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Home Archived Alex Rodriguez To Be Uncomfortable on Yankees' Bench For 4-To-6 Months With...
Alex Rodriguez To Be Uncomfortable on Yankees' Bench For 4-To-6 Months With Injured Hip
Kyle Scanlan
The New York Yankees' announced Monday that Alex Rodriguez will have a surgery on his left hip that will leave the third baseman uncomfortable on the team's bench during games for up to six months after the surgery.
Despite being a three-time Most Valuable Player and being owed around $115 million on the remainder of his contract with the Yankees, Rodriguez was benched by manager Joe Girardi during the teams playoff run this October. He was 3-for- 25 in the playoffs without an extra-base hit and smiled a career low 34 times.
Now the once superstar athlete, who probably had unprotected sex with Madonna, will have arthroscopic surgery to repair a torn labrum, as well as correct bone alignment and a cyst. Rodriguez had a very similar surgery on his right hip in 2009, but he was a much better baseball player then.
The Yankees' third baseman will need four to six weeks to decide which gym shoes to wear to games this year and another two to three months to think of clever nicknames for his teammates while chewing sunflower seeds.
"I'm just worried about Alex's health," said Manager Joe Girardi. "I can't imagine what it's like to think one minute you're going to be sitting on your team's bench perfectly healthy, and the next minute you find out your hip is going to hurt while you sit on your team's bench."
When asked for comment, Rodriguez, the 19-year Major League veteran, summed it up: "I really don't give a shit anymore."
Written by: Kyle Scanlan
Follow Kyle on Twitter: @kylescanlan http://twitter.com/#!/kylescanlan
short stop
third baseman
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http://kylescanlan.com
Kyle Scanlan grew up in rural southern Illinois and has the vocabulary to prove it. Now he lives in the big fancy city of Chicago, because he tells jokes there. When not telling jokes, he's usually drunkenly yelling at his television during NBA games. He is a contributor, editor and creator of The Whiskey Journal. He's not excited about the future.
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\begin{center}
  {\@setfontsize\Large\@xviipt{22}
  \textbf{Data types with symmetries and \\
  polynomial functors over groupoids
  }\par}

  \vskip 22pt

  {\@setfontsize\large\@xivpt{18} Joachim Kock\footnote{Email:
  \href{mailto:kock@mat.uab.cat} {\texttt{\normalshape kock@mat.uab.cat}}}$^,$%
\footnote{The author has profited greatly from related collaborations with
  Andr\'e Joyal, Nicola Gambino, Anders Kock, Imma G\'alvez, Andy Tonks,
  and notably David Gepner.
  Partial support from
  research
  grants
  MTM2009-10359
and
MTM2010-20692
of Spain is gratefully acknowledged.}
}

\vspace{12pt}

{\small \itshape{Departament de matem\`atiques \\
Universitat Aut\`onoma de Barcelona \\[-6pt]
    Spain}}

\end{center}

{\small

   \vskip 20pt
   \hrule height 0.4pt
   \vskip 5pt

  \noindent
  {\bf Abstract}
  \par\medbreak

  \noindent Polynomial functors (over $\kat{Set}$ or other locally cartesian closed
  categories) are useful in the theory of data types, where they are often
  called containers.  They are also useful in algebra, combinatorics, topology,
  and higher category theory, and in this broader perspective the polynomial
  aspect is often prominent and justifies the terminology.  For example,
  Tambara's theorem states that the category of finite polynomial functors is
  the Lawvere theory for commutative semirings \cite{Tambara:CommAlg},
  \cite{Gambino-Kock:0906.4931}.

  \noindent In this talk I will explain how an upgrade of the theory from sets
  to groupoids (or other locally cartesian closed $2$-categories) is useful to
  deal with data types with symmetries, and provides a common generalisation of
  and a clean unifying framework for quotient containers (in the sense of Abbott
  et al.), species and analytic functors (Joyal 1985), as well as the stuff
  types of Baez and Dolan.  The multi-variate setting also includes relations
  and spans, multispans, and stuff operators.  An attractive feature of this
  theory is that with the correct homotopical approach --- homotopy slices,
  homotopy pullbacks, homotopy colimits, etc.~--- the groupoid case looks
  exactly like the set case.

  \noindent After some standard examples, I will illustrate the notion of
  data-types-with-symmetries with examples from perturbative quantum field
  theory, where the symmetries of complicated tree structures of graphs play a
  crucial role, and can be handled elegantly using polynomial functors over
  groupoids.  (These examples, although beyond species, are purely combinatorial
  and can be appreciated without background in quantum field theory.)

  \noindent Locally cartesian closed $2$-categories provide semantics for a
  $2$-truncated version of Martin-L\"of intensional type theory.  For a
  fullfledged type theory, locally cartesian closed $\infty$-categories seem to
  be needed.  The theory of these is being developed by David Gepner and the
  author as a setting for homotopical species, and several of the results
  exposed in this talk are just truncations of $\infty$-results obtained in
  joint work with Gepner.  Details will appear elsewhere.

  \vskip 5pt

  \noindent
  {\it Keywords: \ }
    Polynomial functors, groupoids, data types, symmetries, species, trees.

    \vskip 7pt
    \hrule height 0.4pt
    \par

  \vfill
}

\newcommand{\arxiv}[1]{\href{http://arxiv.org/pdf/#1}{ArXiv:#1}}

\section{Polynomial functors over $\kat{Set}$ and data types}
\label{intro}

\begin{blanko}{Polynomial functors in one variable.}\label{bla:1var}
  In its simplest form, a {\em polynomial functor} is an endofunctor
of $\kat{Set}$ of the form
\begin{equation}
X \mapsto \sum_{b\in B} X^{E_b} .
  \label{1var}
\end{equation}
Here the sum sign is disjoint union of sets,
$X^{E_b}$ denotes the
hom set $\operatorname{Hom}(E_b,X)$, and
$(E_b\mid b\in B)$
is a $B$-indexed family of sets, encoded
conveniently as a single map of sets
$$
E\to B.
$$
Viewed as a data type constructor, $E\to B$ is often called a {\em container}
\cite{Abbott,Abbott-et-al:fossacs03,Abbott-Altenkirch-Ghani:strictly-positive,Abbott-et-al:tlca03,Abbott-Altenkirch-Ghani-McBride:qotient,Altenkirch-Morris:lics09}%
; then $B$ is regarded as a set of shapes,
and the fibre $E_b$ is the set of positions in the shape corresponding to $b$.
The data to be inserted into these positions can be of any type $X$:
the polynomial functor receives a type $X$ (a set)
and returns the new more elaborate type $\sum X^{E_b}$.
Polymorphic functions
correspond to natural transformations of polynomial functors, and
these can be handled in terms of the representing sets $E \to B$ alone,
cf.~\cite{Abbott}, \cite{Gambino-Kock:0906.4931}, and \ref{NT} below.
A fundamental example is the list functor, $X\mapsto \sum_{n\in \mathbb{N}} X^{\fat n}$,
which to a set $X$ associates the set of lists of elements in $X$.
Here $n\in \mathbb{N}$ is the shape, and $\fat n$ denotes the
$n$-element set $\{0,1,\ldots,n-1\}$ of positions in
a length-$n$ list.

