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text:
Q: Crash when trying to remove object from ArrayList im having a problem i havent been able to figure out a solution to. Im making a small game where the object in _sballs ArrayList will be deleted if collided with another object called ball.
the problem i've encountered is that when the collision occur and when i attempt to remove the object from the ArrayList, the application crashes.
for(GObject sballgraphic : _sballs){
            Coordinates sballcoords = sballgraphic.getCoords();
            if(coords.getY() - coords._height > sballcoords.getY() + sballcoords._height && coords.getX() - coords._width > sballcoords.getX() + sballcoords._width){
                _sballs.remove(sballgraphic);
            }
        }

So the code compares the balls coordinates with all the sballs objects to check if there is a collision, then tries to remove the sball.
What is the problem here? :)

A: I'm guessing that the "crash" is a ConcurrentModificationException.
It is happening because you are trying to remove from a collection while you are iterating over it with an iterator (the internal workings of the enhanced for).
Your choices are:

*

*Iterate using an index (old-style for( i=0; i<_sballs.size(); i++ ))

*Iterate using an iterator explicitly, and use the iterator's remove() method.

*Remember which items to remove by putting them into another list, and then use removeAll() after the loop is finished.

A: You cant do such operation because you are modifying the same _sballs onwhich you are iterating too.
ArrayList<GObject> _sballs;
ArrayList<GObject> _sballsForRemove;

for(GObject sballgraphic : _sballs){
            Coordinates sballcoords = sballgraphic.getCoords();
            if(coords.getY() - coords._height > sballcoords.getY() + sballcoords._height && coords.getX() - coords._width > sballcoords.getX() + sballcoords._width){
                _sballsForRemove.add(sballgraphic);
            }
        }
_sballs.removeAll(_sballsForRemove);

redpajama_set_name: RedPajamaStackExchange


text:
The John Ruan House is a historic mansion in the Frankford neighborhood of Philadelphia, Pennsylvania. It was built in 1796 as the home of Dr. John Ruan (June 9, 1771, St. Croix, West Indies – July 2, 1845, Bristol, Pennsylvania), a physician and community leader.

The 2½-story house, the oldest of its size and stature still standing in Frankford, was listed on the National Register of Historic Places in 1985. It was added to the Philadelphia Register of Historic Places on January 3, 1985.

It has been continuously occupied since its erection and was previously the home of the Grand Army of the Republic Civil War Museum and Library.  The museum housed Civil War and Grand Army of the Republic artifacts, books, and memorabilia. It is now privately owned.

References

External links

Grand Army of the Republic Civil War Museum and Library
"History of the Ruan House, Circa 1796" at the Grand Army of the Republic Civil War Museum and Library
Listing at Philadelphia Architects and Buildings

Museums in Philadelphia
Houses on the National Register of Historic Places in Philadelphia
Libraries in Pennsylvania
American Civil War museums in Pennsylvania
Historic house museums in Philadelphia
Georgian architecture in Pennsylvania
Houses completed in 1796
Philadelphia Register of Historic Places
Frankford, Philadelphia
1796 establishments in Pennsylvania

redpajama_set_name: RedPajamaWikipedia


This lovingly maintained second home offers a prime location in the heart of Duck! Short walk to the beach, the sound or the shops and dinning. Sold fully furnished and turn key for rentals or private use. The large, open great room and sun decks allow you to enjoy the views of the sound from many vantage points.
The community pool is just steps away and the large hot tub sits on the top deck overlooking the sound. Large, 15,000 square foot home would allow for an addition in the future or a pool. The 2 car, enclosed garage allows for lots of storage.


