $N\mathrm{Cat}$,
the nerve of the monoidal bicategory $\mathrm{Cat}$ of categories and functors.

The structure of skew-monoidal category was introduced recently by Szlach\'anyi~\cite{Szl2012} in his study of bialgebroids, which are themselves
an extension of the notion of quantum group.

Thus the work presented here lies at the interface of several
mathematical disciplines:
\begin{itemize}[nolistsep]
\item[(a)] algebraic topology, because it involves simplicial sets and nerves;
\item[(b)] combinatorics, in the form of the Catalan numbers;
\item[(c)] quantum groups, through recent work on bialgebroids;
\item[(d)] logic, because of the distinguished role of the Boolean algebra $\mathbf{2}$; and
\item[(e)] category theory.
\end{itemize}

The natural generalisation of the notion of monoid from the level of
sets to categories is that of \emph{monoidal category}~\cite{CWM}. Like a
monoid, a monoidal category ${\mathscr A}$ comes equipped with a a unit element
$I \in {\mathscr A}$ and a multiplication $\otimes \colon {\mathscr A} \times
{\mathscr A} \to {\mathscr A}$ (now a functor, rather than a
function); but unlike a monoid, the basic operations of a monoidal
category do not satisfy associativity and unitality on the nose, but
only up to coherent natural families of isomorphisms:
\begin{equation}\label{eq:constraints}
\begin{gathered}
  \lambda_A \colon I \otimes A \to A \qquad \text{and} \qquad
  \rho_A \colon A \to A \otimes I \quad \text{(for $A \in {\mathscr A}$)}\\
  \alpha_{ABC} \colon (A \otimes B) \otimes
  C \to A \otimes (B \otimes C)\quad \text{(for $A,B,C, \in
    {\mathscr A}$).}
\end{gathered}
\end{equation}
In calling these families coherent, we mean to say that certain
diagrams of derived natural transformations, built from composites of
tensors of $\alpha$'s, $\lambda$'s and $\rho$'s, must commute. The
commutativity of these particular diagrams in fact implies the
commutativity of \emph{all} such diagrams: this is one form of the
coherence theorem for monoidal categories~\cite{CWM}, and justifies the
choice of axioms made. Mac\ Lane's original formulation~\cite{ML1963}
posited five generating axioms, recalled in Section~\ref{sec:skew}
below; Kelly later reduced these to two~\cite{Kelly1964}.

Recently, Szlach\'anyi~\cite{Szl2012} has introduced \emph{skew-monoidal
  categories}. The basic data for a skew-monoidal category are the
same as for a monoidal category, except that the constraint morphisms
in~\eqref{eq:constraints} are no longer required to be invertible. The
axioms which are to be satisfied are Mac\ Lane's original five, rather
than Kelly's two; the choice is substantive, as without invertibility
in~\eqref{eq:constraints}, the two axiomatisations are no longer
equivalent.
Szlach\'anyi's motivation in~\cite{Szl2012} for defining skew-monoidal
structure comes from representation theory: he identifies left
bialgebroids based on a ring $R$ with closed skew-monoidal structures
on the category of left $R$-modules. Lack and Street have placed this
result in a more general context, showing in~\cite{Smswqc} that the quantum
categories of~\cite{QCat} can be captured as
\emph{skew monoidales}---internal skew-monoidal objects---in a
suitable monoidal bicategory.

Whilst these applications justify the use of skew-monoidal structure,
they do not give an intrinsic justification for the form the structure
takes. There are in fact two places in the definition where a
non-obvious choice has been made.  The first concerns the orientation
of the maps in~\eqref{eq:constraints}.  For example, had we taken
$\lambda$ to have components $A \to I \otimes A$, whilst leaving
$\rho$ and $\alpha$ unchanged, we would have obtained (the one-object
case of) Burroni's notion of~\emph{pseudocategory}~\cite{Burroni1971};
on the other hand, if we had reversed the sense of $\alpha$ whilst leaving
$\lambda$ and $\rho$ unchanged, we would have obtained something very close to Grandis'
notion~\cite{Grandis2006} of \emph{d-lax $2$-category}.

The second choice concerns the axioms the maps
in~\eqref{eq:constraints} must satisfy. We have already said that Mac\
Lane's five axioms are no longer equivalent to Kelly's two in the skew
setting; so why, then, should we prefer the former to the latter?  For
monoidal categories, we justified the axioms in terms of a theorem
stating that all diagrams of coherence morphisms commute. In the
skew-monoidal case, we have no such justification, since a general
diagram of skew-monoidal coherence morphisms need \emph{not} commute;
describing the structure these coherence morphisms determine is in fact
quite subtle~\cite{Tosmc}.

The objective of the paper is to provide a perspective on
skew-monoidal structure which, amongst other things, makes it quite
apparent why the choices made above are natural ones.
To do this, we use the Catalan simplicial set $\mathbb C$ mentioned
above. It turns out to be quite easy to describe: it is itself a
nerve, the nerve of the monoidal poset
$(\mathbf 2, \vee, 0)$. In particular, this makes it
$2$-coskeletal---meaning that for $n > 2$, each $n$-simplex boundary
has a unique filler---and thus completely determined by its $0$-, $1$-
and $2$-simplices, of which it has one, two, and five respectively; more
generally, the number of $n$-simplices is the $n$-th Catalan number.

Our perspective, then, is that $\mathbb C$ classifies skew-monoidal
structures in the sense that simplicial
maps from $\mathbb C$ into a suitably-defined nerve of $\mathbf{Cat}$
are precisely skew-monoidal categories. More generally,
skew monoidales in a monoidal bicategory ${\mathscr K}$ are classified by maps
from $\mathbb C$ into the simplicial nerve of ${\mathscr K}$.

The two non-degenerate $2$-simplices in $\mathbb C$ encode the tensor
and unit operations borne by any skew-monoidal category; the
non-degenerate $3$-simplices encode $\alpha$, $\lambda$ and $\rho$,
with the orientations specified above; whilst the non-degenerate
$4$-simplices encode the skew-monoidal axioms. The coskeletality means
that, in particular, the $3$- and $4$-simplices are completely
determined by the $2$-simplices, and it is in this sense that our
perspective justifies the choices of coherence data and axioms for a
skew-monoidal structure.

There is another well-known connection between the Catalan numbers and (skew) monoidal structure,
arising from the fact that the $n$th Catalan number $C_n$ is the number of ways to bracket an
$(n+1)$-fold product.
The connection described in this paper seems to be quite different;
indeed, in the context of this paper $C_n$ involves $n$-fold products.

This work is inspired by an old idea of Michael Johnson on how to capture not
only associativity but also unitality constraints simplicially.  He
reminded us of this in a recent talk~\cite{MJ2013} to the
Australian Category Seminar.

\section{Skew-monoidal categories and skew monoidales}\label{sec:skew}

As in the introduction, a \emph{skew-monoidal category} is a
 category ${\mathscr A}$ equipped with a unit element $I \in {\mathscr A}$, a tensor
product $\otimes \colon {\mathscr A} \times {\mathscr A} \to {\mathscr A}$, and natural families
of (non-invertible) constraint maps $\alpha$, $\lambda$ and $\rho$ as
in~\eqref{eq:constraints}, all subject to the commutativity of the
following diagrams---wherein tensor is denoted by juxtaposition---for
all $a,b,c,d \in {\mathscr A}$:

\begin{equation}\label{axiom1}
\cd[@C-36pt]{
& &(ab)(cd) \ar[drr]^{\alpha}& & \\
((ab)c)d \ar[urr]^{\alpha} \ar[dr]_{\alpha1} & & & & a(b(c(d)) \\
& (a(bc))d \ar[rr]_{\alpha} && a((bc)d) \ar[ur]_{1\alpha}&
}
\end{equation}
\begin{equation}\label{axiom4}
\cd[@!C@C-6pt]{
& (aI)b \ar[r]^{\alpha} & a(Ib) \ar[dr]^{1\lambda} & \\
ab \ar[ur]^{\rho1} \ar[rrr]_{\mathrm{id}} & & & ab
}
\end{equation}
\begin{equation}\label{axiom3}
\cd[@!C@C-12pt]{
& I(ab) \ar[dr]^{\lambda} & \\
(Ia)b \ar[ur]^{\alpha} \ar[rr]_{\lambda1} &  & ab
}
\end{equation}
\begin{equation}\label{axiom2}
\cd[@!C@C-12pt]{
& (ab)I \ar[dr]^{\alpha} & \\
ab \ar[ur]^{\rho} \ar[rr]_{1\rho} &  & a(bI)
}
\end{equation}
\begin{equation}\label{axiom5}
\cd[@!C@C-6pt]{
& II \ar[dr]^{\lambda} & \\
I \ar[ur]^{\rho} \ar[rr]_{\mathrm{id}} && I\rlap{ .}
}
\end{equation}
We may also say that $(\otimes,I,\alpha,\lambda,\rho)$ is a
\emph{skew-monoidal structure} on ${\mathscr A}$.  What we call skew-monoidal
is what Szlach\'anyi~\cite{Szl2012} calls left-monoidal, whilst what he calls
right-monoidal we would call \emph{opskew-monoidal}; an
opskew-monoidal structure on ${\mathscr A}$ is the same as a skew-monoidal
structure on ${\mathscr A}^\mathrm{op}$.