There is another important use of polynomial functors in type theory:
one then regards
$E\to B$ as a signature generating an algebra, namely the initial algebra for
the polynomial functor.  Initial algebras for polynomial functors are inductive
data types, corresponding to W-types
in (extensional) Martin-L\"of type
theory~\cite{Petersson-Synek}, \cite{Moerdijk-Palmgren:Wellfounded}.
Similarly, terminal coalgebras are coinductive data types (sometimes called
M-types), often interpreted as programs or systems (see for
example~\cite{Rutten:universal-coalgebra},
\cite{Hancock-Setzer:CSL2000}).
\end{blanko}

\begin{blanko}{Species and analytic functors.}
A functor is {\em finitary} when it preserves $\omega$-filtered colimits.  For a
polynomial functor this is equivalent to $E\to B$ having finite fibres.
Let
$\mathbb{B}_\omega$ denote the groupoid of finite sets and bijections.  A {\em species}
\cite{Joyal:1981} is a
functor $F:\mathbb{B}_\omega \to \kat{Set}$, written $S \mapsto F[S]$; the set $F[S]$ is to be
thought of as the set of $F$-structures that can be put on the set $S$.  The
{\em extension} of $F$ is the endofunctor
\begin{align}\label{anal}
  \kat{Set} \ & \longrightarrow \ \ \kat{Set}  \\
  X\  & \longmapsto \  \sum_{n\in \mathbb{N}} \frac{F[n] \times X^n}{\operatorname{Aut}(n)} \notag
\end{align}
which is the left Kan extension of $F$ along the (non-full) inclusion $\mathbb{B}_\omega\subset
\kat{Set}$.  A functor of this form is called {\em analytic} \cite{Joyal:foncteurs-analytiques}.
Joyal established an equivalence of
categories between species and analytic functors, and characterised analytic
functors as the finitary functors preserving cofiltered limits and weak
pullbacks ~\cite{Joyal:foncteurs-analytiques}, see also~\cite{Hasegawa}
and~\cite{Adamek-Velebil:TAC21}.
Finitary polynomial functors are precisely the analytic functors
which preserve pullbacks strictly.  In terms of species they correspond to
those for which the symmetric group actions are free.

Monoids in species (under the operation of substitution, which corresponds to
composition of analytic functors) are precisely operads.
Many important polynomial functors
have the structure of monad.  For example,
the list functor has a natural monad structure by concatenation of lists.
Polynomial monads equipped with a cartesian monad map to the list monad are the
same thing as non-symmetric operads~\cite{Leinster:0305049}.
More generally, finitary
polynomial monads correspond to projective operads~\cite{Kock:0807} (i.e.~such
that every epi to it splits).
\end{blanko}

\begin{blanko}{Polynomial functors in many variables.}
  Following \cite{Gambino-Kock:0906.4931}, a {\em polynomial} is a diagram of sets
\begin{equation}\label{IEBJ}
I \stackrel s \longleftarrow E \stackrel p \longrightarrow B \stackrel t
\longrightarrow J ,
\end{equation}
and
the associated {\em polynomial functor} (or the {\em extension} of the polynomial)
is given by the composite
\begin{equation}\label{poly}
\kat{Set}_{/I} \overset{\upperstar{s}}\longrightarrow
\kat{Set}_{/E} \overset{\lowerstar{p}}\longrightarrow
\kat{Set}_{/B} \overset{\lowershriek{t}}\longrightarrow
\kat{Set}_{/J} \,,
\end{equation}
where $\upperstar{s} $ is pullback along $s$,
$\lowerstar{p} $ is the right adjoint
to pullback (called dependent product),
and $\lowershriek{t}$ is left adjoint to
pullback (called dependent sum).
For a map $f:B\to A$ we have the three explicit formulae
\begin{align}
\upperstar{f}  (X_a\mid a\in A) &= ( X_{f(b)}\mid b\in B)
  \label{pbk}\\[3pt]
\lowershriek{f} (Y_b \mid b\in B) &= (\underset{b\in B_a}{\textstyle{\sum}} Y_b \mid a\in A)
  \label{sum}\\
\lowerstar{f}  (Y_b \mid b\in B) &= (\underset{b\in B_a}{\textstyle{\prod}} Y_b \mid a\in A) \,,
  \label{prod}
\end{align}
giving altogether the following formula for \eqref{poly}
$$
(X_i \mid i\in I) \longmapsto (\sum_{b\in B_j} \prod_{e\in E_b} X_{s(e)}  \mid
j\in J) ,
$$
which specialises to \eqref{1var} when $I=J=1$.

The multi-variate polynomial functors correspond to {\em indexed
containers}~\cite{Altenkirch-Morris:lics09}, and their initial algebras
are sometimes called {\em general tree types}~\cite[Ch.~16]{Nordstrom-Petersson-Smith}.

From the abstract description in terms of adjoints, it follows that the notion
of polynomial functor (and most of the
theory) makes sense in any locally cartesian closed category, and polynomial
functors are the most natural class of functors between slices of such
categories.  They have been characterised intrinsically~\cite{Kock-Kock:1005.4236}
as the local fibred right adjoints.
\end{blanko}

\begin{blanko}{Incorporating symmetries.}
  A container is a rigid data structure: it does  not allow for data
to be permuted in any way among the positions of a given shape.
In many situations it is desirable to allow for permutation, so that
certain positions within a shape become indistinguishable.
In quantum physics, the principle of indistinguishable particles
imposes such symmetry at a fundamental level.
A fundamental
example is the multiset data type,
whose extension is the functor
\begin{equation}\label{exp}
X\mapsto \sum_{n\in\mathbb{N}} \frac{X^{\fat n}}{\operatorname{Aut}(\fat n)},
\end{equation}
which is analytic but not polynomial.

In order to account for such data types with symmetries,
Abbott et al.~\cite{Abbott-Altenkirch-Ghani-McBride:qotient}  (see also
Gylterud~\cite{Gylterud})
have extended the container formalism by adding the symmetries `by hand':
for each shape (element $b$ in $B$) there
is now associated a group of symmetries of the fibre $E_b$, and data inserted
into the corresponding positions is quotiented out by this group action.
It is not difficult to see (cf.~also \cite{Adamek-Velebil:TAC21})
that in the finitary case, this is precisely the notion of species
and analytic functors.

In fact it has been in the air for some time (see for example
\cite{Fiore:FOSSACS2005}, and more recently \cite{Carette-Uszkay},
\cite{Yorgey:2010}) that species should be a good framework for data type
theory.  It is the contention of the present contribution that polynomial
functors over groupoids provide a clean unifying framework: in the setting of
groupoids, the essential distinction between `analytic' and `polynomial'
evaporates (\ref{anal=poly}), and the functors can be represented by diagrams
with combinatorial content \eqref{IEBJ} just as polynomials over sets, as we
proceed to explain.