text:
AY closes on 72,000 s/f in leases at upgraded Paramount
by REW December 6, 2018 02807
A new multi-story, award-winning Art Deco building lobby has been officially completed at the iconic Midtown Manhattan office property 1501 Broadway, also known as the Paramount Building, in the Theater District.
The introduction of the new lobby represents the final stage of a $50 million capital improvement program completed by ownership that added state-of-the-art amenities to the 33-story building's historical charm.
The completed renovations have increased tenant interest in 1501 Broadway as the New York office for Avison Young, the exclusive office leasing agent for the building, is announcing approximately 72,000 s/f of new leasing activity at the property, all closing within weeks of the introduction of the lobby. The transactions include the following:
• A new, 18,680 s/f 10-year lease on the partial 10th floor for KIPP Foundation & KIPP NYC, a non-profit charter school organization. The tenant is moving from their previous location at 520 Eighth Avenue and their new space at 1501 Broadway also features an outdoor terrace.
• An 18,332 s/f lease extension and expansion across the entire ninth floor for the American Federation of Musicians of the United States and Canada. The prominent organization, the labor union representing professional musicians throughout both countries, has also been a longstanding tenant at 1501 Broadway.
A new, 16,657-square-foot, 15-year lease on the 11th floor for EC New York, a leading school for teaching English to adults. The organization is moving from their previous location at 1450 Broadway.
• A 17,656 s/f lease expansion for engineering firm and long-term tenant Hardesty & Hanover, LLC, across the entire sixth floor of the building. This expansion will cover the remaining 12 years of their lease and will bring their total footprint in the building to over 52,000 s/f.
Avison Young has served as the exclusive leasing agent for 1501 Broadway since 2012. The team representing ownership in these recent transactions includes Principal and Tri-State President Arthur J. Mirante, II, Principal Michael Gottlieb, and Associate Joseph Gervino.
1501 broadwayAvison Youngparamount buildingShare0
Leases: Koeppel Rosen leases second floor of NoMad building, Savitt Partners bring in media group
Vanderbilt trio to tackle newest SL Green rebuild
112-113 South Street | Seaport – SOLD!

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

The starting point of the work \cite{CMMV17} was to study from a fibrational point of view the notion of \emph{regular span}, introduced by Yoneda in \cite{Y60} as a formal categorical setting in order to reformulate the classical theory of $\ensuremath{\mathsf{Ext}}^n$ functors. The results in \cite{CMMV17} reveal that a regular span $S$
is nothing but a cartesian functor with codomain a product projection:
\[
\xymatrix@!=4ex{
X \ar[dr]_{S_2} \ar[rr]^S & & A \times B \ar[dl]^{pr_2} \\
& B
}
\]
which is in addition an internal opfibration in $\ensuremath{\mathsf{Cat}}/B$ (i.e.~a \emph{fibrewise opfibration}---see Definition \ref{def:fw}). This enlights also the difference between regular spans and two-sided fibrations, which in turn were characterized in \cite{BournPenon} as internal opfibrations, with codomain a product projection, in the 2-category $\ensuremath{\mathsf{Fib}}(B)$ of fibrations over $B$.

This fibrational interpretation makes it possible to reformulate Yoneda's Classification Theorem of \cite{Y60} as the result of a canonical factorization, yielding a reflection of regular spans into profunctors, i.e.\ two-sided \emph{discrete} fibrations. Actually, in \cite{CMMV17} it is shown that such a factorization exists not only for regular spans, but for any fibrewise opfibration $p$ with codomain a split fibration. Moreover, it turns out that this factorization, performed via a coidentifier (Lemma 3.12 in [\emph{loc.~cit.}]), is the internal version in $\ensuremath{\mathsf{Fib}}(B)$ of the \emph{comprehensive factorization} introduced in \cite{StW} for \ensuremath{\mathsf{Cat}}. In other words, $p$ is the composite of an initial morphism in $\ensuremath{\mathsf{Fib}}(B)$ with an internal discrete opfibration, which is the same as a fibrewise opfibration whose fibres are discrete (Corollary 2.9 in \cite{CMMV17}).

It is natural to ask if, for fibrewise opfibrations, replacing the coidentifier with a coinverter in the construction above, we get as a comparison a fibrewise opfibration whose fibres are groupoids. To this end, we need first to detect some sufficient conditions to transfer the construction of coidentifiers and coinverters from a 2-category \ensuremath{\mathcal{C}}\ to the 2-category $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ of internal fibrations over a fixed object $B$ (see Proposition \ref{prop:coinv} and Proposition \ref{prop:coid}). This happens when the 2-monad $R\colon\ensuremath{\mathcal{C}}/B\to\ensuremath{\mathcal{C}}/B$, whose pseudo-algebras define internal fibrations (in the sense of Street \cite{Street74}), preserves coidentifiers and coinverters of identees. Under this assumption $(\dagger)$, the coinverter (coidentifier) $q$ of the identee $\kappa$ of a fibration $f$
\begin{equation} \label{fact}
\xymatrix@!=5ex{
K \ar@/^2ex/[r]^{d_0}_{}="1" \ar@/_2ex/[r]_{d_1}^{}="2" \ar@{=>}"1";"2"^{\kappa} & A \ar[d]_{f} \ar[r]^{q} & Q \ar[dl]^{s} \\
& B
}
\end{equation}
induces a  comparison morphism $s$, which is still a fibration in \ensuremath{\mathcal{C}}\  (see Corollary \ref{cor:fib.to.frac} and Corollary \ref{cor:fib.to.sfrac}).