More generally, we can consider skew monoidales in a monoidal
bicategory. A monoidal bicategory is a one-object tricategory in the
sense of~\cite{GPS}; it thus comprises a bicategory ${\mathscr B}$ equipped with a
unit object $I$ and tensor product homomorphism $\otimes \colon {\mathscr B}
\times {\mathscr B} \to {\mathscr B}$ which is associative and unital only up to
pseudonatural equivalences $\mathfrak a$, $\mathfrak l$ and $\mathfrak
r$. The coherence of these equivalences is witnessed by invertible
modifications $\pi, \mu, \sigma$ and $\tau$, whose components are
$2$-cells with boundaries those of the
axioms~\eqref{axiom1}--\eqref{axiom2} above, and an invertible
$2$-cell $\theta$ whose boundary is that of \eqref{axiom5}. The
modifications $\pi, \mu, \sigma$ and $\tau$ are as in~\cite{GPS},
though we write $\sigma$ and $\tau$ for what there are called
$\lambda$ and $\rho$; whilst $\theta \colon \mathfrak r_I \circ \mathfrak l_I \Rightarrow 1_I \colon I \to I$ can be defined from the
remaining coherence data as the composite:
\begin{equation*}
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  node[above=0.1cm,pos=0.495] {$\mathfrak a$} (200.0000,1012.3622);
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  node[above right=0.055cm,pos=0.35] {$\mathfrak l 1$}(40.0000,1012.3622)
  (200.0000,1012.3622) .. controls (186.7856,972.7190) and (147.3782,963.6344) ..
  node[above right=0.07cm,pos=0.43] {$\mathfrak r 1$}(122.1590,956.2645)
  (112.8208,953.3027) .. controls (105.0036,950.4894) and (100.0000,947.3640) ..
  (100.0000,942.3622) .. controls (100.0000,927.5293) and (140.0000,927.5293) ..
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  node[above=0.1cm,pos=0.57] {$\mathfrak r$} (0.2778,1002.3622) .. controls (-5.7407,1007.0671) and (-10.8148,1012.6400) ..
  node[above right=0.12cm,at end]{\!\!$\mathfrak r$}(-28.0000,1012.6400);
  \path[fill=black] (40,1012.3622) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text2987) {$\sigma$};
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\end{tikzpicture}
\quad \text.
\end{equation*}

Here, and elsewhere in this paper, we use string notation to display
composite $2$-cells in a bicategory, with objects represented by
regions, $1$-cells by strings, and generating $2$-cells by
vertices. We orient our string diagrams with 1-cells proceeding down
the page and 2-cells proceeding from left to right.  If a 1-cell $\xi$
belongs to a specified adjoint equivalence, then we will denote its
specified adjoint pseudoinverse by $\xi^\centerdot$, and as usual with
adjunctions, will draw the unit and counit of the adjoint equivalence
in string diagrams as simple caps and cups. In representing the
monoidal structure of a bicategory, we we notate the tensor product
$\otimes$ by juxtaposition, notate the structural 1-cells $\mathfrak
a, \mathfrak l, \mathfrak r$ and 2-cells $\pi, \nu, \lambda, \rho$
explicitly, and use string crossings to notate pseudonaturality
constraint 2-cells, and also instances of the pseudofunctoriality of
$\otimes$ of the form $(f \otimes 1) \circ (1 \otimes g) \cong (1
\otimes g) \circ (f \otimes 1)$ (the interchange isomorphisms).

With our notational conventions now established, we now define a
\emph{skew monoidale} in a monoidal bicategory ${\mathscr B}$ to be given by an object
$A \in {\mathscr B}$, unit and multiplication morphisms $i \colon I \to A$ and $t
\colon A \otimes A \to A$, and (non-invertible) coherence $2$-cells
\begin{equation*}
 \cd{
    (A \otimes A) \otimes A \rtwocell{drr}{\alpha} \ar[rr]^{\mathfrak a} \ar[d]_{t \otimes A} & &
    A \otimes (A \otimes A) \ar[d]^{A \otimes t} \\
    A \otimes A \ar[r]_t & A & A \otimes A \ar[l]^t
  }
  \ \  \text{and} \ \
  \cd{
    I \otimes A \ar[d]_{i \otimes A} \rtwocell{dr}{\lambda}
    \ar[r]^-{\mathfrak l} &
    A \rtwocell{dr}{\rho} \ar@{=}[d] \ar[r]^-{\mathfrak r}
    & A \otimes I \ar[d]^{A \otimes i} \\
    A \otimes A \ar[r]_-t & A & A \otimes A \ar[l]^-t
  }
\end{equation*}
subject to the following five axioms, the appropriate analogues of~\eqref{axiom1}--\eqref{axiom5}.
\begin{align*}
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\quad\ \  &= \quad\ \
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    minimum width=5mm, fill=white, inner sep=0.25mm] (text3309) {$\alpha$    };
  \path[fill=black] (139.03424,1031.976) node[circle, draw, line width=0.65pt,
    minimum width=5mm, fill=white, inner sep=0.25mm] (text3317) {$\alpha$  };
\end{tikzpicture}}}\\[6pt]
\vcenter{\hbox{\begin{tikzpicture}[y=0.8pt, x=0.7pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (20.0000,912.3622) .. controls (43.4793,912.3622) and (63.3478,901.7100) ..
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  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (67.0808,898.9849) .. controls (78.1440,914.6333) and (153.3281,912.3622) ..
  node[above left=0.12cm,at end] {$t$\!}(180.0000,912.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (35.0870,872.9554) .. controls (37.6957,884.2598) and (62.6957,891.0578) ..
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  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (36.3058,870.5159) .. controls (50.8051,868.6788) and (66.2099,872.3069) ..
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  (87.8061,879.1556) .. controls (102.1676,883.9328) and (115.8114,889.0641) ..
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  (67.1072,894.2441) .. controls (74.8719,875.7718) and (111.4903,864.4336) ..
  node[above left=0.05cm,pos=0.7] {$\mathfrak a$}(164.3696,863.8187);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (130.6522,892.5796) .. controls (138.3381,891.9647) and (163.9007,873.7454) ..
  node[below right=0.03cm,pos=0.5] {$1 \mathfrak l$}(165.8990,866.5206);
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  \path[fill=black] (128.51724,892.49457) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text4138) {$1 \lambda$     };
  \path[fill=black] (166.03291,864.05157) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text4142) {$\mu$   };
\end{tikzpicture}}} \quad \ &= \ \quad
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  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (30.0000,1022.3622) --
  node[above right=0.12cm,at start] {\!$t$}node[above left=0.12cm,at end] {$t$\!}(150.0000,1022.3622);
\end{tikzpicture}}}\\[6pt]
\vcenter{\hbox{\begin{tikzpicture}[y=0.7pt, x=0.7pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
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  node[above right=0.12cm,at start] {\!$t$}(60.9223,1006.2135);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (30.0000,972.3622) .. controls (52.9363,972.3622) and (72.2458,974.7838) ..
  node[above right=0.12cm,at start] {\!\!$(i 1) 1$}(87.1746,980.7689)
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  node[above right=0.03cm,pos=0.38] {$i 1$}(117.7983,1009.8696)
  (63.7389,1000.0316) .. controls (69.5802,988.6564) and (102.3454,972.3622) ..
  node[above left=0.12cm,at end] {$\mathfrak a$\!}(150.0000,972.3622)
  (64.9686,1004.3523) .. controls (78.6374,1006.2162) and (94.1546,1003.1361) ..
  node[above=0.07cm,pos=0.5] {$1t$}(109.2174,999.6320)
  (114.4978,998.3882) .. controls (127.3619,995.3357) and (139.6875,992.3622) ..
  node[above left=0.12cm,at end] {$1 t$\!}(150.0000,992.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (120.0000,1012.3622) --
  node[above left=0.12cm,at end] {$\mathfrak l$\!}(150.0000,1012.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (63.9470,1007.7837) .. controls (75.0666,1019.6587) and (106.1606,1024.4623) ..
  node[below=0.05cm,pos=0.5] {$t$}(117.7486,1013.9325);
  \path[fill=black] (62.102421,1003.787) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3566) {$\alpha$
     };
  \path[fill=black] (118.9784,1012.3953) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3570) {$\lambda$
   };
\end{tikzpicture}}} \quad \ &= \ \quad
\vcenter{\hbox{\begin{tikzpicture}[y=0.7pt, x=0.7pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (30.0000,912.3622) .. controls (50.6075,912.3622) and (57.8287,917.4085) ..
  node[above right=0.12cm,at start] {\!\!$(i 1) 1$}(59.9808,919.8680);
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  node[above right=0.12cm,at start] {\!$t 1$}(60.4529,924.6762);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (103.7058,920.3556) .. controls (106.1653,914.5142) and (134.3086,912.3622) ..
  node[above left=0.12cm,at end] {$\mathfrak a$\!}(150.0000,912.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (30.0000,952.3622) .. controls (76.4799,952.3622) and (98.0970,945.8975) ..
  node[above right=0.12cm,at start] {\!$t$}(114.2384,940.3189)(119.5946,938.4494) .. controls (129.3527,935.0470) and (137.7583,932.3622) ..
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  node[above left=0.12cm,at end] {$\mathfrak l$\!}(150.0000,952.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (62.2980,922.3160) --
  node[above=0.09cm,pos=0.5] {$\mathfrak l 1$}(100.3701,922.3160);
  \path[fill=black] (60.869564,922.14478) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3884) {$\lambda 1$
     };
  \path[fill=black] (102.13043,922.14478) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3888) {$\sigma$
   };
\end{tikzpicture}}}\\[6pt]
\vcenter{\hbox{\begin{tikzpicture}[y=0.7pt, x=0.7pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
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  node[above left=0.12cm,at end] {$\mathfrak a$\!}(150.0000,922.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (45.6001,918.6729) .. controls (63.5753,918.6729) and (75.6670,917.2210) ..
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  (30.0000,882.3622) .. controls (49.1430,882.2684) and (61.6720,886.9527) ..
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  (74.8228,897.2828) .. controls (80.1942,902.3978) and (83.9960,908.2524) ..
  (87.2443,913.8769) .. controls (93.7170,925.0845) and (97.9916,935.3785) ..
  node[above right=0.055cm,pos=0.45] {$t 1$}(108.1057,937.0861);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (110.7769,939.5952) .. controls (116.0033,940.8250) and (132.7835,942.3622) ..
  node[above left=0.12cm,at end] {$1 t$\!}(150.0000,942.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (110.4694,942.8316) .. controls (113.5438,951.7473) and (131.8377,962.3622) ..
  node[above left=0.12cm,at end] {$t$\!}(150.0000,962.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (46.1157,921.3937) .. controls (51.6495,932.1540) and (83.9305,950.6003) ..
    node[above right=0.055cm,pos=0.45] {$t$}(108.5255,941.3772);
  \path[fill=black] (42.733845,918.9342) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3236) {$\rho$ };
  \path[fill=black] (109.44783,939.83997) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3240) {$\alpha$ };
\end{tikzpicture}}} \quad \ &= \ \quad
\vcenter{\hbox{\begin{tikzpicture}[y=0.7pt, x=0.7pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (30.0000,1022.3622) --
  node[above right=0.12cm,at start] {\!$t$}node[above left=0.12cm,at end] {$t$\!}(150.0000,1022.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (150.0000,942.3622) .. controls (126.0199,942.3622) and (109.2231,948.0580) ..
  node[above left=0.12cm,at start] {$\mathfrak r$\!}(106.1488,953.5919);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (150.0000,982.3622) .. controls (124.4568,981.7473) and (110.0000,967.8961) ..
  node[above left=0.12cm,at start] {$\mathfrak a$\!}(105.6959,957.7506)
  (150.0000,962.3622) .. controls (139.9401,962.2395) and (129.7698,965.4711) ..
  node[above left=0.12cm,at start] {$1 i$\!}(118.1710,969.3606)
  (112.6296,971.2200) .. controls (96.3251,976.6597) and (77.0420,982.5314) ..
  node[below=0.05cm,pos=0.4] {$1(1 i)$}(51.5372,982.2002);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (150.0000,1002.3622) .. controls (111.2519,1002.3622) and (62.6049,999.7407) ..
  node[above left=0.12cm,at start] {$1 t$\!}(51.2298,985.2911);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (103.2991,956.1342) .. controls (87.6198,956.4416) and (57.1834,965.0499) ..
    node[above left=0.055cm,pos=0.45] {$1 \mathfrak r$}(51.3421,978.5771);
  \path[fill=black] (48.575161,981.65149) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3352) {$1\rho$
     };
  \path[fill=black] (104.83627,955.82672) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3356) {$\tau$
   };
\end{tikzpicture}}}\\[6pt]
\vcenter{\hbox{\begin{tikzpicture}[y=0.7pt, x=0.7pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (34.0000,882.3622) .. controls (53.5806,882.2663) and (65.0541,887.1694) ..
  node[above right=0.12cm,at start] {\!$i$}(72.8455,894.0826)
  (76.6782,897.9452) .. controls (81.0329,902.8997) and (84.1354,908.4937) ..
  node[above right=0.04cm,pos=0.77] {$i 1$}(87.2443,913.8769) .. controls (93.7170,925.0845) and (97.9916,935.3785) .. (108.1057,937.0861)
  (45.9075,915.2580) .. controls (54.5158,896.5043) and (126.9177,877.7506) ..
  node[above=0.07cm,pos=0.5] {$\mathfrak r$}(140.1620,884.8217)
  (45.6001,918.6729) .. controls (62.8702,918.6729) and (75.6953,918.1544) ..
  node[above=0.09cm,pos=0.5] {$1 i$}(86.3824,917.4146)
  (91.8786,917.0018) .. controls (105.3036,915.9160) and (115.6132,914.5265) ..
  node[above=0.09cm,pos=0.59] {$1 i$}(128.3871,913.5523)
  (133.7731,913.1782) .. controls (141.7041,912.6822) and (150.7402,912.3622) ..
  node[above left=0.12cm,at end] {$i$\!}(162.0000,912.3622)
  (111.0843,939.2547) .. controls (131.3752,934.0283) and (133.2198,896.5374) ..
  node[below right=0.05cm,pos=0.34] {$\mathfrak l$}(139.5471,888.5440);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (46.1157,921.3937) .. controls (51.6495,932.1540) and (83.9305,950.6003) ..
  node[below left=0.05cm,pos=0.59] {$t$}(108.5255,941.3772);
  \path[fill=black] (42.733845,918.9342) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3236) {$\rho$
     };
  \path[fill=black] (109.44783,939.83997) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3240) {$\lambda$
     };
  \path[fill=black] (141.11392,887.26813) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text4834) {$\theta$
   };
\end{tikzpicture}}} \quad \ &= \ \quad
\vcenter{\hbox{\begin{tikzpicture}[y=0.7pt, x=0.7pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (30.0000,1022.3622) --
  node[above right=0.12cm,at start] {\!$i$}node[above left=0.12cm,at end] {$i$\!}(150.0000,1022.3622);\end{tikzpicture}}}
\rlap{ .}\end{align*}