From the viewpoint of species, there are other reasons for this upgrade anyway.
In fact, it was soon realised by combinatorists that the 1985 notion of analytic
functors is not optimal for enumerative purposes: taking
cardinality simply does not yield
the exponential generating functions central to enumerative combinatorics!
(It does so if the analytic functor is polynomial.) In
fact, the Species Book~\cite{Bergeron-Labelle-Leroux} does not mention analytic
functors at all.

The issue was sorted out by Baez and Dolan~\cite{Baez-Dolan:finset-feynman}: the
problem is that dividing out by the group action in \eqref{anal} is a bad
quotient from the viewpoint of homotopy theory, and does not behave well with
respect to cardinality.  To get the correct cardinalities, it is necessary to
use {\em homotopy quotients}, and the result is then no longer a set but a
groupoid, and the cardinality has to be {\em homotopy cardinality}.  So it is
necessary to work from the beginning with groupoids instead of sets.  Baez and
Dolan introduced {\em species in groupoids} (\ref{stuff}), dubbing them {\em
stuff types}, showed that homotopy cardinality gives the correct generating
functions, and illustrated the usefulness of the broader generality by showing
how the types needed for a combinatorial description of the quantum harmonic
oscillator are stuff types, not classical
species~\cite{Baez-Dolan:finset-feynman}.

Joint work with David Gepner closes the circle by observing that over
groupoids, species/analytic functors are the same thing as discrete finitary
polynomial functors (\ref{anal=poly});
hence the neat formalism of polynomials provides a natural unifying
framework for (quotient) containers and species.
\end{blanko}

\section{Polynomial functors over groupoids}

A groupoid is a category in which all arrows are invertible.  A useful intuition
for the present purposes is that groupoids are `sets fattened with symmetries'.
From the correct homotopical viewpoint groupoids behave very much like sets.  We
are interested in groupoids up to equivalence, and for this reason many familiar
$1$-categorical notions, such as pullback and fibre, are not appropriate, as
they are not invariant under equivalence.  The good notions are the
corresponding {\em homotopy} notions, which we briefly recall.  They can all be
deduced from the beautiful simplicial machinery developed by
Joyal~\cite{Joyal:qCat+Kan,Joyal:CRM} to generalise the theory of categories to
quasi-categories (called $\infty$-categories by Lurie~\cite{Lurie:HTT}).  Since
the $2$-category $\kat{Grpd}$ of groupoids has only invertible $2$-cells, it is an
example of a quasi-category.  From now on when we say $2$-category we shall mean
`$2$-category with only invertible $2$-cells'.

\begin{blanko}{Slices.}
  If $I$ is a groupoid, the {\em homotopy\xspace slice} $\kat{Grpd}_{/I}$ is the $2$-category
  of {\em projective cones} with base $I$ (cf.~\cite{Joyal:qCat+Kan}): its
  objects are maps $X \to I$; its arrows
are triangles with a $2$-cell
$$
\xymatrix@C=3ex{
X \ar[rr] \ar[rd] & \ar@{}[d]|{\Rightarrow} & Y \ar[ld] \\
&I&
}$$
and $2$-arrows are diagrams
$$
\xymatrix@C=3ex{
X \ar@{}[rr]|{\Uparrow}
\ar@/_0.5pc/[rr]\ar@/^0.5pc/[rr] \ar[rd] & \ar@{}[d]|{\Rightarrow^\Rightarrow}&
Y \ar[ld] \\
&I&
}$$
commuting with the structure triangles.
More generally, if $d:T \to \kat{Grpd}$ is any diagram, there is a $2$-category
$\kat{Grpd}_{/d}$ of projective cones with base $d$.

A {\em homotopy\xspace terminal object} in a $2$-category $\mathscr{C}$ is an object $t$ such that
for any other object $x$, the groupoid $\mathscr{C}(x,t)$ is contractible,
i.e.~equivalent to a point.
More general homotopy\xspace limits are defined in the usual way using homotopy\xspace slices:
the homotopy\xspace limit of a functor $d:T \to \kat{Grpd}$ is by definition a homotopy\xspace terminal
object in the homotopy\xspace slice $\kat{Grpd}_{/d}$.
Homotopy limits are unique up to equivalence.
\end{blanko}

\begin{blanko}{Pullbacks and fibres.}
  Given a diagram of groupoids $X,Y,S$ indicated by the solid arrows,
$$
\xymatrix{
X\times_S Y \drpullback \ar@{-->}[r]\ar@{-->}[d]
&Y\ar[d]^-{g}\\X\ar[r]_-{f}&S
}
$$
the {\em homotopy\xspace pullback} is the homotopy\xspace limit, i.e.~given as a  homotopy\xspace terminal
object in a a certain slice $2$-category of projective cones over the solid
diagrams of the shape in question, and as such it is determined uniquely up to
equivalence.
A specific model is
the groupoid $X\times_S Y$ whose objects
are triples
$(x,y,\phi)$
 with $x \in X$, $y\in Y$ and  $\phi:fx\to gy$ an arrow of $S$, and whose
 arrows are pairs $(\alpha,\beta):(x,y,\phi)\to(x',y',\phi')$ consisting
 of
 $\alpha: x \to x'$ an arrow in $X$ and $\beta: y \to y'$ an arrow in $Y$
 such that the following diagram commutes in $S$
$$
\xymatrix{
fx\ar[r]^-\phi\ar[d]_-{f(\alpha)}&gy\ar[d]^-{g(\beta)}\\
fx'\ar[r]_-{\phi'}&gy'.
}
$$
(One should note that if $f$ or $g$ is a fibration then the na\"ive
set-theoretic pullback is equivalent to the homotopy\xspace pullback.)

The \emph{homotopy\xspace fibre} $E_b$ of a morphism $p:E\to B$ over an object $b$ in $B$ is
the homotopy\xspace pullback of $p$ along the  inclusion map
$\xymatrix{1\ar[r]^-{\name b}&B}$:
$$
\xymatrix{
E_b\drpullback\ar[r]\ar[d]
&E\ar[d]^-{p}\\1\ar[r]_-{\name b}&B.
}
$$
(Note that the homotopy fibre $E_b$ is not in general a
subgroupoid of $E$, although the map $E_b \to E$ is always faithful.
But again, if $p$ is a fibration then the set-theoretic fibre is equivalent to
the homotopy\xspace fibre.)
\end{blanko}

\begin{blanko}{Homotopy quotients.}
  Whenever a group $G$ acts on a set or a groupoid $X$, the
  {\em homotopy\xspace quotient} $X/G$ is the groupoid obtained by gluing in a path
  (i.e.~an arrow)
  between $x$ and $y$ for each $g\in G$ such that $gx=y$.
  More formally it is the total space of the Grothendieck construction
  of the presheaf $G \to \kat{Grpd}$ that the action constitutes;
  it is a special case
  of a homotopy\xspace colimit.
  (The notation $X/\!/G$ is often used~\cite{Baez-Dolan:finset-feynman}.)
  If $G$ acts on the point groupoid $1$, then
  $1/G$ is the groupoid with one object and vertex group $G$.