It is important to point out that, for such an $s$, as for any other isofibration, having groupoidal fibres is the same as being conservative, as we show in Corollary \ref{cor:isofib.cons=gpd(isofib)} in the general context of a finitely complete 2-category.

Now we have two facts. First, we prove that the above described behaviour of coinverters (coidentifiers) for  fibrations (and opfibrations) holds in \ensuremath{\mathsf{Cat}}\ and in $\ensuremath{\mathsf{Fib}}(B)$, for any  $B$, precisely because in these cases, condition $(\dagger)$ and its dual $(\dagger')$ are fulfilled (Corollary \ref{cor:cat.dag} and Proposition \ref{prop:dag.to.fib}).

On the other hand,  it is known that any functor can be factorized through a conservative functor, thanks to a factorization system for \ensuremath{\mathsf{Cat}}\ which is obtained by a (possibly transfinite) iteration of the invertee/coinverter construction. Actually, we show in Corollary \ref{cor:isofib.cons=gpd(isofib)} that for any isofibration, the coinverter of the invertee is the same as the coinverter of the identee, so that our factorization $f=sq$ in diagram (\ref{fact}) realizes the first step of the above mentioned  construction. Moreover, this first step is sufficient to produce the desired factorization for each fibration (respectively opfibration) $f$ in \ensuremath{\mathsf{Cat}}: considering the construction (\ref{fact}) for such an $f$, $s$ turns out to be a fibration (resp. opfibration)  in groupoids, i.e.\ conservative (Proposition \ref{prop:isocompr.cat}).

The same phenomenon occurs for the factorization system in \ensuremath{\mathsf{Cat}}\ given by (sequence of coidentifiers, discrete functor), which, when restricted to fibrations (opfibrations), reduces to a single application of the identee/coidentifier construction and it realizes the comprehensive factorization. This result is proved in Proposition  \ref{prop:comprehensive.cat} in the case of a 2-category $\ensuremath{\mathsf{Cat}}(\ensuremath{\mathcal{E}})$ of internal categories where the construction of the comprehensive factorization of any functor provided in \cite{StV} is still valid, as, for example, when \ensuremath{\mathcal{E}}\ is a finitely cocomplete locally cartesian closed category.

In the last part of the paper, we extend the above results to  fibrewise opfibrations in $\ensuremath{\mathsf{Fib}}(B)$, relying on the pseudo-functorial interpretation  of opfibrations in \ensuremath{\mathsf{Cat}}.
This way,  we prove in Proposition \ref{prop:refl.gpd.fib} that every fibrewise (resp.~internal) opfibration $p$ in $\ensuremath{\mathsf{Fib}}(B)$ admits a factorization
\[
\xymatrix@!=5ex{
A \ar@/^3ex/[rr]^{p} \ar[dr]_{f} \ar[r]_-{q'} & Q' \ar[d]_(.4){h'} \ar[r]_-{s'} & C \ar[dl]^{g} \\
& B
}
\]
where $q'$ is the coinverter of the identee of $p$ and $s'$ is a fibrewise (resp.~internal) opfibration in groupoids in $\ensuremath{\mathsf{Fib}}(B)$. Such a factorization coincides with the one given by (sequence of coinverters, conservative functor) in \ensuremath{\mathsf{Cat}}.

The analogous results hold when replacing coinverters with coidentifiers and opfibrations in groupoids with discrete opfibrations (Proposition \ref{prop:comprehensive.int}).

\section{Review of internal fibrations}

Let \ensuremath{\mathcal{C}}\ be a finitely complete 2-category \cite{Street76}. For a fixed object $B$ in \ensuremath{\mathcal{C}}, we shall denote by $\ensuremath{\mathcal{C}}/B$ the comma 2-category over $B$ and by $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$ the pseudo-comma 2-category over $B$.