Note that a skew monoidale in the monoidal
bicategory $(\mathbf{Cat}, \times, 1)$ is precisely a skew-monoidal
category, whilst a skew monoidale in the $2$-cell dual
$(\mathbf{Cat}^\mathrm{co}, \times, 1)$ is an opskew-monoidal category.

\section{Nerves of monoidal bicategories}\label{sec:nerves}
Before describing the simplicial set $\mathbb C$ that
classifies skew-monoidal categories, and more generally,
skew monoidales in a monoidal bicategory, we will first describe the
nerve construction by which we will assign a simplicial set $\mathrm N
{\mathscr B}$ to a given monoidal bicategory ${\mathscr B}$; the classification of
skew monoidales in ${\mathscr B}$ will then be in terms of simplicial maps
$\mathbb C \to \mathrm N{\mathscr B}$.

First let us recall some basic definitions.  We write $\Delta$
for the simplicial category; the objects are $[n] = \{0,\dots,n\}$ for
$n\ge 0$ and the morphisms are order-preserving functions.  Objects
$X$ of $\mathrm{SSet} = [\Delta^{\mathrm{op}}, \mathrm{Set}]$ are
called {\em simplicial sets}; we write $X_n$ for $X([n])$ and call its
elements \emph{$n$-simplices} of $X$. We use the notation $d_{i}
\colon X_n \to X_{n-1}$ and $s_i \colon X_n \to X_{n+1}$ for the face
and degeneracy maps, induced by acting on $X$ by the maps $\delta_i
\colon [n-1]\to [n]$ and $\sigma_{i} \colon [n+1]\to [n]$ of $\Delta$,
the respective injections and surjections for which
$\delta_{i}^{-1}(i) = \emptyset$ and $\sigma_i^{-1}(i) = \{i, i+1\}$.
An $(n+1)$-simplex $x$ is called {\em degenerate} when it is in the image
of some $s_i$, and \emph{non-degenerate} otherwise.

A simplicial set is called \emph{$r$-coskeletal} when it lies in the
image of the right Kan extension functor $[(\Delta^{(r)})^\mathrm{op},
\mathrm{Set}] \to [\Delta^\mathrm{op}, \mathrm{Set}]$, where
$\Delta^{(r)} \subset \Delta$ is the full subcategory on those $[n]$
with $n\le r$.  In elementary terms, a simplicial set is
$r$-coskeletal when every $n$-boundary with $n > r$ has a unique
filler; here, an \emph{$n$-boundary} in a simplicial set is a
collection of $(n-1)$-simplices $(x_0, \dots, x_n)$ satisfying $d_j(x_i)
 = d_i(x_{j+1})$ for all $0 \leqslant i \leqslant j < n$;
a \emph{filler} for such a boundary is an $n$-simplex $x$ with $d_i(x)
 = x_i$ for $i = 0, \dots, n$.