  If $p:X\to B$ is a morphism of groupoids, for $b\in B$ the
  `inclusion' of the homotopy\xspace fibre $X_b \to X$ is faithful but not full in general.  But
  $\operatorname{Aut}(b)$ acts on $X_b$ canonically, and the homotopy\xspace quotient $ X_b /\operatorname{Aut}(b) $
  provides exactly the missing arrows, so as to make the natural map $X_b
  /\operatorname{Aut}(b) \to X$ fully faithful.  Since every object $x\in X$ must map to some
  connected component of $B$, we find the equivalence
\begin{align}
X &\simeq
\sum_{b\in \pi_0 B} X_b/\operatorname{Aut}(b) =: \grint{b}{B}{X_b},
\end{align}
expressing $X$ as the homotopy\xspace sum of the fibres, or equivalently as a family
of groupoids (indexed by $\pi_0(B)$ and with a group action in each).
 Given morphisms of groupoids $Y \stackrel p \to B \stackrel f \to A$, we have
 the following `Fubini formula':
  $$
  \grint{b}{B}{Y_b} \simeq \grint{a}{A}{ \grint{b}{B_a}{Y_b}}
  $$
  \end{blanko}
which is actually the formula for the `dependent-sum' functor
$\lowershriek{f} : \kat{Grpd}_{/B} \to \kat{Grpd}_{/A}$ given by postcomposition.
In family notation the formula reads
$$
\lowershriek{f}(Y_b \mid b\in B) \ = \ (\textstyle{\grint{b}{B_a}{Y_b}} \mid a\in A) \,,
$$
just as Formula~\eqref{sum} in the $\kat{Set}$ case.

Pullback along $f:B\to A$, denoted $\upperstar{f}$,
is right adjoint to $\lowershriek{f}$.  This means
of course homotopy\xspace adjoint, and amounts to a natural equivalence of mapping
groupoids
$\kat{Grpd}_{/A}(\lowershriek{f} Y, X) \simeq \kat{Grpd}_{/B}(Y,\upperstar{f} X) .$
The proof relies on the universal property of the pullback.
One may note the following formula for pullback, in family notation:
$$
\upperstar{f}(X_a\mid a\in A) \ = \ (X_{f(b)} \mid b\in B) ,
$$
again completely analogous to the $\kat{Set}$ case (Formula~\eqref{pbk}).

The $2$-category of groupoids is locally cartesian closed.
This means that the pullback functor in turn has a right adjoint
$\lowerstar{f} : \kat{Grpd}_{/B} \to \kat{Grpd}_{/A}$.  The general formula is
an end formula;
for $Y \to B$, the
fibre of $\lowerstar{f} Y$ over $a\in A$ can be described explicitly as the
mapping groupoid
$$
(\lowerstar{f} Y)_a = \kat{Grpd}_{/B}( B_a , Y) .
$$
(A more explicit formula will be derived in the discrete case below.)

\begin{blanko}{Polynomial functors.}
  A {\em polynomial} is a diagram of groupoids
  $$
I \stackrel s \longleftarrow E \stackrel p \longrightarrow B \stackrel t
\longrightarrow J .
$$
The associated {\em polynomial functor} (or the {\em extension} of the polynomial)
is given as the composite
$$
\kat{Grpd}_{/I} \overset{\upperstar{s}}\longrightarrow
\kat{Grpd}_{/E} \overset{\lowerstar{p}}\longrightarrow
\kat{Grpd}_{/B} \overset{\lowershriek{t}}\longrightarrow
\kat{Grpd}_{/J}  .
$$
\end{blanko}

\begin{blanko}{Beck--Chevalley, distributivity, and composition.}
  Given a homotopy\xspace pullback square
  $$
  \xymatrix{
  \cdot \drpullback \ar[r]^{\psi}
  \ar[d]_{\alpha} & \cdot  \ar[d]^{\beta} \\
  \cdot \ar[r]_{\varphi}& \cdot
  }
  $$
  there are natural equivalences of functors
  $$
  \lowershriek{\alpha} \circ \upperstar{\psi}  \isopil
  \upperstar{\varphi} \circ \lowershriek{\beta}
  \qquad \text{ and } \qquad
    \upperstar{\beta} \circ \lowerstar{\varphi}
    \isopil \lowerstar{\psi} \circ \upperstar{\alpha} ,
    $$
   usually called the Beck--Chevalley conditions.
A more subtle feature of the theory is distributivity, which in this setting is
an equivalence saying how to distribute dependent products over dependent sums
(and which can be interpreted as a type-theoretic form of the axiom of
choice~\cite{Martin-Loef-1984}).
We shall not need the details here.  See \cite{Gambino-Kock:0906.4931} for the
classical case, and
Weber~\cite{Weber:1106.1983} for a deeper treatment.
The Beck--Chevalley conditions and distributivity yield a formula for
composing polynomial functors~\cite{Gambino-Kock:0906.4931}.
\end{blanko}

\begin{blanko}{Natural transformations.}\label{NT}
  Just as in the classical case~\cite{Gambino-Kock:0906.4931}, homotopy\xspace cartesian
  natural transformations $P'\Rightarrow P$ of polynomial functors (in one
  variable) correspond precisely to
  homotopy\xspace cartesian diagrams
  $$
  \xymatrix{
  E' \ar[r]\ar[d]\drpullback & B' \ar[d]\\
  E \ar[r] & B .
  }$$
  This is an easy consequence of Beck--Chevalley.
  Showing more generally that arbitrary natural transformations are given
  essentially uniquely by diagrams
  $$
  \xymatrix{
  E' \ar[r] & B' \ar@{=}[d]\\
  \cdot \ar[u]\ar[r]\ar[d]\drpullback & B'\ar[d]\\
  E \ar[r] & B
  }$$
  is a bit more involved and depends on a homotopy\xspace  version of the Yoneda lemma.
  (At the time of this writing, this result is not as precise as in the
  $1$-dimensional case of \cite{Gambino-Kock:0906.4931}.)
\end{blanko}

\begin{blanko}{Spans and stuff operators.}
  Spans of groupoids are the special case of groupoid polynomials where the
  middle map is the identity (or an equivalence).  These constitute a
  categorification of matrix algebra, and were called {\em stuff operators} by
  Baez and Dolan~\cite{Baez-Dolan:finset-feynman}; they have been used to give
  groupoid models for certain aspects of Hecke algebras and Hall
  algebras~\cite{Baez-Hoffnung-Walker:0908.4305}.
\end{blanko}

\section{Exactness; combinatorial polynomial functors}

The following results from \cite{Gepner-Kock} are actually proved in the much
richer setting
of \mbox{$\infty$-groupoids}, but the proofs work also for $1$-groupoids.  We now
suppress
the clumsy `homotopy\xspace' everywhere, although of course all limits and colimits mentioned
refer to the homotopy notions.