We shall denote as follows the (strict) comma objects in \ensuremath{\mathcal{C}}\ of identities, along identities on the left and on the right respectively, and iso-comma along identities:
\begin{equation} \label{diag:commas}
\begin{aligned}
\xymatrix@!=4ex{
B/B \ar[d]_{d_0} \ar[r]^-{d_1} & B \ar[d]^{1} & B/f \ar[r]^-{d_1} \ar[d]_{Rf} & A \ar[d]^f & f/B \ar[r]^-{Lf} \ar[d]_{d_0} & B \ar[d]^{1} & f/_{\cong} B \ar[r]^-{If} \ar[d]_{w_f} & B \ar[d]^1 \\
B \ar[r]_{1} \ar@{}[ur]|(.3){}="1" \ar@{}[ur]|(.7){}="2" \ar@{=>}"1";"2"_{\mu_B} & B & B \ar[r]_{1} \ar@{}[ur]|(.3){}="3" \ar@{}[ur]|(.7){}="4" \ar@{=>}"3";"4"_{\varphi_f} & B & A \ar[r]_{f} \ar@{}[ur]|(.3){}="5" \ar@{}[ur]|(.7){}="6" \ar@{=>}"5";"6"_{\psi_f} & B & A \ar[r]_{f} \ar@{}[ur]|(.3){}="7" \ar@{}[ur]|(.7){}="8" \ar@{=>}"7";"8"_{\omega_f}^{\sim} & B
}
\end{aligned}
\end{equation}
One can extend the assignment $f\mapsto Lf$ to 1-cells and 2-cells in $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$ in the following way, yielding a 2-functor:
\begin{itemize}
 \item on 1-cells:
 \[
 \xymatrix@!=6ex{
 A \ar[dr]_f^{}="1" \ar[r]^t & A' \ar[d]^{f'} \ar@{}[dl]|(.15){}="2" \ar@{=>}"2";"1"_{\theta}^{\sim} & \mapsto & f/B \ar[dr]_{Lf}^{}="1" \ar[r]^{Lt} & f'/B \ar[d]^{Lf'} \ar@{}[dl]|(.15){}="2" \ar@{=>}"2";"1"_{1} \\
 & B & & & B
 }
 \]
 where $Lt$ is determined by the equations $d_0(Lt)=td_0$ and $\psi_{f'}(Lt)=\psi_f\cdot\theta d_0$;
 \item on 2-cells:
 \[
 \xymatrix@!=6ex{
 A \ar[dr]_f^{}="1" \ar@/^4ex/[r]^{t_1}_{}="3" \ar[r]_{t_2}^{}="4" \ar@{=>}"3";"4"^{\xi} & A' \ar[d]^{f'} \ar@{}[dl]|(.15){}="2" \ar@{=>}"2";"1"^{\theta_2} & \mapsto & f/B \ar[dr]_{Lf}^{}="1" \ar@/^4ex/[r]^{Lt_1}_{}="3" \ar[r]_{Lt_2}^{}="4" \ar@{=>}"3";"4"^{L\xi}  & f'/B \ar[d]^{Lf'} \ar@{}[dl]|(.15){}="2" \ar@{=>}"2";"1"^{1} \\
 & B & & & B
 }
 \]
 where $L\xi$ is determined by the equations $(Lf')(L\xi)=1$ and $d_0(L\xi)=\xi d_0$.
\end{itemize}