As we noted above, a monoidal bicategory is a one-object tricategory
in the sense of~\cite{GPS}. There are several known constructions of
nerves for tricategories; the one of interest to us is essentially
Street's $\omega$-categorical nerve~\cite{aos}, restricted from dimension
$\omega$ to dimension $3$, and generalised from strict to weak
$3$-categories. An explicit description of this nerve is given
in~\cite{GRoT}; we now reproduce the details for the case of a monoidal
bicategory ${\mathscr B}$. For such a ${\mathscr B}$, the \emph{nerve} $\mathrm N{\mathscr B}$ is the simplicial
set defined as follows:
\begin{itemize}[nolistsep]
\item There is a unique $0$-simplex, denoted $\star$.
\item A $1$-simplex is an object $A_{01}$ of ${\mathscr B}$; its two faces are
  necessarily $\star$.
\item A $2$-simplex is given by objects $A_{12}, A_{02}, A_{01}$ of
  ${\mathscr B}$ together with a $1$-cell ${A}_{012} \colon {A}_{12} \otimes {A}_{01} \to
  {A}_{02}$; its three faces are $A_{12}$, $A_{02}$, and $A_{01}$.
\item A $3$-simplex is given by:
\begin{itemize}[noitemsep]
\item Objects ${A}_{ij}$ for each $0 \leqslant i < j
  \leqslant 3$;
\item $1$-cells ${A}_{ijk} \colon {A}_{jk} \otimes {A}_{ij} \to {A}_{ik}$ for
  each $0 \leqslant i < j < k \leqslant 3$;
\item A $2$-cell
  \begin{equation*}
 \cd{
    ({A}_{23} \otimes {A}_{12}) \otimes {A}_{01} \rtwocell{drr}{{A}_{0123}} \ar[rr]^{\mathfrak a} \ar[d]_{{A}_{123} \otimes 1} & &
    {A}_{23} \otimes ({A}_{12} \otimes {A}_{01}) \ar[d]^{1 \otimes {A}_{012}} \\
    {A}_{13} \otimes {A}_{01} \ar[r]_{{A}_{013}} & {A}_{03} & {A}_{23}
    \otimes {A}_{02}\rlap{ ;} \ar[l]^{{A}_{023}}
  }
  \end{equation*}
\end{itemize}
its four faces are $A_{123}$, $A_{023}$, $A_{013}$ and $A_{012}$.
\item A $4$-simplex is given by:
\begin{itemize}[noitemsep]
\item Objects ${A}_{ij}$ for each $0 \leqslant i < j
  \leqslant 4$;
\item  $1$-cells ${A}_{ijk} \colon {A}_{jk} \otimes {A}_{ij} \to {A}_{ik}$ for
  each $0 \leqslant i < j < k \leqslant 4$;
\item $2$-cells ${A}_{ijk\ell} \colon {A}_{ij\ell} \circ ({A}_{jk\ell}
  \otimes 1) \Rightarrow {A}_{ik\ell} \circ (1 \otimes {A}_{ijk}) \circ
  \mathfrak a$ for each $0 \leqslant i < j < k < \ell \leqslant 4$
\end{itemize}
such that the $2$-cell equality
\begin{gather*}
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    node[above right=0.15cm,at start] {$\!\!({A}_{234}  1)  1$}(48.0259,1000.9550);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (0.5761,1020.4270) .. controls (32.7994,1020.6795) and (38.4294,1009.3612) ..
    node[above right=0.17cm,at start] {$\!\!{A}_{124}  1$}(48.0259,1003.0477);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (49.8969,1000.5044) .. controls (58.2307,980.6554) and (151.0215,979.4784) ..
    node[above right=0.25cm,pos=0.4] {$\mathfrak a  1$}(166.1738,987.8122);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (49.6135,1003.1578) .. controls (60.9777,1006.6933) and (72.9109,1017.9473) ..
    node[below left=0.05cm,pos=0.57]{${A}_{134}  1$} (80.4466,1027.1123);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (0.5761,1042.0698) .. controls (14.4657,1042.3223) and (75.2981,1040.0698) ..
    node[above right=0.15cm,at start] {$\!\!{A}_{014}$} (80.8539,1029.7127);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (82.2825,1030.8845) .. controls (90.1821,1054.0933) and (173.6677,1051.1703)
    .. node[above left=0.12cm,at end] {${A}_{034}$}(214.7867,1051.3122);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (141.5590,1019.2256) .. controls (143.0743,1011.6494) and (166.0258,992.5776)
    .. node[below right=0.04cm] {$1  \mathfrak a$}(166.0258,992.5776);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (141.6203,1021.3685) .. controls (141.6203,1021.3685) and (163.4571,1014.0062)
    .. node[above left=0.12cm,at end] {$1  (1  {A}_{012})\!\!$}(214.7868,1014.7638);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (141.4111,1023.0061) .. controls (141.4111,1023.0061) and (142.5493,1033.5038)
    .. node[above left=0.12cm,at end] {$1  {A}_{023}$}(214.7868,1033.7564);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (167.5590,988.4831) .. controls (167.5590,988.4831) and (177.6142,980.4754) ..
    node[above left=0.12cm,at end] {$\mathfrak a$} (214.7868,979.9703);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (167.9162,991.3403) .. controls (172.8729,995.3375) and (185.9479,996.5546) ..
    node[above left=0.12cm,at end] {$\mathfrak a$}(214.7867,996.5546);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (82.5536,1028.3546) .. controls (96.1906,1037.3648) and (132.5520,1035.8496)
    .. node[below right=0.12cm,pos=0.42] {$1  {A}_{013}$} (139.8756,1022.0412);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (82.7084,1025.8212) .. controls (84.1609,1008.5230) and (138.1030,990.5459) ..
    node[above left=0.07cm,pos=0.6] {$\mathfrak a$}(166.0786,990.2620)
    (49.8145,1001.8228) .. controls (60.2802,1001.0663) and (79.7259,1003.0374) ..
    node[above right=0.08cm,pos=0.4,rotate=-9] {$(1  {A}_{123})  1$}(98.0839,1006.4270)
    (105.9744,1007.9878) .. controls (121.2679,1011.2244) and (134.6866,1015.3648) ..
    node[below left=0.08cm,pos=0.7,rotate=-15] {$1  ({A}_{123}  1)$}(139.8779,1019.5827);
  \path[fill=black] (166.83624,990.0094) node[circle, draw, line width=0.65pt,
    minimum width=5mm, fill=white, inner sep=0.25mm] (text3313) {$\ \
      \pi\ \ $    };
  \path[fill=black] (48.501251,1001.512) node[circle, draw, line width=0.65pt,
    minimum width=5mm, fill=white, inner sep=0.25mm] (text3305) {${A}_{1234}  1$    };
  \path[fill=black] (81.600487,1027.7435) node[circle, draw, line width=0.65pt,
    minimum width=5mm, fill=white, inner sep=0.25mm] (text3309) {${A}_{0134}$    };
  \path[fill=black] (145.38065,1021.0309) node[circle, draw, line width=0.65pt,
    minimum width=5mm, fill=white, inner sep=0.25mm] (text3317) {$1  {A}_{0123}$  };
\end{tikzpicture}}} \\ = \\
\vcenter{\hbox{\begin{tikzpicture}[y=1pt, x=1pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (0.5761,1010.9221) .. controls (22.7994,1011.1746) and (33.5330,1012.5294) ..
    node[above right=0.12cm,at start] {\!${A}_{124}  1$}(49.1780,1027.2420);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (0.5761,1042.0698) .. controls (14.4657,1042.3223) and (43.6151,1040.3578) ..
    node[above right=0.12cm,at start] {\!${A}_{014}$}(49.1709,1030.0007);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (51.1755,1030.5965) .. controls (67.4280,1056.6856) and (126.7968,1041.6654)
    .. node[below right=0.14cm] {${A}_{024}$} (139.8511,1033.1665);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (142.8500,1030.5854) .. controls (142.8500,1030.5854) and (184.5612,1028.3914)
    .. node[above left=0.12cm,at end] {$1  {A}_{023}$}(225.8909,1029.1490);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (142.6407,1033.9511) .. controls (142.6407,1033.9511) and (172.9964,1047.0248)
    .. node[above left=0.12cm,at end] {${A}_{034}$}(225.8909,1046.9894);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
    (51.8895,1026.1092) .. controls (64.3077,997.3630) and (142.1189,975.3128) ..
    node[above left=0.12cm,at end] {$\mathfrak a$}(225.8909,975.6050)
    (52.8868,1028.6427) .. controls (64.4224,1023.3508) and (87.4860,1019.3533) ..
    node[below right=0.1cm,pos=0.53] {$1  {A}_{012}$} (113.0660,1016.4776)
    (121.4833,1015.5795) .. controls (135.8648,1014.1237) and (150.7159,1013.0048) ..
    node[above=0.1cm] {$1  {A}_{012}$} (164.5150,1012.1936)
    (173.4370,1011.7035) .. controls (191.3809,1010.7870) and (206.7880,1010.4220) ..
    node[above left=0.12cm,at end] {$1  (1  {A}_{012})\!\!$} (225.8909,1010.5363)
    (143.3647,1028.4425) .. controls (173.9707,1013.3776) and (183.0558,992.0340) ..
    node[above left=0.12cm,at end] {$\mathfrak a$}(225.8909,992.6100)
    (1.1521,981.0038) .. controls (30.9106,980.4472) and (58.1432,987.9204) ..
    node[above right=0.12cm,at start] {$\!\!({A}_{234}  1)  1$}(81.9942,998.1170)
    (88.1242,1000.8286) .. controls (108.0234,1009.9228) and (125.3927,1020.6737) ..
    node[above right=0.1cm,pos=0.22] {${A}_{234}  1$}
    (139.6961,1029.7577);
  \path[fill=black] (49.917496,1028.8956) node[circle, draw, line width=0.65pt,
    minimum width=5mm, fill=white, inner sep=0.25mm] (text3309) {${A}_{0124}$    };
  \path[fill=black] (139.03424,1031.976) node[circle, draw, line width=0.65pt,
    minimum width=5mm, fill=white, inner sep=0.25mm] (text3317) {${A}_{0234}$  };
\end{tikzpicture}}}
\end{gather*}
holds. The five faces of this simplex are $A_{1234}$, $A_{0234}$,
$A_{0134}$, $A_{0124}$ and $A_{0123}$.
\item Higher-dimensional simplices are determined by the requirement
  that $\mathrm{N}{\mathscr B}$ be $4$-coskeletal.
\end{itemize}
It remains to describe the degeneracy operators. The degeneracy of the
unique $0$-simplex is the unit object $I \in {\mathscr B}$; the two
degeneracies $s_0(A), s_1(A)$ of a
$1$-simplex $A \in {\mathscr B}$ are the unit constraints $\mathfrak
r^\centerdot \colon A \otimes I \to A$ and $\mathfrak l \colon I
\otimes A \to A$; the three degeneracies $s_0(\gamma), s_1(\gamma)$ and $s_2(\gamma)$ of a $2$-simplex
$\gamma \colon B \otimes C \to A$ are the respective $2$-cells
\begin{equation*}
\vcenter{\hbox{\begin{tikzpicture}[y=0.8pt, x=0.8pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (0.0000,902.3622) .. controls (22.2057,902.3622) and (24.4296,862.1105) ..
  node[above right=0.12cm,at start] {$\!\mathfrak r^\centerdot$}(54.6378,854.8261) .. controls (70.4902,851.0034) and (80.9687,860.6367) .. (63.1815,866.8469)
  (0.0000,882.3622) .. controls (6.5599,882.4187) and (12.3710,883.9537) ..
  node[above right=0.12cm,at start] {$\!\gamma 1$}(18.0721,886.3023)(22.7007,888.3801) .. controls (40.8765,897.1386) and (59.6979,912.3622) ..
  node[above left=0.12cm,at end] {$\gamma$\!}(100.0000,912.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (59.1409,868.1591) .. controls (29.0888,882.3013) and (70.2005,892.3622) ..
  node[above left=0.12cm,at end] {$1 \mathfrak r^\centerdot$\!}(100.0000,892.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (62.6764,869.3723) .. controls (70.2525,874.1706) and (91.3989,872.3622) ..
  node[above left=0.12cm,at end] {$\mathfrak a$\!}(100.0000,872.3622);
  \path[fill=black] (60.710678,868.80756) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text14966) {$\tau$   };
 \end{tikzpicture}}}
\qquad
\vcenter{\hbox{\begin{tikzpicture}[y=0.8pt, x=0.8pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (0.0000,892.3622) .. controls (36.8530,892.1096) and (33.3645,857.8413) ..
  node[above right=0.12cm,at start] {\!$\mathfrak r^\centerdot 1$}(56.4056,853.3108) .. controls (75.1348,849.6282) and (85.9633,861.3740) .. (63.1815,866.8469);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (0.0000,912.3622) .. controls (29.3000,912.6147) and (43.6628,912.3622) ..
  node[above left=0.12cm,at end] {$\gamma$\!}node[above right=0.12cm,at start] {\!$\gamma$}(100.0000,912.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (62.5254,872.1096) .. controls (63.3845,886.3127) and (70.2005,892.3622) ..
  node[above left=0.12cm,at end] {$1 \mathfrak l$\!}(100.0000,892.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (62.6764,869.3723) .. controls (70.2525,874.1706) and (91.3989,872.3622) ..
  node[above left=0.12cm,at end] {$\mathfrak a$\!}(100.0000,872.3622);
  \path[fill=black] (60.963215,870.82788) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text14966) {$\mu^{\!\scriptscriptstyle -1}$
   };
 \end{tikzpicture}}}\qquad\text{and}\qquad
\vcenter{\hbox{\begin{tikzpicture}[y=0.8pt, x=0.8pt,yscale=-1, inner sep=0pt, outer sep=0pt, every text node part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (43.0799,884.7365) .. controls (43.9391,898.9396) and (70.2005,912.3622) ..
  node[above left=0.12cm,at end] {$\mathfrak l$\!}(100.0000,912.3622)
  (0.0000,902.3622) .. controls (26.8739,902.5938) and (38.5571,900.6887) ..
  node[above right=0.12cm,at start] {\!$\gamma$}(48.0003,898.4810)
  (54.0398,897.0009) .. controls (63.6207,894.6257) and (73.8763,892.3622) ..
  node[above left=0.12cm,at end] {$1 \gamma$\!}(100.0000,892.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (43.2310,880.4840) .. controls (49.3612,869.8661) and (91.3989,872.3622) ..
  node[above left=0.12cm,at end] {$\mathfrak a$\!}(100.0000,872.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt] (0.0000,882.3622) --
  node[above right=0.12cm,at start] {\!$\mathfrak l 1$}(40.0000,882.3622);
  \path[fill=black] (41.767765,882.36218) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text14966) {$\sigma$
     };
 \end{tikzpicture}}}\rlap{ .}
\end{equation*}
The four degeneracies of a $3$-simplex are
simply the assertions of certain $2$-cell equalities; that these hold
is a consequence of the axioms for a monoidal bicategory. Higher
degeneracies are determined by coskeletality.

\section{The Catalan simplicial set}\label{Cssan}

As mentioned in the introduction, the \emph{Catalan simplicial set} is the nerve of the monoidal category $(\mathbf 2, \vee, 0)$.
The category $\mathbf 2$ has two objects $0$, $1$ and a single morphism $0\to 1$. The tensor for the monoidal structure is disjunction and the unit is $0$.
This is actually a strict monoidal category and so can be regarded as a one-object 2-category $\Sigma \mathbf 2$.
The Catalan simplicial set $\mathbb C$ is the nerve of this 2-category; its $n$-simplices are normal lax functors $[n] \to \Sigma \mathbf 2$ as in~\cite{aos}.
In section \ref{sec:CatalanClassifies} we will show that $\mathbb C$ classifies skew monoidales in a monoidal bicategory.