\begin{thm}
  (Gepner-Kock~\cite{Gepner-Kock}.)  A functor $\kat{Grpd}_{/I} \to
  \kat{Grpd}_{/J}$ is polynomial if and only if it is accessible and preserves
  conical limits.
\end{thm}
By {\em conical limit} we mean
limit over a diagram with a terminal vertex.
Recall that a functor is accessible \cite[Ch.~5]{Lurie:HTT} when it preserves $\kappa$-filtered
colimits for some regular cardinal $\kappa$.  The regular cardinal here is
explicitly characterised:

\begin{prop}(\cite{Gepner-Kock})
  A polynomial functor given by $I \leftarrow E \stackrel p \to E \to J$
  preserves $\kappa$-filtered colimits if and only if $p$ has $\kappa$-compact
  fibres.
\end{prop}
An important case is $\kappa=\omega$.  A groupoid is $\omega$-compact
when it has finitely many components (i.e.~$\pi_0(X)$ is a finite set) and all
vertex groups are finitely presented.

\begin{blanko}{Discreteness.}
  For many data types occurring in practice (including species and all the
  examples below), although they may have symmetries, the positions in each
  shape form a {\em discrete} groupoid, i.e.~a groupoid equivalent to a set.  In the
  polynomial formalism this amounts to the middle map $p:E\to B$ having discrete
  fibres.  In this case, the dependent product formula simplifies to
$$
(\lowerstar{p} Y)_b = \prod_{e\in \pi_0(E_b)} Y_e ,
$$
  in analogy with \eqref{prod}, and hence all the formulae look exactly like the $\kat{Set}$ case.

  The
  corresponding exactness condition is preservation of sifted colimits.
  A {\em $\kappa$-sifted colimit} is a colimit over a diagram $D$ whose diagonal
  $D\to D^S$ is cofinal for every set $S$ of cardinality $< \kappa$ \cite[Ch.~5]{Lurie:HTT}.
\end{blanko}

\begin{prop}(\cite{Gepner-Kock})
  A polynomial functor given by $I \leftarrow E \stackrel p \to E \to J$
  preserves $\kappa$-sifted colimits if and only if $p$ has $\kappa$-compact
  discrete
  fibres.
\end{prop}

\begin{blanko}{Combinatorial polynomial functors.}
  We call a polynomial functor $I \leftarrow E \stackrel p \to B \to J$ {\em
  combinatorial} if the fibres of $p$ are
  equivalent to finite sets (i.e.~are $\omega$-compact discrete).
\end{blanko}

\begin{blanko}{Species in groupoids (stuff types).}\label{stuff}
  A Baez-Dolan {\em stuff type} \cite{Baez-Dolan:finset-feynman}
is a map of groupoids  $F\to\mathbb{B}_\omega$.
We prefer the name {\em species in groupoids}.  (A classical species is
when the map has discrete fibres, or equivalently is faithful.)
Its extension is the left homotopy\xspace Kan extension
of $n\mapsto F_n$ along $\mathbb{B}_\omega \subset \kat{Grpd}$:
\begin{align*}
  \kat{Grpd} \ & \longrightarrow \  \kat{Grpd}  \\
  X \ & \longmapsto  \ \sum_{n\in \pi_0(\mathbb{B}_\omega)=\mathbb{N}} \frac{F_n \times
  X^n}{\operatorname{Aut}(n)}\,.
\end{align*}
(That's a homotopy quotient of course.)

This functor is polynomial~\cite{Gepner-Kock}: the representing groupoid map is the top
row in the pullback
$$
\xymatrix{
E\ar[r]\ar[d]\drpullback & F \ar[d]\\
\mathbb{B}_\omega' \ar[r] & \mathbb{B}_\omega .
}
$$
This map has finite discrete fibres since $\mathbb{B}_\omega'\to\mathbb{B}_\omega$
has.  (Here $\mathbb{B}_\omega'$ is the groupoid of finite pointed sets.)
Conversely, given a groupoid polynomial $E \to F$ with finite discrete
fibres,
the `classifying map' $F \to \mathbb{B}_\omega$ (obtained since
$\mathbb{B}_\omega'\to \mathbb{B}_\omega$ classifies finite discrete fibrations)
yields a species in groupoids.
One can check that the extension of the polynomial  agrees with the
extension of the species.
In conclusion:
\end{blanko}

\begin{prop}\label{anal=poly} (\cite{Gepner-Kock})
  Combinatorial polynomial functors $\kat{Grpd}\to\kat{Grpd}$ are the same thing as
  analytic functors (in the sense of Baez-Dolan).
\end{prop}

Combining these results we get a `Joyal theorem':

\begin{cor} (\cite{Gepner-Kock})
  A functor $\kat{Grpd}\to\kat{Grpd}$ is analytic (in the sense of Baez-Dolan)
  if and only if it preserves $\omega$-sifted
  colimits and conical limits.
\end{cor}

\begin{blanko}{Generalised species.}
  The relationship between polynomial functors and the generalised species of
  \cite{Fiore-Gambino-Hyland-Winskel:Esp} has been sketched by Gambino and the
  author (unpublished).  A generalised species depends on two categories $I$ and
  $J$, and has as extension a generalised analytic functor $\kat{PrSh}(I) \to
  \kat{PrSh}(J)$; this generalises the 1985 notion but not the Baez-Dolan notion.  If
  $I$ and $J$ are groupoids, these generalised analytic functors correspond to
  the {\em `classical'} extension of combinatorial polynomials over groupoids,
  i.e.~involving $\pi_0$ on quotients.
\end{blanko}

\begin{blanko}{Examples.}
  Groupoid polynomials encode data types in groupoids.  For example,
  $\mathbb{B}'_\omega \to \mathbb{B}_\omega$ encodes the multiset data type: the groupoid $\mathbb{B}_\omega$
  of finite sets and bijections is the
  groupoid of shapes --- the shape of a multiset is really the set indexing its
  elements, not just its size.  There are $\mathbb{N}$-many isoclasses;
  the isomorphisms should be interpreted as propositional equality.
  The fibre over $S\in \mathbb{B}_\omega$ is the discrete groupoid of
  positions in $S$, i.e.~a uniform prescription of positions in multisets
  indexed by $S$. Indeed,  since $\mathbb{B}'_\omega \to \mathbb{B}_\omega$ is a fibration,
  the fibre is canonically identified with the set $S$ itself --- note its natural
  $\operatorname{Aut}(S)$-action.  The discreteness of the fibre means that propositional
  equality
  between positions can be regarded as definitional equality.
The extension of this quotient container is naturally an endofunctor
$\kat{Grpd}\to\kat{Grpd}$.  But one obtains an endofunctor $\kat{Set}\to\kat{Set}$ (in this case
precisely \eqref{exp}) by
precomposing with the natural inclusion $\kat{Set}\to\kat{Grpd}$ and postcomposing
with $\pi_0:\kat{Grpd}\to\kat{Set}$.  The first is harmless.  The second corresponds
to collapsing all isomorphisms to identity, i.e.~interpreting propositional
equality as definitional equality.  If the argument is a set, the only
collapse is the passage from homotopy quotient to na\"ive quotient (of
actions on sets).