One can also define a pseudo 2-natural tranformation $u\colon 1_{\ensuremath{\mathcal{C}}\ensuremath{/\!/} B}\to L$ and a (strict) 2-natural transformation $m\colon L^2\to L$ as follows:
\begin{itemize}
 \item on objects: for each $f\colon A \to B$,
 \[
 \xymatrix@!=6ex{
 A \ar[dr]_f^{}="1" \ar[r]^-{u_f} & f/B \ar[d]^{Lf} \ar@{}[dl]|(.15){}="2" \ar@{=>}"2";"1"_{1} & & Lf/B \ar[dr]_{L^2f}^{}="1" \ar[r]^-{m_f} & f/B \ar[d]^{Lf} \ar@{}[dl]|(.15){}="2" \ar@{=>}"2";"1"_{1} \\
 & B & & & B
 }
 \]
 where $u_f$ and $m_f$ are determined by the equations
\[
\left\{
\begin{array}{l}
d_0u_f=1 \\
\psi_fu_f=1
\end{array}
\right.
\quad\mbox{and}\quad
\left\{
\begin{array}{l}
d_0m_f=d_0d_0 \\
\psi_fm_f=\psi_{Lf}\cdot\psi_fd_0
\end{array}
\right.
\]
respectively.
 \item on 1-cells: for each $(t,\theta)$ as above,
\[
\xymatrix{
A \ar[d]_{t} \ar[r]^-{u_f} & f/B \ar[d]^{Lt} \\
A' \ar[r]_{u_{f'}} \ar@{}[ur]|(.3){}="1" \ar@{}[ur]|(.7){}="2" \ar@{=>}"2";"1"_{u_{(t,\theta)}}^{\sim} & f'/B
}
\]
is uniquely determined by the equations $d_0u_{(t,\theta)}=1$ and $(Lf')u_{(t,\theta)}=\theta$, while $m_{(t,\theta)}=1$.
\end{itemize}
One can check that these data satisfy the properties
\begin{enumerate}
 \item[M1] $m\ast uL=1=m\ast Lu$;
 \item[M2] $m\ast mL=m\ast Lm$.
\end{enumerate}
Hence the triple $(L,u,m)$ forms a 2-monad on $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$, which is not strict only because of the invertible 2-cells $u_{(t,\theta)}$ for each $(t,\theta)$.

Let us now consider, for each $f\colon A \to B$, the 2-cell $\kappa_f\colon u_fd_0\to 1_{f/B}$ determined by the equations $d_0\kappa_f=1$ and $(Lf)\kappa_f=\psi_f$, and define
\[
\xymatrix{
f/B \ar@/^3ex/[rr]^{Lu_f}_{}="1" \ar@/_3ex/[rr]_{u_{Lf}}^{}="2" \ar@{=>}"1";"2"_{\lambda_f} & & Lf/B
}
\]
as the unique 2-cell such that $d_0\lambda_f=\kappa_f$ and $(L^2f)\lambda_f=1$.

It is tedious but straightforward to check that the collection of all $\lambda_f$ for each $f$ gives rise to a modification $\lambda\colon Lu\to uL$ between natural tranformations in $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$. Moreover, $\lambda$ satisfies the following properties:
\begin{enumerate}
 \item[KZ1] $\lambda\ast u=1$;
 \item[KZ2] $m\ast\lambda=1$;
 \item[KZ3] $m\ast Lm\ast\lambda L=1$.
\end{enumerate}
Since M1 (=KZ0) already holds, the data $(L,u,m,\lambda)$ give rise to a KZ-doctrine in the sense of Definition 1.1 in \cite{Kock}. In other words, this structure provides a lax-idempotent 2-monad.

Let us observe that, in fact, $L\colon \ensuremath{\mathcal{C}}\ensuremath{/\!/} B \to \ensuremath{\mathcal{C}}\ensuremath{/\!/} B$ factors through the inclusion of $\ensuremath{\mathcal{C}}/B$ in $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$, and the 1-cell components of $u$ are such that $u_{(t,1)}=1$, so that the above 2-monad on $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$ restricts to a \emph{strict} 2-monad on $\ensuremath{\mathcal{C}}/B$, which is also part of a KZ-doctrine by the same $\lambda$. We will adopt the same notation for both monads as far as no confusion arises.
\medskip

Like $L$, also the 2-functors $R$ and $I$ on $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$, defined by the corresponding comma squares in (\ref{diag:commas}), can be endowed with a structure of 2-monad, which is colax-idempotent in the case of $R$ and pseudo-idempotent in the case of $I$. In both cases, these structures restrict to strict 2-monads $(R,v,n,\rho)$ and $(I,i,l,\iota)$ on $\ensuremath{\mathcal{C}}/B$.