Now consider item (f{}f{}f{}) in Stanley's list~\cite{StanleyEC2} of Catalan sets. These are relations $R \subseteq [n] \times [n]$ that are reflexive, symmetric and have the interpolation property:
\begin{equation*}
(i,k) \text{ and } i\le j \le k \text{ implies } (i,j) \text{ and } (j,k)\ .
\end{equation*}
Let $\mathbb{K}_n$ be the set of such relations. Each $\mathbb K_n$ has $C_n$ elements.

\begin{proposition}\label{catalanbyrelations}
The assignment sending each normal lax functor $F \colon [n] \to \Sigma \mathbf 2$ to
$$ \{(i,j)\ :\ F(i\le j) = 0\ \vee\ F(j\le i) = 0\}\ $$
is a bijection $N\Sigma\mathbf 2_n \cong \mathbb{K}_n$. Thus $|\mathbb C_n| = C_n$.
\end{proposition}
\begin{proof}
It is easy to see that the relation above is symmetric and reflexive. Suppose that $i\leq j \leq k$ and $(i,k)$ is in the set above. Then we find that $F(j\le k).F(i\le j) \Rightarrow F(i\le k)$ has codomain 0 and so must be the identity on 0. Then the domain is 0 and thus $F(j\le k) = F(i\le j) = 0$.
Now notice that since $\Sigma \mathbf 2$ has a single object and is locally posetal, normal lax functors $[n] \to \Sigma \mathbf 2$ are completely determined by their action on 1-cells.
The assignment is a bijection because each $F$ is completely determined by which maps $i\le j$ go to 0.
$\square$
\end{proof}

\begin{corollary} The number of non-degenerate simplices of each dimension in
$\mathbb C$ is given by the Motzkin sequence
$$1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, \dots \ .$$
\end{corollary}
\begin{proof}
The Catalan numbers can be obtained from the Motzkin numbers~\cite{oeis} by taking
cardinalities in
$$\mathbb C_n \cong \displaystyle\sum\limits_{m=0}^{n} \binom{n}{m} \times \mathrm{nd}_m\mathbb C$$
where $\mathrm{nd}_m\mathbb C$ is the set of non-degenerate m-simplices in $\mathbb C$.
$\square$
\end{proof}

The Catalan simplicial set can also be described using ideals.
The category of ordered sets and order-preserving functions is denoted by $\mathrm{Ord}$.
For ordered sets $M$ and $N$, an {\em ideal} $A : M\to N$ is a subset
$A\subseteq N\times M$ (that is, a relation from $M$ to $N$) such that
$$q\le j, (j,i)\in A, i\le p \text{\ implies\ } (q,p)\in A \ .$$
Composition of ideals is composition of relations.
We have a 2-category $\mathrm{Idl}$ of ordered sets, ideals and inclusions (for example, see~\cite{CS}).
The identity ideal $1_M$ of $M$ is $\{(j,i):j\le i\}$.
Each order-preserving function $\xi : M \to N$ gives rise to ideals
$\xi_* : M \to N$ and $\xi^* : N \to M$ defined by
$$\xi_* = \{(j,i): j \le \xi(i)\} \quad \text{and} \quad \xi^* = \{(i,j): \xi(i)\le j\} \ .$$
Then $1_M\le \xi^* \xi_*$ and $\xi_* \xi^* \le 1_N$.
This means $\xi_* \dashv \xi^*$ in $\mathrm{Idl}$.
This defines functors $(-)_* : \mathrm{Ord} \to \mathrm{Idl}$ and
$(-)^* : \mathrm{Ord}^{\mathrm{op}} \to \mathrm{Idl}$, both the identity
on objects. This is all familiar ${\mathscr V}$-category theory with ${\mathscr V} = \mathbf 2$.

Let us put
$$\mathbb{S}_{n} = \{B\in \mathrm{Idl}([n],[n]) : 1_{[n]} \le B \} \ .$$
Then each $\xi : [m] \to [n]$ gives a function $\mathbb{S}_{\xi} : \mathbb{S}_{n} \to \mathbb{S}_{m}$
defined by
$$\mathbb{S}_{\xi}(B) = {\xi}^* B {\xi}_* = \xi^{-1}(B) = \{(p,q) : (\xi(p),\xi(q)) \in B \} \ .$$
Notice that $1_{[n]} \le B$ implies $1_{[n]} \le {\xi}^* {\xi}_* \le {\xi}^* B {\xi}_* $.
Thus we have defined a simplicial set
$\mathbb{S} : \Delta^{\mathrm{op}} \to \mathrm{Set}$,
indeed, a simplicial ordered set $\mathbb{S} : \Delta^{\mathrm{op}} \to \mathrm{Ord}$.

There is an isomorphism $\mathbb S \cong \mathbb C$. These ideals can
be enumerated using Young diagrams (of which there are a Catalan
number); this provides an alternative proof that there are a Catalan
number of simplices at each dimension of $\mathbb C$.

\begin{remark}
Each Tamari lattice~\cite{TamariBk} has a Catalan number of elements. It is not too hard to define the Tamari order on each $\mathbb C_n$. It is natural to ask whether the simplicial structure on $\mathbb C$ preserves the Tamari order. This is not the case: we have $\rho \le s_1(i)$ in $\mathbb C_3$ but $d_1(\rho) = s_1(c) \nleq i = d_1(s_1(i))$.
\end{remark}

\section{The Catalan simplicial set classifies \\
skew monoidales}\label{sec:CatalanClassifies}

Once we have enumerated the simplices of $\mathbb C$ in low
dimensions, it will become clear that the image of every map
$F\colon\mathbb C \to N{\mathscr B}$ picks out essentially the data and axioms
for a skew monoidale in ${\mathscr B}$. More precisely, such maps are in
bijection with skew monoidales in ${\mathscr B}'$ (defined below). First, it is
important to recognise that, since $N{\mathscr B}$ is 4-coskeletal, every
simplicial map $F\colon\mathbb C \to N{\mathscr B}$ is completely determined by
its image on the 4-truncation of its domain. In fact, if two such
simplicial maps are equal on their 3-truncation, then they are equal.

We now investigate the non-degenerate n-simplices in $\mathbb C$ for
$n\leq4$. It is convenient to note that all simplices above dimension
2 are uniquely determined by their faces. As such, every $n$-simplex
$a$ can be identified with the $(n+1)$-tuple $(d_0(a),d_2(a), \dots,
d_{n}(a))$ for $n \geq 2$.  \renewcommand{\aa}{a}
\renewcommand{\ll}{\ell} \newcommand{r} \newcommand{\kk}{k}{r} \newcommand{\kk}{k}