The data type of cyclic lists is groupoid polynomial, represented by
$\mathbb{C}'_\omega\to\mathbb{C}_\omega$,
where $\mathbb{C}_\omega$ is the groupoid of finite cyclically ordered sets, and $\mathbb{C}'_\omega$ is the
groupoid of pointed cyclically ordered finite sets.
From \ref{bla:1var}, the list data type is represented by $\mathbb{N}'\to\mathbb{N}$, interpreted as
the groupoids of linearly ordered finite
sets and pointed ditto.  The diagram of groupoids
\[
\xymatrix{
\mathbb{N}' \ar[r]\ar[d]\drpullback & \mathbb{N} \ar[d] \\
\mathbb{C}'_\omega \ar[r]\ar[d]\drpullback & \mathbb{C}_\omega \ar[d] \\
\mathbb{B}_\omega' \ar[r] & \mathbb{B}_\omega
}\]
now represents
the cartesian natural transformations, or polymorphic functions,
from lists to cyclic lists to multisets.
\end{blanko}

\section{Trees}

W-types in Martin-L\"of type theory correspond to initial algebras of polynomial
functors (cf.~\cite{Moerdijk-Palmgren:Wellfounded} and \cite{Gambino-Hyland}
for the extensional case, and
\cite{Awodey-Gambino-Sojakova:1201.3898}
for the fully intensional case).  The initial algebra for $1+P$ can also be
described as the set of operations for the free monad on $P$, which in turn
is the set of {\em $P$-trees}.
$P$-trees (for $P$ a polynomial functor over
$\kat{Set}$ or any lccc) are always rigid, i.e.~have no symmetries.
Abstract trees, on the other hand, admit symmetries, so they are
not $P$-trees for any $\kat{Set}$-polynomial functor $P$, and they are neither W-types nor
containers in the classical sense.
Instead, according to \cite{Kock:0807}, abstract trees {\em are}
themselves polynomial functors.
It is convenient to take the following
characterisation of trees as a definition:

\begin{blanko}{Trees.}\label{polytree-def}
  (\cite{Kock:0807}) A
  {\em (finite) tree} is a diagram of finite sets
    $$
    A \stackrel s \longleftarrow  M \stackrel p \longrightarrow  N  \stackrel
    t \longrightarrow  A
$$
satisfying the following three conditions:

  (1) $t$ is injective

  (2) $s$ is injective with singleton complement (called  {\em
  root} and denoted $1$).

  \noindent With $A=1+M$,
  define the walk-to-the-root function
  $\sigma: A \to A$ by $1\mapsto 1$ and $e\mapsto t(p(e))$ for
  $e\in M$.

  (3)  $\forall x\in A : \exists k\in \mathbb{N} : \sigma^{k}(x)=1$.

  The elements of $A$ are called {\em edges}.  The elements of $N$
  are called {\em nodes}.  For $b\in N$, the edge $t(b)$ is called
  the {\em output edge} of the node.  That $t$ is injective is just to
  say that each edge is the output edge of at most one node.  For
  $b\in N$, the elements of the fibre $M_b$ are
  called {\em input edges} of $b$.  Hence the whole set
  $M=\sum_{b\in N} M_b$ can be thought of as the set of
  nodes-with-a-marked-input-edge, i.e.~pairs $(b,e)$ where $b$ is a
  node and $e$ is an input edge of $b$.  The map $s$ returns the
  marked edge.  Condition (2) says that every edge is the input edge
  of a unique node, except the root edge.
  Condition (3) says that if you walk towards the root, in a finite
  number of steps you arrive there.
  The edges not in the image of $t$ are called {\em leaves}.
\end{blanko}

\begin{blanko}{Decorated trees: $P$-trees} (\cite{Kock:0807}; see also
  \cite{Kock:1109.5785,Kock:graphs-and-trees,Kock-Joyal-Batanin-Mascari:0706})
  An efficient way of encoding and manipulating decorations of trees
  is in terms of polynomial endofunctors.  Let $P$ be a polynomial
  endofunctor given by $I \overset d\leftarrow E \overset q\to B \overset c\to I$.
  A {\em $P$-tree} is a diagram
  \begin{equation}\label{Ptree}
  \xymatrix @!C=16pt {
  A\ar[d]
  & M\ar[l]
  \drpullback\ar[r] \ar[d]& N \ar[d] \ar[r]
  &A\ar[d] \\
  I & \ar[l] E  \ar[r] & B \ar[r]&I \,, \\
}\end{equation}
  where the top row is a tree.  The squares are commutative up to
  isomorphism, and it is important that the $2$-cells be specified as part of
  the structure.  Unfolding the definition, we see that a $P$-tree is a tree
  whose edges are decorated in $I$, whose nodes are decorated in $B$, and with
  the additional structure of an equivalence $M_n \simeq E_b$ for each node $n \in
  N$ with decoration $b \in B$ (this is essentially just a bijection, since the
  fibres are discrete), an iso in $I$ between the decoration of an edge
  $m\in M_n$ and the corresponding $d(e)$, and finally an iso in $I$ between the
  decoration of the output edge of $n$ and $c(b)$.
\end{blanko}

\begin{blanko}{Examples of $P$-trees.}
  Natural numbers are $P$-trees for the identity monad $P(X)=X$, and
  are also the set of operations of the list monad.
  Planar finite trees are $P$-trees for $P$
  the list monad, and are also the set of operations of the free-non-symmetric-operad
  monad~\cite{Leinster:0305049}.  These two examples are the first entries of a canonical sequence
  of inductive data types underlying several approaches to higher
  category theory, the {\em opetopes}: opetopes in dimension $n$
  are $P$-trees for
  $P$ a $\kat{Set}$-polynomial functor whose operations are
  $(n-1)$-opetopes~\cite{Kock-Joyal-Batanin-Mascari:0706};  hence opetopes are
  higher-dimensional trees.

  Abstract finite trees are $P$-trees for
  the multiset functor
  $
  1 \leftarrow \mathbb{B}'_\omega \to \mathbb{B}_\omega \to 1,
  $
  but cannot be realised as $P$-trees for any $\kat{Set}$-polynomial $P$.
\end{blanko}

\begin{blanko}{Trees of Feynman graphs.}
  In the so-called BPHZ renormalisation of perturbative quantum field theories,
  one is concerned with nestings of \linebreak 1-particle irreducible (1PI)
  Feynman graphs, i.e.~graphs~\cite{Joyal-Kock:0908.2675} for which no single
  edge removal disconnects.  Kreimer~\cite{Kreimer:9707029} discovered that the
  BPHZ procedure is encoded in a Hopf algebra of (non-planar) rooted trees,
  expressing the nesting of graphs.
  \begin{center}
  \begin{texdraw}
  \bsegment
	\linewd 1
      \move (-3 0) \lvec (20 0) \Onedot \lvec (68 40)
      \move (56 30) \Onedot \lvec (56 -30) \Onedot
      \move (32 10) \Onedot \lvec (32 -10) \Onedot
      \move (20 0) \lvec (68 -40)

      \move (56 9) \Onedot
      \move (56 -9) \Onedot
      \move (56 0) \larc r:9 sd:-90 ed:90