One of the most important features of KZ-doctrines is that the corresponding (pseudo-)algebra structures are unique up to isomorphism for each object and they are characterized as right (pseudo-)inverse left adjoint to the unit component of the monad. Applying this observation and its dual to the special cases of the 2-functors $L$, $R$ and $I$ described above, one can characterize (pseudo-)fibrations (and dually opfibrations) and isofibrations in \ensuremath{\mathcal{C}}.

\begin{Proposition} \label{prop:fib}
For a morphism $f\colon A \to B$ in \ensuremath{\mathcal{C}}\ the following conditions are equivalent and define an internal \emph{fibration} (respectively \emph{pseudo-fibration}):
\begin{enumerate}
\item
\begin{itemize}
\item[(i)] For all $X$ in \ensuremath{\mathcal{C}}, $\ensuremath{\mathcal{C}}(X,f)\colon\ensuremath{\mathcal{C}}(X,A)\to\ensuremath{\mathcal{C}}(X,B)$ is a fibration (respectively pseudo-fibration) in \ensuremath{\mathsf{Cat}};
\item[(ii)] for all $g\colon Y \to X$, the commutative square below is a morphism of fibrations (respectively pseudo-fibrations) in \ensuremath{\mathsf{Cat}}:
\[
\xymatrix{
\ensuremath{\mathcal{C}}(X,A) \ar[r]^-{\ensuremath{\mathcal{C}}(g,A)} \ar[d]_{\ensuremath{\mathcal{C}}(X,f)} & \ensuremath{\mathcal{C}}(Y,A) \ar[d]^{\ensuremath{\mathcal{C}}(Y,f)} \\
\ensuremath{\mathcal{C}}(X,B) \ar[r]_-{\ensuremath{\mathcal{C}}(g,B)} & \ensuremath{\mathcal{C}}(Y,B)
}
\]
\end{itemize}
\item $f$ admits a structure of pseudo-algebra for the 2-monad $R\colon \ensuremath{\mathcal{C}}/ B\to\ensuremath{\mathcal{C}}/ B$ (respectively $R\colon \ensuremath{\mathcal{C}}\ensuremath{/\!/} B\to\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$);
\item The morphism $v_f\colon f \to Rf$ admits a right adjoint in $\ensuremath{\mathcal{C}}/ B$ (respectively $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$);
\item (Chevalley criterion) The morphism $f_1\colon A/A\to B/f$, determined by the equations
\[
\left\{
\begin{array}{l}
(Rf)f_1=fd_0 \\
d_1f_1=d_1 \\
\varphi_ff_1=f\mu_A
\end{array}
\right.
\]
admits a right adjoint in \ensuremath{\mathcal{C}}\ with counit an identity (respectively isomorphism).
\end{enumerate}
\end{Proposition}

In practice, given an internal fibration according to the above definition 2, it is convenient to fix a corresponding pseudo-algebra structure once and for all (which in \ensuremath{\mathsf{Cat}}\ means to fix a cleavage). Accordingly, throughout the paper, $\ensuremath{\mathsf{Fib}}_{\ensuremath{\mathcal{C}}}(B)$ will denote the 2-category whose objects are pseudo-algebras for the monad $R\colon\ensuremath{\mathcal{C}}/ B\to\ensuremath{\mathcal{C}}/ B$, whose 1-cells are strict pseudo-algebra morphisms, and with the obvious 2-cells (we shall write just $\ensuremath{\mathsf{Fib}}(B)$ for $\ensuremath{\mathcal{C}}=\ensuremath{\mathsf{Cat}}$).

\begin{Remark}
{\rm
The definition of internal fibration (resp. pseudo-fibration) in a representable 2-category appears in the form 2 of Proposition \ref{prop:fib} in the works of Street \cite{Street74,StreetBI}. The characterizations 1 and 3 in Proposition \ref{prop:fib} are well-known and already present in the literature (see, for example, \cite{Weber}). As for the Chevalley criterion, it was first proved by Gray \cite{Gray} for fibrations in \ensuremath{\mathsf{Cat}}, while an internal version of it (for opfibrations) appears in \cite[Proposition 9]{Street74}, asking for the unit to be an isomorphism. As the following example shows, such a condition does not characterize opfibrations. In fact, it characterizes pseudo-opfibrations (see 3.17 in \cite{StreetBI}). This is the reason why we consider the characterization 4.\ also for internal (strict) fibrations. Since we could not find a proof of the latter in the literature, we provide it in the appendix for the sake of completeness.
}
\end{Remark}

\begin{Example}{\em
Let $\ensuremath{\mathcal{C}}=\ensuremath{\mathsf{Cat}}$ and consider any functor $f\colon 1\to B$, where $1$ is the terminal category and $B$ is the groupoid with two objects and exactly one isomorphism between them. It is easy to see that $1/1\cong 1$, $f/B\cong B$, and these isomorphisms make the induced functor $f_1\colon 1/1\to f/B$ of the dual of Proposition \ref{prop:fib} 4.\ isomorphic to $f$. By uniqueness, the terminal functor $t\colon B \to 1$ is the only possible left adjoint to $f$, and $tf=1_1$, hence one can choose $1_{tf}$ as a counit. On the other hand, $ft$ is not equal, but isomorphic to $1_B$, so that $f$ and $t$ are actually adjoint with unit an isomorphism. But $f$  is not an opfibration, just a pseudo-opfibration.
}
\end{Example}