\begin{itemize}
\item There is a single 0-simplex, call it $\star$.
\item There is a single non-degenerate 1-simplex $c$ whose two faces are necessarily $\star$.
\item There are two non-degenerate 2-simplices:
\begin{align*}
t &= (c,c,c) \\
i &= (s_0(\star),c, s_0(\star))
\end{align*}
\item There are four non-degenerate 3-simplices:
\begin{align*}
\aa &= (t,t,t,t) \\
\ll &= (i,s_1(c),t,s_1(c)) \\
r} \newcommand{\kk}{k &= (s_0(c),t,s_0(c),i) \\
\kk &= (i, s_1(c), s_0(c), i)
\end{align*}
\item There are nine non-degenerate 4-simplices:
\begin{equation*}
\begin{tabular}{ l l l }
$A1 = (\aa,\aa,\aa,\aa,\aa)$ \hspace{0.7cm} &
$A6 = (s_0(i),r} \newcommand{\kk}{k, \kk, \ll, s_2(i))$ \\
$A2 = (r} \newcommand{\kk}{k,s_1(t),\aa,s_1(t),\ll)$ &
$A7 = (\kk,r} \newcommand{\kk}{k, s_0s_1(c), \ll, \kk)$  \\
$A3 = (r} \newcommand{\kk}{k,r} \newcommand{\kk}{k,s_2(t),\aa,s_2(t))$ &
$A8 = (\ll, s_1(t), s_0(t), \ll, \kk)$ \\
$A4 = (s_0(t),\aa,s_0(t),\ll,\ll)$ \hspace{0.7cm} &
$A9 = (\kk, r} \newcommand{\kk}{k, s_2(t), s_1(t), r} \newcommand{\kk}{k)$ \\
$A5 = (s_1(i),s_2(i), \kk, s_0(i), s_1(i))$ &
\end{tabular}
\end{equation*}
\end{itemize}
The image of $F$ on the 4-truncation of $\mathbb C$ consists of the following.
\begin{itemize}[nolistsep]
\item A single object $F(c) = A$.
\item Two 1-cells
\begin{equation*}
\cd[]{A\otimes A \ar[r]^-{F(t)} & A}\quad\text{and}\quad\cd[]{I\otimes I \ar[r]^-{F(i)} & A}\ .
\end{equation*}
\item Four 2-cells
\begin{equation*}
 \cd{
    (A \otimes A) \otimes A \rtwocell{drr}{F(\aa)} \ar[rr]^{\mathfrak a} \ar[d]_{F(t) \otimes A} & &
    A \otimes (A \otimes A) \ar[d]^{A \otimes F(t)} \\
    A \otimes A \ar[r]_{F(t)} & A & A
    \otimes A \ar[l]^{F(t)}
  }
\end{equation*}
\begin{equation*}
 \cd{
    (A \otimes I) \otimes I \rtwocell{drr}{F(r} \newcommand{\kk}{k)} \ar[rr]^{\mathfrak a} \ar[d]_{\mathfrak r^\centerdot \otimes 1} & &
    A \otimes (I \otimes I) \ar[d]^{A \otimes F(i)} \\
    A \otimes I \ar[r]_{\mathfrak r^\centerdot} & A & A
    \otimes A \ar[l]^{F(t)}
  }
\end{equation*}
\begin{equation*}
 \cd{
    (I \otimes I) \otimes A \rtwocell{drr}{F(\ll)} \ar[rr]^{\mathfrak a} \ar[d]_{F(i) \otimes 1} & &
    I \otimes (I \otimes A) \ar[d]^{1 \otimes \mathfrak l} \\
    A \otimes A \ar[r]_{F(t)} & A & I
    \otimes A \ar[l]^{\mathfrak l}
  }
\end{equation*}
\begin{equation*}
 \cd{
    (I \otimes I) \otimes I \rtwocell{drr}{F(\kk)} \ar[rr]^{\mathfrak a} \ar[d]_{F(i) \otimes 1} & &
    I \otimes (I \otimes I) \ar[d]^{1 \otimes F(i)} \\
    A \otimes I \ar[r]_{\mathfrak r^\centerdot} & A & I
    \otimes A \ar[l]^{\mathfrak l}
  }
\end{equation*}
\item And nine equalities:
\end{itemize}
\begin{equation}\label{stringaxiom1}
        \def\eye{t}\def\mathfrak r^\centerdot{t}\def\abe{t}
        \def\mathfrak l}\def\ace{\tee}\def\ade{\tee{t}\def\ace{t}\def\ade{t}
        \def\eye}\def\acd{\mathfrak l{t}\def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{t}
        \def\tee{t}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{t}
        \defs_2\tee}\def\abce{F\rr{F\aa}\def\acde{F\aa}
        \defF\kk{F\aa}\def\abce{F\aa}
        \defF\rr}\def\acde{s_1\tee{F\aa}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom2}
        \def\eye{\mathfrak l}\def\mathfrak r^\centerdot{t}\def\abe{t}
        \def\mathfrak l}\def\ace{\tee}\def\ade{\tee{\mathfrak l}\def\ace{t}\def\ade{t}
        \def\eye}\def\acd{\mathfrak l{i}\def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{\mathfrak r^\centerdot}
        \def\tee{t}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{\mathfrak r^\centerdot}
        \defs_2\tee}\def\abce{F\rr{F\aa}\def\acde{s_1(t)}
        \defF\kk{F\ll}\def\abce{s_1(t)}
        \defF\rr}\def\acde{s_1\tee{Fr} \newcommand{\kk}{k}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom3}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{t}\def\tee{\mathfrak r^\centerdot}\def\ade{t}
        \def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{\mathfrak r^\centerdot}\def\ace{t}\def\abe{\mathfrak r^\centerdot}
        \def\eye}\def\acd{\mathfrak l{\mathfrak r^\centerdot}\def\mathfrak l}\def\ace{\tee}\def\ade{\tee{t}
        \def\mathfrak r^\centerdot{\mathfrak r^\centerdot}
        \def\eye{i}
        \defs_2\tee}\def\abce{F\rr{s_2t}\def\abce{Fr} \newcommand{\kk}{k}
        \defF\rr}\def\acde{s_1\tee{s_2t}\def\acde{F\aa}
        \defF\kk{Fr} \newcommand{\kk}{k}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom4}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{i}\def\tee{\mathfrak l}\def\ade{\mathfrak l}
        \def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{t}\def\ace{t}\def\abe{t}
        \def\eye}\def\acd{\mathfrak l{\mathfrak l}\def\mathfrak l}\def\ace{\tee}\def\ade{\tee{\mathfrak l}
        \def\mathfrak r^\centerdot{t}
        \def\eye{t}
        \defs_2\tee}\def\abce{F\rr{s_0t}\def\abce{F\aa}
        \defF\rr}\def\acde{s_1\tee{F\ll}\def\acde{F\ll}
        \defF\kk{s_0t}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom5}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{\mathfrak l}\def\tee{i}\def\ade{\mathfrak l}
        \def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{i}\def\ace{i}\def\abe{\mathfrak l}
        \def\eye}\def\acd{\mathfrak l{\mathfrak l}\def\mathfrak l}\def\ace{\tee}\def\ade{\tee{i}
        \def\mathfrak r^\centerdot{i}
        \def\eye{\mathfrak l}
        \defs_2\tee}\def\abce{F\rr{F\kk}\def\abce{s_2(i)}
        \defF\rr}\def\acde{s_1\tee{s_1i}\def\acde{s_0(i)}
        \defF\kk{s_1i}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom6}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{i}\def\tee{t}\def\ade{\mathfrak l}
        \def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{\mathfrak r^\centerdot}\def\ace{t}\def\abe{\mathfrak r^\centerdot}
        \def\eye}\def\acd{\mathfrak l{\mathfrak r^\centerdot}\def\mathfrak l}\def\ace{\tee}\def\ade{\tee{\mathfrak l}
        \def\mathfrak r^\centerdot{i}
        \def\eye{i}
        \defs_2\tee}\def\abce{F\rr{F\kk}\def\abce{Fr} \newcommand{\kk}{k}
        \defF\rr}\def\acde{s_1\tee{s_2i}\def\acde{F\ll}
        \defF\kk{s_0i}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom7}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{i}\def\tee{\mathfrak l}\def\ade{\mathfrak l}
        \def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{\mathfrak r^\centerdot}\def\ace{t}\def\abe{\mathfrak r^\centerdot}
        \def\eye}\def\acd{\mathfrak l{i}\def\mathfrak l}\def\ace{\tee}\def\ade{\tee{\mathfrak l}
        \def\mathfrak r^\centerdot{\mathfrak r^\centerdot}
        \def\eye{i}
        \defs_2\tee}\def\abce{F\rr{s_0s_1c}\def\abce{Fr} \newcommand{\kk}{k}
        \defF\rr}\def\acde{s_1\tee{F\kk}\def\acde{F\ll}
        \defF\kk{F\kk}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom8}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{i}\def\tee{\mathfrak l}\def\ade{\mathfrak l}
        \def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{\mathfrak r^\centerdot}\def\ace{t}\def\abe{t}
        \def\eye}\def\acd{\mathfrak l{\mathfrak l}\def\mathfrak l}\def\ace{\tee}\def\ade{\tee{\mathfrak l}
        \def\mathfrak r^\centerdot{t}
        \def\eye{\mathfrak l}
        \defs_2\tee}\def\abce{F\rr{s_0t}\def\abce{s_1t}
        \defF\rr}\def\acde{s_1\tee{F\kk}\def\acde{F\ll}
        \defF\kk{F\ll}
        \stringpent
\end{equation}
\begin{equation}\label{stringaxiom9}
        \def\mathfrak r^\centerdot}\def\bde{\tee}\def\ade{t{\mathfrak r^\centerdot}\def\tee{t}\def\ade{t}
        \def\mathfrak r^\centerdot}\def\ace{\tee}\def\abe{\mathfrak r^\centerdot{\mathfrak r^\centerdot}\def\ace{t}\def\abe{\mathfrak r^\centerdot}
        \def\eye}\def\acd{\mathfrak l{i}\def\mathfrak l}\def\ace{\tee}\def\ade{\tee{\mathfrak l}
        \def\mathfrak r^\centerdot{\mathfrak r^\centerdot}
        \def\eye{i}
        \defs_2\tee}\def\abce{F\rr{s_2t}\def\abce{Fr} \newcommand{\kk}{k}
        \defF\rr}\def\acde{s_1\tee{Fr} \newcommand{\kk}{k}\def\acde{s_1t}
        \defF\kk{F\kk}
        \stringpent
\end{equation}

The similarity with skew monoidales in ${\mathscr B}$ is immediately clear.
There is however one problem: the unit map for a skew monoidale is of the form $I \to A$ but $F(i)$ is a map $I\otimes I \to F(c)$.
Similarly, the left and right unit constraints for a skew monoidale have different domains and codomains than $F(r} \newcommand{\kk}{k)$ and $F(\ll)$.
This disparity means that the structure given by $F$ is not strictly that of a skew monoidale.
However, the difference amounts to the fact that $I\otimes I$ does not equal $I$.

We address this problem by considering the monoidal bicategory ${\mathscr B}'$ with the same tensor as ${\mathscr B}$ but whose unit object is $I\otimes I$. The unit maps for this monoidal structure are
\begin{equation*}
\cd[]
{
 (I\otimes I)\otimes A \ar[r]^-{\mathfrak l_I \otimes A} & I\otimes A \ar[r]^-{\mathfrak l_A} & A
}
\quad\text{and}\quad
\cd[]
{
 A \otimes (I\otimes I) \ar[r]^-{A\otimes \mathfrak r^\centerdot_I} & A\otimes I \ar[r]^-{\mathfrak r_A} & A
}\ .
\end{equation*}
The invertible modifications $\pi, \mu, \sigma$ and $\tau$ are also altered accordingly.
In this case the identity functor on ${\mathscr B}$ becomes a strong monoidal functor ${\mathscr B} \to {\mathscr B}'$.
Having modified the unit object in ${\mathscr B}$ we can construct a bijection between simplicial maps $F\colon \mathbb C \to N{\mathscr B}$ and skew monoidales in ${\mathscr B}'$.

\begin{remark}
In the fundamental example ${\mathscr B} = \mathrm{Cat}$, there is a further bijection between skew monoidales in $\mathrm{Cat}'$ and skew monoidales in $\mathrm{Cat}$; that is, skew-monoidal categories.
\end{remark}

The first thing to notice is that the equality in \eqref{stringaxiom5}
together with the monoidal bicategory axioms force $F(\kk)$ to be
equal to the 2-cell
\begin{equation} \label{kappastring}
  \vcenter{\hbox{\begin{tikzpicture}[y=0.8pt, x=0.9pt,yscale=-1, inner
        sep=0pt, outer sep=0pt, every text node
        part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (60.5482,860.1807) .. controls (97.7329,834.6731) and (20.2790,833.4806) ..
  node[above=0.07cm,pos=0.5] {$\mathfrak r^\centerdot 1$}(21.6649,878.0631) .. controls (22.2692,897.5009) and (50.8157,910.5973) .. (77.9793,909.6016)
  (83.8468,909.1607) .. controls (91.1546,908.3220) and (98.2018,906.3945) ..
  node[above left=0.055cm,pos=0.55] {$\mathfrak r$}(104.3947,903.2213)
  (10.0000,912.3622) .. controls (28.6404,912.3622) and (47.9703,914.2230) ..
  node[above right=0.12cm,at start] {\!\!$Fi.1$}(68.7323,916.3393)
  (77.0089,917.1848) .. controls (95.0409,919.0225) and (114.1847,920.8922) ..
  node[below=0.074cm,pos=0.5] {$Fi$}(134.9003,921.7992)
  (140.5333,922.0201) .. controls (146.8711,922.2390) and (153.3558,922.3622) ..
  node[above left=0.12cm,at end] {$1.Fi$\!\!}(160.0000,922.3622)
  (10.0000,932.3622) .. controls (46.3655,932.3622) and (117.0711,908.0690) ..
  node[above right=0.12cm,at start] {\!$\mathfrak r^\centerdot$}(60.5546,892.8673)
  (61.6193,889.4751) .. controls (137.9573,830.7177) and (122.5843,941.9527) ..
  node[above left=0.12cm,at end] {$\mathfrak l$\!}(160.0000,942.3622)
  (59.5990,867.2517) .. controls (61.2888,870.8982) and (66.6120,874.7013) ..
  node[above right=0.05cm,pos=0.5,rotate=-35]{$1 \mathfrak l$}(73.0425,878.5617)
  (78.0172,881.4569) .. controls (88.8715,887.6317) and (100.9341,893.8888) ..
  node[above right=0.035cm,pos=0.4]{$\mathfrak l$}(104.0457,899.8292);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (62.3769,864.7264) .. controls (128.2265,830.2570) and (130.2744,900.0073) ..
  node[above left=0.12cm,at end] {$\mathfrak a$\!}(160.0000,902.3622);
  \path[fill=black] (52.527935,864.72632) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3790) {$\mu^{\scriptscriptstyle -1}$  };
  \path[fill=black] (55.101635,890.83112) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3790-1) {$\theta^{\scriptscriptstyle -1}$   };
  \path[fill=black] (104.09403,901.18524) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3790-6) {$\theta$};
      \end{tikzpicture}}}
\end{equation}
and thus completely specified by the coherence data of ${\mathscr B}$.