      \writeps{1 0 0 setrgbcolor}
      \lpatt (1 3)
      \move (28 -1) \freeEllipsis{13}{20}{0}
      \move (58 0) \freeEllipsis{11}{16}{0}
      \move (43 0) \freeEllipsis{35}{41}{0}

  \esegment

  \move(150 -20)
  \bsegment
  	\linewd 1
  \move (0 0) \Onedot \lvec (-12 20) \Onedot
  \move (0 0) \lvec (12 20) \Onedot
  \esegment

\move (260 -35)

  \bsegment

    \move (0 0)
    \lvec (0 20) \Onedot \lvec (-4 64)
    \move (0 20) \lvec (6 62)
    \move (0 20) \linewd 1
    \lvec (25 42) \linewd 0.5
    \Onedot \lvec (20 60) \move (25 42) \lvec (30 60)

    \move (0 20) \linewd 1 \lvec (-25 40) \linewd 0.5

    \bsegment
      \move (0 0) \Onedot \lvec (-16 25) \move (0 0) \lvec (-5 30)
      \move (0 0) \lvec (6 29)
    \esegment

\move (6 17) \trekant
\setunitscale{0.8} \rmove (12 0) \bsegment \fcir f:0 r:1.5 \esegment \setunitscale{1}
\move (-44 33) \trekant
\move (33 38) \tokant

\htext (-4 3){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}
\htext (-16 25){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}
\htext (-4 70){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}
\htext (7 67){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}

\htext (-42 70){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}
\htext (-30 75){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}
\htext (-18 74){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}

\htext (20 65){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}
\htext (30 65){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$}}
\htext (12 38){{\@setsize\footnotesize{11\p@}\xpt\@xpt $2$}}

\move (52 11)
\bsegment
\htext (0 0){{\@setsize\footnotesize{11\p@}\xpt\@xpt $2$ :}} \move (10 0) \tovert
\htext (0 -12){{\@setsize\footnotesize{11\p@}\xpt\@xpt $3$ :}} \move (10 -12) \trevert
\esegment
\esegment

\end{texdraw}
\end{center}
In the picture the combinatorial tree in the middle expresses the nesting of
1PI subgraphs on the left; such trees are sufficient in Kreimer's Hopf-algebra
approach to BPHZ, but do not capture the symmetries of the graph.
To this end, further
decoration is needed in the tree, as partially indicated on the right.
First of all, each node in the tree should be decorated by the 1PI graph
it corresponds to in the nesting, and second,
the tree should have leaves (input
slots) corresponding to the vertices of the graph.
The decorated tree should be
regarded as a
recipe for reconstructing the graph by inserting the decorating graphs into
the vertices of the graphs of parent nodes.  The numbers on the edges
indicate the  type constraint of each substitution: the outer interface of
a graph must match the local interface of the vertex it is substituted into.
But the type constraints on the tree decoration are not enough to reconstruct the
graph, because for example the small graph
\raisebox{-5pt}{\begin{texdraw}\trekant\end{texdraw}} decorating
the left-hand node could be substituted into various different vertices of the
graph
\raisebox{-5pt}{\begin{texdraw}
  \trekant  \setunitscale{0.8} \rmove (12 0) \bsegment \fcir f:0 r:1.5 \esegment \setunitscale{1}
\end{texdraw}}.

  The solution found in \cite{Kock:graphs-and-trees} is to consider $P$-trees,
  for $P$ the polynomial endofunctor given by $I \stackrel s\leftarrow E
  \stackrel p\to B \stackrel t\to I$, where $I$ is the groupoid of interaction
  labels for the theory (in this case the one-vertex graphs
  \raisebox{2pt}{\begin{texdraw} \tovert \end{texdraw}} and
  \raisebox{-1pt}{\begin{texdraw} \trevert \end{texdraw}}\ ) and $B$ is the
  groupoid of connected 1PI graphs of the theory, and $E$ is the groupoid of such 1PI
  graphs with a marked vertex.  The map $s$ returns the one-vertex subgraph at
  the mark, $p$ forgets the mark, and $t$ returns the outer interface of the
  graph, i.e.~the graph obtained by contracting everything to a point, but
  keeping the external lines.  A $P$-tree is hence a diagram like \eqref{Ptree}
  with specified $2$-cells.  These $2$-cells carry much of the structure: for
  example the $2$-cell on the right says that the 1PI graph decorating a given
  node must have the same outline as the decoration of the outgoing edge of the
  node --- or more precisely, and more realistically: an isomorphism is specified (it's
  a bijection between external lines of one-vertex graphs).  Similarly, the
  left-hand $2$-cell specifies for each node-with-a-marked-incoming-edge $x'\in
  M$, an isomorphism between the one-vertex graph decorating that edge and the
  marked vertex of the graph decorating the marked node $x'$.  Hence the
  structure of a $P$-tree is a complete recipe not only for which graphs should
  be substituted into which vertices, but also {\em how}: specific bijections
  prescribe which external lines should be identified with which lines in the receiving
  graph.
  In fact, {\em there is an equivalence of groupoids between nested graphs and
  $P$-trees}~\cite{Kock:graphs-and-trees}.  This is exploited
  in~\cite{Galvez-Kock-Tonks:FdB}
  to establish algebraic identities concerning
  graphs by interpreting them as
  homotopy\xspace cardinalities of equivalences of groupoids of decorated trees.

  Notice that the polynomial functor $P$ is combinatorial, since each
  graph has a discrete finite set of vertices.  It is not a species in the
  classical sense though: the classifying map $B \to \mathbb{B}_\omega$ sends a graph
  to its set of vertices, and since a graph may have nontrivial automorphisms that fix
  every
  vertex, this map does
  not have discrete fibres.
\end{blanko}

\section{Outlook}

  A $2$-category is called {\em locally cartesian closed} when for every arrow
  $f:B\to A$, we have the string of adjoint functors
  $
  \lowershriek{f} \dashv \upperstar{f} \dashv \lowerstar{f} .
  $  This structure formally implies the Beck-Chevalley equivalences and
  distributivity, which are the minimal requirements for a reasonable theory of
  polynomial functors.  The theory of strength can be copied almost verbatim from
  \cite{Gambino-Kock:0906.4931},
  and it seems that the
  representation theorem of \cite{Gambino-Kock:0906.4931} also carries over.