The equivalent conditions of Proposition \ref{prop:fib} may be easily adapted to define \emph{internal opfibrations} (respectively pseudo-opfibrations), for example replacing, in 2.~and 3., the monad $R$ with the monad $L$. For the reader's convenience, throughout the paper, most of the results are stated in terms of fibrations. The corresponding results for opfibrations can be obtained accordingly.

Before giving the characterization of isofibrations, we point out the following property which is specific for iso-comma squares and will be useful later on.

\begin{Lemma} \label{lemma:w_equiv}
For each $f\colon A \to B$ in \ensuremath{\mathcal{C}}, the 1-cells $i_f$ and $w_f$ form an adjoint equivalence with, in particular, $w_fi_f=1_A$. This yields also an equivalence in $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$ with $(i_f,1)$ and $(w_f,\omega_f)$ as adjoint pair.
\end{Lemma}

\begin{proof}
$w_fi_f=1_A$ by definition of $i_f$. Then let $\epsilon_f\colon i_fw_f\to 1\colon f/_{\cong}B\to f/_{\cong}B$ be the unique 2-cell such that $(If)\epsilon_f=\omega_f$ and $w_f\epsilon_f=1_{w_f}$. It is easy to check that the inverse $\epsilon_f^{-1}$ can be defined by the equations $(If)\epsilon_f^{-1}=\omega_f^{-1}$ and $w_f\epsilon_f^{-1}=1_{w_f}$, and this completes the proof of the first assertion (triangle identities follow easily). For the second one it suffices to recall that $\omega_fi_f=1_f$ and use the first part of the Lemma.
\end{proof}

\begin{Proposition} \label{prop:isofib}
For morphism $f\colon A \to B$ in \ensuremath{\mathcal{C}}\ the following conditions are equivalent and define an internal \emph{isofibration}:
\begin{enumerate}
\item  For all $X$ in \ensuremath{\mathcal{C}}, $\ensuremath{\mathcal{C}}(X,f)\colon\ensuremath{\mathcal{C}}(X,A)\to\ensuremath{\mathcal{C}}(X,B)$ is an isofibration  in \ensuremath{\mathsf{Cat}};
\item $f$ admits a structure of pseudo-algebra for the 2-monad $I\colon \ensuremath{\mathcal{C}}/ B\to\ensuremath{\mathcal{C}}/ B$;
\item the morphism $i_f\colon f \to If$ admits a right adjoint in $\ensuremath{\mathcal{C}}/ B$.
\end{enumerate}
\end{Proposition}

\begin{Remark} \label{rem:monads}
{\em
\begin{enumerate}
\item The ``pseudo'' version of Proposition \ref{prop:isofib} does not make sense, of course. In fact, every functor is an isofibration up to isomorphism, so that the ``pseudo'' version of condition 1. is empty. This is reflected in the fact that, thanks to Lemma \ref{lemma:w_equiv}, each $f$ always admits a pseudo-algebra structure for $I\colon \ensuremath{\mathcal{C}}\ensuremath{/\!/} B \to \ensuremath{\mathcal{C}}\ensuremath{/\!/} B$, given by $(w_f,\omega_f)$. Condition 2.\ says that, when $f$ is an isofibration, one can replace $(w_f,\omega_f)$ with another left adjoint (the pseudo-algebra structure) to $(i_f,1)$, in order to obtain an adjoint equivalence which actually lives in $\ensuremath{\mathcal{C}}/B$.
\item By the general theory of monads, for each $f$ in \ensuremath{\mathcal{C}}, $Lf$, $Rf$ and $If$ are free algebras for the corresponding monads (whose structure is given by the multiplication component at $f$) and it is immediate to see that these are actually strict algebras, which represent split opfibrations (respectively fibrations and isofibrations) internally.
\end{enumerate}
}
\end{Remark}