We now compare the data comprising the image of $F$ with the data
for a skew monoidale in ${\mathscr B}'$. At dimensions 0 and 1 these data are
exactly equal: a single object $F(c) = A$ together with two 1-cells
$F(t) \colon A \otimes A \to A$ and $F(i) \colon I \otimes I \to
A$. At dimension two, the $2$-cell $F(\aa)$ has the same form as the
associativity constraint $\alpha$ for a skew monoidale; whilst, as
observed above, $F(\kk)$ is necessarily of the form~\eqref{kappastring}.  On the other
hand, the data $F(\ll)$ and $F(r} \newcommand{\kk}{k)$ give rise to left and right unit
constraints $\lambda$ and $\rho$ for a skew monoidale in ${\mathscr B}'$ upon forming the
composites
\begin{equation*}
  \vcenter{\hbox{\begin{tikzpicture}[y=0.8pt, x=0.9pt,yscale=-1, inner
        sep=0pt, outer sep=0pt, every text node
        part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (77.4746,873.6249) .. controls (65.7563,893.8888) and (37.6777,911.3520) ..
  node[below right=0.06cm,pos=0.6] {$\mathfrak l$}(22.5254,894.9370)
  (119.7970,902.3622) .. controls (89.8639,902.6202) and (75.3129,897.3809) ..
  node[above left=0.12cm,at start] {$\mathfrak l$\!}(63.5684,893.3012)
  (57.8098,891.3693) .. controls (48.0652,888.3122) and (38.8450,886.9959) ..
  node[above right=0.06cm,pos=0.4] {$1\mathfrak l$}(21.7678,891.8571);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (77.5761,869.1198) .. controls (65.5753,863.5577) and (30.6214,880.5944) ..
  node[above left=0.06cm,pos=0.5] {$\mathfrak a$}(22.2728,888.9282);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (120.0495,882.3622) .. controls (94.0270,882.3622) and (96.0094,871.2505) ..
  node[above left=0.12cm,at start] {$\mathfrak l 1$\!}(80.0495,871.2505);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (-10.0000,882.3622) .. controls (-0.5398,882.3622) and (13.5355,882.8673) ..
  node[above right=0.12cm,at start] {\!\!$Fi. 1$}(19.0914,888.9282);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (-10.0000,902.3622) .. controls (-0.8054,902.3622) and (14.5457,900.5944) ..
  node[above right=0.12cm,at start] {\!$Ft$}(18.8388,894.0284);
  \path[fill=black] (78.568542,870.78729) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3387) {$\sigma^{\scriptscriptstyle -1}$     };
  \path[fill=black] (19.697973,892.25299) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3391) {$F\ll$   };
      \end{tikzpicture}}} \qquad
\text{and} \qquad
  \vcenter{\hbox{\begin{tikzpicture}[y=0.8pt, x=0.9pt,yscale=-1, inner
        sep=0pt, outer sep=0pt, every text node
        part/.style={font=\scriptsize} ]
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (106.0000,862.3622) -- node[above left=0.12cm,at end] {$1 \mathfrak r$\!}(130.0000,862.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (77.5241,882.8673) -- node[above left=0.12cm,at end] {$1.Fi$\!\!} (130.0000,882.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (77.5241,886.7048) .. controls (85.1003,895.0386) and (105.7563,902.3622) ..
  node[above left=0.12cm,at end] {$Ft$\!}(130.0000,902.3622);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (103.9848,865.4840) .. controls (99.6916,869.7771) and (81.5698,871.5449) ..
  node[above left=0.05cm,pos=0.5] {$\mathfrak a$}(77.7817,878.8685);
  \path[draw=black,line join=miter,line cap=butt,line width=0.650pt]
  (103.7323,860.6857) .. controls (93.8833,844.1683) and (32.0387,832.8929) ..
  node[above=0.05cm,pos=0.66] {$\mathfrak r$}(33.0291,868.4666) .. controls (33.5742,888.0447) and (62.3769,897.3038) ..
  node[below=0.05cm,pos=0.8] {$\mathfrak r^\centerdot$}(73.2361,886.1921)
  (72.7259,882.0107) .. controls (66.4734,880.4919) and (47.8665,880.0812) ..
  node[above right=0.05cm,pos=0.66,rotate=-10] {$\mathfrak r^\centerdot 1$}(47.6192,867.2444) .. controls (47.4461,858.2562) and (59.4891,852.3199) .. (74.8220,848.5084)
  (85.9387,846.1826) .. controls (102.4316,843.2893) and (120.1558,842.3313) ..
  node[above left=0.12cm,at end] {$\mathfrak r$\!}(130.0000,842.3622);
  \path[fill=black] (75.508904,883.41418) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3622) {$Fr} \newcommand{\kk}{k$    };
  \path[fill=black] (104.80333,862.45349) node[circle, draw, line width=0.65pt, minimum width=5mm, fill=white, inner sep=0.25mm] (text3626) {$\tau^{\scriptscriptstyle -1}$   };
      \end{tikzpicture}}}\ .
\end{equation*}
The assignments $F(\ll) \mapsto \lambda$ and $F(r} \newcommand{\kk}{k) \mapsto \rho$ are
in fact bijective, the former since it is given by composing with an
invertible $2$-cell, and the latter since it is given by composition
with an invertible $2$-cell followed by transposition under
adjunction. Thus the two-dimensional data of $F$ and of a skew
monoidale in ${\mathscr B}'$ are in bijective correspondence.

Finally, we find after some calculation that, with respect to the
$\alpha$, $\lambda$ and $\rho$ defined above, equations
\eqref{stringaxiom1}, \eqref{stringaxiom2}, \eqref{stringaxiom3},
\eqref{stringaxiom4}, \eqref{stringaxiom6} and \eqref{stringaxiom7}
express precisely the five axioms for a skew monoidale in ${\mathscr B}'$;
equation \eqref{stringaxiom5} specifies $F(\kk)$ and nothing more;
whilst equations \eqref{stringaxiom8} and \eqref{stringaxiom9} are
both equalities which follow using only the axioms for a monoidal
bicategory. We have thus shown:

\begin{theorem}
There is a bijection between simplicial maps $\mathbb C \to N{\mathscr B}$ and skew monoidales in ${\mathscr B}'$.
\end{theorem}

\begin{remark}
If we consider a bicategory ${\mathscr K}$ (not necessarily monoidal), and let $N{\mathscr K}$ be its nerve, then simplicial maps $\mathbb C \to N{\mathscr K}$ are monads in ${\mathscr K}$.
\end{remark}

\begin{remark}
  The assignation ${\mathscr B} \mapsto N({\mathscr B})$ sending a monoidal bicategory
  to its nerve can be extended to a functor $N \colon
  \mathrm{MonBicat}_s \to \mathrm{SSet}$, where $\mathrm{MonBicat}_s$
  is the category of monoidal bicategories and morphisms which
  strictly preserve all the structure. When seen in this way, the
  nerve functor has a left adjoint $\Phi$; it follows that $\Phi(
  \mathbb{C})$ is the free monoidal bicategory containing a skew
  monoidale.
\end{remark}

redpajama_set_name: RedPajamaArXiv


text:
Xesús Antonio Gulías Lamas (Beariz, Orense, 29 de mayo de 1948) es un escritor y antropólogo español.

Biografía
Estudió en el Seminario de Orense y en el Instituto Nacional de Misiones Extranjeras (IEME) en Madrid, donde también cursó antropología.

Su interés por la cantería y por la arquitectura medieval lo llevó a escribir el Diccionario do Verbo dos Canteiros (En español: Diccionario del verbo de los canteros)  en el año 2000.

Gran conocedor de la cultura tradicional gallega y más especialmente, la relacionada con el arte de la cartería. Muestra de ello es el libro, "O Hórreo en Galicia" publicado en 2013 por la editorial Ir Indo.

En 2013 también publicó el libro "Remendos dunha sotana" en el que rememora todos los recuerdos de sus días de niñez y juventud en el Seminario de Ourense. El libro lleno de anécdotas repasa a compañeros y profesores a lo largo de toda una década.

Su compromiso con Beariz y con la Terra de Montes lo guiaron siempre en la investigación de su historia y en la recogida de sus leyendas y tradiciones.

Ha sido presidente de la Asociación de Amigos da Terra de Montes (2003-2005) y ha recibido la medalla de oro de esta Asociación en el año 2009.

El 16 de octubre de 2015, fue nombrado de Hijo Predilecto del Ayuntamiento de Beariz.

Obras

 Beariz de Montes, aspecto antropolóxico, histórico e cultural dun pobo.
 O Castelo do Castro do Castro de Montes.
 Evolución socio-económica das parroquias de Lebozán, Beariz e Xirazga.
 Polos camiños de Beariz de Montes.
 Do castro de Magros a San Pedro.
 Nas ribeiras do Lérez.
 A parroquia de Beariz no Tumbo Grande de Acibeiro.
 Xuíces meiriños inéditos da Terra de Montes.
 Historia para unha romanza de cego.
 Un nome para a eternidade.
 Os ronseis da pedra.
 Un sol de eternidade.
 Xotitle e o indiano.
 Polos vieiros da Pedra.
 O románico e as marcas dos canteiros no cuadrante noroccidental de Galicia.
 A lenda do Churruchao.
 Lenda da Serpe da fonte de Ricobanco.
 O Castelo de Outes e os monumentos medievais do Barbanza.
 A fortaleza da Barreira.
 Hórreos de Galicia.
 Diccionario do Verbo dos canteiros

Referencias

Enlaces externos
 Arte de Galicia
 Castro de Montes
 Castelo de Outes
 Cerviño e o Cruceiro de Xirazga
 Xente de Beariz

Antropólogos de España del siglo XX
Antropólogos de España del siglo XXI
Nacidos en Beariz
Escritores de Galicia