While locally cartesian closed categories provides semantics for an extensional
version of Martin-L\"of type theory~\cite{Seely:lccc},
\cite{Clairambault-Dybjer:1112.3456}, and locally cartesian closed $2$-categories capture
some $2$-truncated version (\cite{Hofmann-Streicher:98},
\cite{Garner:0808.2122}), recent insight of Homotopy Type Theory
strongly
suggests that in the long run, the case of $\infty$-groupoids and other locally
cartesian closed $\infty$-categories will be the real meat for type theory.
Large parts of the $\infty$-theory of polynomial functors,
as well as aspects of the theory of
locally cartesian closed
$\infty$-categories geared towards Homotopy Type Theory
have already been
worked out in joint work with David Gepner, and
will appear
elsewhere~\cite{Gepner-Kock},~\cite{Gepner-Kock:univalence}.
Nevertheless the groupoid case is interesting in its own right,
since
it already covers important applications: in particular
for many purposes of combinatorial nature, $1$-groupoids are all that is
needed in order to handle symmetry issues.  Time will tell whether for
purposes of program semantics the groupoid level is enough too
--- otherwise it is a good stepping stone into the $\infty$-world.

redpajama_set_name: RedPajamaArXiv


1. The Department for Transport (DfT) annually estimates the number of reported drink drive accidents: i.e. those reported injury road accidents involving drivers with illegal alcohol levels (above the current drink-drive limit of 80 milligrams (mg) of alcohol per 100 millilitres (ml) of blood or 35 micrograms per 100ml of breath). DfT published GB estimates in Reported Road Casualties Great Britain 2013 in September 2014. Scotland estimates are presented in Table 22. Because of the uncertainty involved figures are rounded to the nearest ten.
2. The DfT's publication outlines the estimation methods in detail. It draws on Stats 19 reported road accident data (where motor vehicle drivers or riders failed or refused to provide a sample of breath) and Procurators Fiscal (and Coroners in England and Wales) data on blood alcohol levels of drivers who died within 12 hours of being injured in a road accident. The estimates include allowances for the numbers of cases where drivers or riders are not breath tested due to the accident being a hit and run accident. Drink drive casualties are defined here as any casualties resulting from a drink drive accident.
3. Estimates for 2013 are not yet available because of the timing of the provision of the data regarding blood alcohol levels of fatalities from Procurators Fiscal (and Coroners in England and Wales) to DfT. At this stage the sample of 2013 data is insufficient to allow a breakdown by country.


text:
Vera Schmitz
CAR: "If everything goes as planned more than 40,000 children will be protected"
Vera and the team are working on a vaccination campaign in Central African Republic. Here she blogs about what's involved as they prepare to protect tens of thousands of children from deadly diseases...
A few weeks ago I started my eighth assignment with Doctors Without Borders - right in the heart of Africa, the Central African Republic.
The project I will spend my coming weeks and months in is quite different from my previous missions - this time I am part of a team running a preventive vaccination campaign.
The target group will include all children under five years old in a region in southern Central African Republic - that's an estimated target of nearly 40,000 children! All this will be done in three consecutive rounds, since the protection against the different diseases is much more effective with a second or third vaccination.
If everything goes as planned more than 40,000 children will be protected against eight frequent and dangerous diseases.
Travelling to remote villages. Photo: Vera Schmitt / MSF.
However, much remains to be done by then. Before starting a vaccination campaign, especially one of this size, it is important to plan really well. We'll be orgainising as much as possible and down to the smallest detail, since everything which has not been planned before leads to delays later on and is often difficult to remedy during a campaign.
In the last two weeks we have already completed an important part of this planning, visiting the whole region. We went to almost every small village, for some of them it takes up to 3.5 hours to reach – one way!
Overcoming these limitations to access of health care is one of the top objectives. This is especially true for the vaccination campaign itself, but also for our visits in advance. The area in which we are working is densely wooded. The roads on which we are travelling can often not even be called "roads", but rather narrow paths.
This is one of the better bridges! Photo: Vera Schmitz / MSF
There are many small creeks and ponds, with bridges which are not always strong enough to carry the load of a car or aren't wide enough to cross. As far and as much as we can, we drive with our land cruisers - and all those villages that we can't reach like this, we go either by motorbike or a small boat. Sometime though, you have to rely on your own feet.
On the road. Photo: Vera Schmitz / MSF
During this planning phase, however, we have not only overcome the limitations given by nature, but sometimes even national boundaries. During our research, for example, we have encountered a village which extends both to the territory of the Central African Republic and to that of the Republic of the Congo.
In order to ensure that that part of the population which is living in the Congolese area can also benefit from the vaccination campaign, we spoke with the village leaders of both parts as well as with the border officials to explain our actions and intentions.
Meeting the responsible and respected persons on site is always of great importance and an essential part of our visits. In all those countless villages we have visited during the last two weeks, we have whenever possible visited the local village leader. Acceptance, communication and cooperation is incredibly important, so that on the planned days of the vaccination campaign, the population is well informed and does indeed bring their children to vaccination.
In some of the larger villages there is usually a health centre, which often gives us valuable information and is going to support us during the actual vaccination campaign.
One of the villages we visited. Photo: Vera Schmitt / MSF.
Informing the local population about the campaign in advance was only one of the reasons to visit the region. Another objective was to gather information and to understand the situation on the ground as much as possible...
Most of the places are not mapped, nor were the streets and paths on the way. Where is the village, how do we get there (By car? By Boat? By Foot?) And how long is the journey?
How big is the population, will a day be enough or will it take two or more to reach all of the children, or can we perhaps send a reduced team, while the other half is proceeding to the next village?
Where exactly can we vaccinate? Rain is coming soon, so the shade among the large mango trees is only the last option, if there is no other possibility. But luckily we mostly find a church or a school which is suitable for our purpose.
Then – are there villagers who can read and write well enough to help us fill the vaccination cards? – Illiteracy unfortunately is quite common.
Who can help in the sensitisation a few days before the vaccination so that all mothers know when they should come and where?
All this information now still has to be merged - which is the actual beginning of microplanning of the campaign. Which team travels to which village and when? How many vaccine doses must be taken and how many vaccine carriers (cool boxes) and ice packs do we need? If the cold chain breaks, this would be a disaster, since the vaccinations are ineffective if they exceed a certain temperature.
Still the team is not complete, in the coming days there are still various posts to be recruited and following this different trainings.
So there is still a lot to do - but the prospect that soon 40,000 children will be protected from diseases such as measles, polio, pneumonia, tetanus, whooping cough, hepatitis and diphtheria is quite enough motivation for some more days' hard work.
PS: The landscape in which we are traveling is an extra bonus. Nearly untouched and often beautiful.
unmapped
More from Vera
When people flee to seek refuge
Iraq: Life after the war
Starting from scratch: an emergency response in the Democratic Republic of Congo
All posts from Vera
Five myths about working for MSF
Nursing in Gaza: "He was hit by a single bullet, passing through both his lower legs"
Spear trauma in South Sudan
Snakebite in South Sudan: "There's only one way to save the girl's life..."
A doctor in DRC: The finish line
A doctor in DRC: Bolomba, soon a memory
A doctor in DRC: Arrival in Bolomba
A doctor in DRC: Along the Ikelemb river
A doctor in DRC: "Preparations are in full swing"

redpajama_set_name: RedPajamaCommonCrawl


in develop mode, if i switch from a Nikon d800 raw to Nikon d750 raw (and viceversa) LR freeze for 10-40s. No problem if I scrolling many images from same camera at all.
In library mode i haven't issue at all.
For this problem i downgrade to v7.2, after this release this bug shows up.
Do you have large no of develop presets?
Can you share a rough number?
Ok. Thanks for the information. We will investigate the issue further.