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text:
Home › Compendium › Third Place › The Third Place: Dealing with the Wealthy (Pt. I of II)
The Third Place: Dealing with the Wealthy (Pt. I of II)
There is a justifiable Intent for doing away with most forms of Taxation such as Income Taxation. I had discussed the topic of taxes before in The Work-Standard (2nd Ed.) where I provided some convenient arguments for deemphasizing Taxation in favor of the Arbeit and Geld created from the LER Process. When I advocated for eliminating Income Taxation, I was essentially arguing in favor of a different methodology to accommodate the other factors introduced by the Work-Standard. Under the Work Theory of Money (WTM), Wages and Prices, despite continuing to relay information about the national economy, are both dependent on the Quality of Arbeit (QW) and Quality of Geld (QM). Issuances of Paygrades are treated as a matter of the Council State because the employers and employees of different Enterprises and Industries now had to focus more on their individual and collective contributions of Arbeit and Geld to the Life-Energy Reserve. The amount of Actual Geld allocated toward the Paygrades by the Council State is impacted by the overall performance of the VCS Economy and the Attrition/Inaction Rate, in addition to the usual qualifications like "years of service" and "number of dependents."
Without Income Taxation, as the reasoning from Social-Democrats and Social Liberals (Read: Progressives) went, there would be nothing to deter people from becoming wealthier than others. How we intend to interpret their positions depends on whether we are evaluating them from the standpoints of Production for Profit, Production for Utility, or Production for Dasein. Production for Profit and Production for Utility represent the two sides in the issue of "Wealth Inequality."
In Production for Profit, we would have to argue that by promoting a "Progressive Taxation System," there will be fewer 'Incentives' for the Market/Mixed Economy to generate a higher Quantity of Kapital to surpass the concurring Quantity of Schuld. The upper classes, because they have more Kapital, can afford to pay down their Schuld, whereas the same cannot always be said for the lower classes. To have the upper classes spend more of their Kapital in taxes will leave them with less Kapital to maintain Kapital Accumulation.
Conversely, in Production for Utility, the "Progressive Taxation System" will create a Wealth Redistribution effect that will trickle-down a fixed Quantity of Kapital to the lower classes to mitigate their Quantity of Schuld. As stated earlier, the lower classes are less likely to ensure that their Quantity of Kapital will surpass their Quantity of Schuld as the upper classes. This Wealth Redistribution occurs in the form of Welfare Capitalist programs created by the Parliament.
When we evaluate the other two Modes of Production in Production for Dasein, we have to realize that the 'Incentives' for the "Progressive Taxation System" revolve around getting the upper classes to spend more of their Kapital. They have to spend more Kapital in order to ensure that the lower classes will have less Schuld. The Parliament's Welfare Capitalist programs function as the vehicle through which the transfer is conducted. The so-called "Wealth Equality" emerges as an "Equilibrium" in Kapital Accumulation between the upper and lower classes: the Quantity of Kapital and the Quantity of Schuld generated by both classes would then be on par with each other. Going back to the Work Theory of Money (WTM), Income Taxes are incapable of directly targeting the LER Process. What it will target instead are two accounts of Actual Geld:
A "Personal Income Tax" on the Actual Geld that Vocations and Professions receive from their Paygrades each week.
A "Corporate Income Tax" on the Actual Geld that Enterprises and Industries receive from selling goods and services.
It takes Actual Geld to contribute Actual Arbeit, just as it takes Actual Arbeit to generate Actual Geld. Income Taxation not only reduces the possibility for Life-Energy Charging (LEC) and leaves larger Enterprises and Industries with less Actual Geld, it is also an inefficient, wasteful solution at addressing the question of what to do with the Actual Geld owned by the wealthy. There are already other methods provided by the Work-Standard in resolving this particular matter. I can think of six different options on what the wealthy could be doing in Production for Dasein:
The first option is to have the wealthy invest their Actual Geld into some NSFIs at the Kontor (Financial Office), giving the Totality and the State the ability to create more Actual Arbeit.
The second option involves the wealthy spending their Actual Geld on goods and services from Small Businesses and Cooperatives, which will also support the Totality and State.
The third option entails the wealthy allocating their Actual Geld toward specific Personal Properties that, when attached to a Domain within the Work-World, can be turned into a Productive Property as part of a new Enterprise.
The fourth option is for the State to provide the wealthy with opportunities to invest their Actual Geld into research and development projects at the Kontor.
The fifth option is for the State to help the wealthy spend their Actual Geld on causes that benefit the national interests of the Totality.
And the sixth option, which will be focus of this Section, is for the wealthy to invest their Actual Geld into the affairs of the SSE by working with its Student Government.
The fifth and sixth options tend to overlap with each other. The Student Government might be interested in promoting certain initiatives that do not necessarily require the intervention of the Council State. A likewise occurrence can also emerge from the Council State, only this time the SSE will not be involved. In both cases, the Council State and the Student Government are trying to provide new Intents for the wealthy to do something with their Actual Geld. It should be noted that not everyone who is wealthy will have political-economic motivations for their Actual Geld. Sometimes, there might be those who believe that they should spend their wealth on causes that appeal to them. That would be the opportune moment for the Council State or the Student Government to take the initiative and turn the cause itself into a national policy issue.
On their own, the Actual Geld of the wealthy is nothing without the Totality and the State. It matters very little as to how much Actual Geld somebody has in their personal accounts. They can spend their wealth on promoting certain causes which benefit the Totality and State, but their efforts alone will not take them far. The Council State and the Student Government's involvement will provide them with access to manpower and resources, which can include the participation of religious congregations. That too is where Production for Dasein needs to distinguish itself from the other two Modes of Production. In essence, the Actual Geld of the wealthy should be spent in a manner that will lead to the flourishing of the Totality and without losing their wealth while doing so. Such acts of charity must not be allowed to devolve into ways of evading taxation (as in Production for Utility) or being a poor substitute for the Council State (as in Production for Profit).
The question that needs to be ascertained now is an in-depth discussion of how each of the six options can be applied in practice. In the next half of this Entry, I will address the Explicit and Implicit Intents behind the wealthy's spending habits in the first and second Modes of Production and how they might change in Production for Dasein. Then I will determine why the wealthy should be inclined in investing their wealth in any one of the six options.
‹ Patagonia's founder giving away his wealth shows his stance that 'every billionaire is a policy failure,' CEO says — Fortune
Update (16 September 2022) ›
Categories: Third Place
Tags: Economics, Finance, Philosophy, Politics, Psychology, Socialist Finance, Socialist Student Economy, Taxes, Third Place, Total Educational Effort, Total Mobilization, Wealth Inequality, Work-Standard

redpajama_set_name: RedPajamaCommonCrawl


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New York Transit Museum – Tourist Attractions in New York City
New York Transit Museum New York
Home » New York Transit Museum » info
Getting around NYC | Getting around New York City by MTA Subway and Buses | New York MTA
New York Transit Museum | New York Transit Museum Gallery & Store
The New York Transit Museum is a museum which displays historical artifacts of the New York City Subway and bus systems; it is located in the unused Court Street subway station in Brooklyn Heights, Brooklyn, New York City. (On the web: http://mta.info/mta/museum/index.html)
There is a smaller Annex in Grand Central Terminal, Manhattan.
On July 4, 1976, the New York City Transit Exhibit opened in the old station as part of the United States Bicentennial celebration, with one subway token for admittance. Old cars which had been preserved, as well as models and other exhibits were displayed. Plans were to close it after the celebration, but it proved to be so popular that it remained open and eventually became a permanent museum.
The Transit Museum entrance is located at the corner of Boerum Place and Schermerhorn Street. The Museum includes subway memorabilia and other exhibits including heritage signage, models and dioramas of subway, bus and other equipment, and lectures and seminars.
At the subway station level, two tracks contain many examples of New York City Subway equipment. the south track is used to display IND and BMT equipment and the north track is for IRT subway cars and BMT El cars (which are roughly the same size). To elimate the gap between the IRT/El cars, wooden platform extenders have been installed. Some of the museum fleet is operable, and is used in subway excursions run by the Museum and other parties (notably by NYCT Division C as a fundraiser benefiting The March of Dimes) on various parts of the system. Tickets for the Museum excursions (called "Nostalgia Trains") are sold in advance while tickets for the March of Dimes excursions are sold on the day of the excursion. In mid-2005 the March of Dimes excursions were suspended although no official reason has been given.
There are examples of most of the kinds of subway equipment ever used on the New York City Subway, including the predecessor BMT and IRT private companies. Notably absent are the three BMT experimental trains, which were scrapped before the Transit Authority began saving historic equipment. There are also several examples of elevated cars, including the BU cars, elevated gate cars that can be ridden on prearranged excursions.
There are also some Museum cars that are kept in the Coney Island Complex that are either awaiting refurbishment, undergoing refurbishment or just not currently being displayed. These cars can be viewed from passing F trains on the IND Culver Line between Avenue X and Neptune Avenue, and from N trains on the BMT Sea Beach Line between Stillwell Avenue and 86th Street.
The Museum also has a sizable retired bus fleet. However, there is no area set aside for their permanent exhibition. They are stored in Bus depots around the city and brought out for special events, such as the Museums' annual "Bus Festival."
The New York Transit Museum
130 Livingston Street,
10th Floor, Brooklyn, NY 11201
Architects of the NYC Subway, Heins & LaFarge: The Tradition of Great Public Works, Part I
19 March 2007 – 08 July 2007
The exhibition located in the NY Transit Museum in the Shuttle Passage features over sixty historic artifacts, architectural drawings, and vintage and contemporary photographs from the collections of the New York Transit Museum, The New-York Historical Society, the Episcopal Diocese of New York, the Bronx Zoo / Wildlife Conservation Center, and private collections
Architects of the New York City Subway Part II : Squire Vickers and the Subway's Modern Age
30 July 2007 – 28 October 2007
New York Transit Museum Gallery Annex and Store at Grand Central Terminal
Between 1906 and 1940, Squire J. Vickers and his team of architects designed more than 300 subway stations for the Dual Contracts and the IND. An accomplished painter, Vickers' subway work reflects his changing aesthetic, moving seamlessly from Arts and Craft to the Machine Age style. Continuing the series from Part I, the exhibit contains additional drawings, paintings, objects, and archival photos from the Museum's and other collections to show how Squire Vickers integrated a more contemporary style into subway stations and structures.
Guest Cars
July 2007 – December 2007
New York Transit Museum, Brooklyn Heights
Each season, the Transit Museum displays a "guest" car, highlighting vintage and current vehicles that aren't normally included in our permanent exhibit. This summer a pair of cars from one of the last Money Trains will be on display in the Transit Museum. These cars ran throughout the system at night collecting revenue from stations and delivering it to the "Money Room" formerly at Jay Street in Brooklyn. As a preview for "Show Me the Money," a new exhibit opening this fall about NYC Transit's legendary Money Room and Money Train operations (which ended in January 2006), Revenue Cars 0R 714 and IR 714 will be on view. These two workhorses were originally R-22 passenger cars 7194 and 7422, built in 1957-58. They were converted in 1988-as a collection car and a crew car-repainted in typical work train "safety" yellow and black, and operated in revenue-collection service through their final trips.
Tuesday – Friday 10 a.m. to 4 p.m.
Saturday and Sunday Noon to 5 p.m.
Closed Mondays and major holidays
Children 3 – 17 years of age $3
Senior Citizens (62+) $3
Seniors Free Wednesdays
Museum members: Free
By Subway:
2 3 4 5 to Borough Hall.
M R to Court Street
A C G to Hoyt-Schermerhorn Street
A C F to Jay Street/Borough Hall.
B25, B26, B37, B38, B41, B45, B52, B57, B61, B65, B67, B75 all stop within 1 to 2 blocks of the museum.
From Manhattan: take the Brooklyn Bridge, proceed straight on to Adams Street. Adams will become Boerum Place. Go two blocks to Schermerhorn Street.
Next story Yeshiva University Museum New York City
Previous story Metropolitan Museum New York City


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