We recall from \cite{Street74} the following properties, which will be used later on.

\begin{Lemma} \label{lem:pbfib}
Fibrations are pullback stable.
\end{Lemma}

\begin{Corollary} \label{cor:p0p1}
Given a comma square
\[
\xymatrix{
f/g \ar[d]_{\overline{g}} \ar[r]^-{\overline{f}} & C \ar[d]^{g} \\
A \ar[r]_-{f} \ar@{}[ur]|(.3){}="1" \ar@{}[ur]|(.7){}="2" \ar@{=>}"1";"2" & B
}
\]
then $\overline{f}$ is an opfibration and $\overline{g}$ is a fibration. In particular, the canonical morphisms $d_0$ and $d_1$ of diagram (\ref{diag:commas}) are a fibration and an opfibration respectively.
\end{Corollary}

Applying the observation 2.\ of Remark \ref{rem:monads} to the 2-functor $I$ we see that, for each $f\colon A \to B$, its image $If$ is actually an isofibration. Restricting to pseudo-fibrations, we get the following result.

\begin{Lemma} \label{lemma:pseudo.to.fib}
If $f\colon A \to B$ is an internal pseudo-fibration in \ensuremath{\mathcal{C}}, then $If$ is an internal fibration.
\end{Lemma}

\begin{proof}
First recall from Lemma \ref{lemma:w_equiv} that $f$ and $If$ are equivalent as objects in $\ensuremath{\mathcal{C}}\ensuremath{/\!/} B$, hence $If$ is a pseudo-fibration, since $f$ is. Moreover, $If$ is an isofibration, by Remark \ref{rem:monads} 2. Then, by the representability of the notions involved, i.e.\ point 1.\ in Propositions \ref{prop:fib} and \ref{prop:isofib}, the thesis follows from the well-known fact that, in \ensuremath{\mathsf{Cat}}, any pseudo-fibration which is also an isofibration is actually a fibration.
\end{proof}

In fact, $I$ sends also morphisms (and 2-cells) of pseudo-fibrations to morphisms (and 2-cells) of fibrations, so that the following result holds.

\begin{Proposition} \label{prop:I.pres.fib}
The restriction of $I\colon\ensuremath{\mathcal{C}}\ensuremath{/\!/} B\to \ensuremath{\mathcal{C}}\ensuremath{/\!/} B$ to the full sub-2-category of pseudo-fibrations can be factorized as
\[
\xymatrix{
\ensuremath{\mathsf{PsFib}}(B) \ar@{^{(}->}[rr] \ar[d] & & \ensuremath{\mathcal{C}}\ensuremath{/\!/} B \ar[d]^{I} \\
\ensuremath{\mathsf{Fib}}(B) \ar@{^{(}->}[r] & \ensuremath{\mathcal{C}}/B \ar[r] & \ensuremath{\mathcal{C}}\ensuremath{/\!/} B.
}
\]
\end{Proposition}

Let us observe that, when $f$ is a pseudo-fibration, the adjoint equivalence $(w_f,\omega_f)\dashv (i_f,1)$ actually lives in $\ensuremath{\mathsf{PsFib}}(B)$.

\section{Coinverters and coidentifiers in $\ensuremath{\mathsf{Fib}}(B)$}

From now on, let \ensuremath{\mathcal{C}}\ be a finitely complete 2-category with coidentifiers and (strict) coinverters of reflexive 2-cells, whose definition we recall for the sake of completeness (the reader may refer to \cite{Kelly} for example).

\begin{Definition}
The coidentifier (coinverter) of a 2-cell $\alpha$ is a 1-cell $q$ such that:
\begin{enumerate}
 \item $q\alpha$ is an identity (isomorphism);
 \item for any other 1-cell $f$ such that $f\alpha$ is an identity (isomorphism), there exists a unique 1-cell $t$ with $tq=f$;
 \item for any 2-cell $\beta\colon g \to h$ such that $g\alpha$ and $h\alpha$ are identities (isomorphisms), there exists a unique 2-cell $\gamma$ with $\gamma q=\beta$;
\end{enumerate}
\end{Definition}

In this paper we will consider in particular coidentifiers (coinverters) of identees (invertees). Given a 1-cell $f$, we denote with $(K,\kappa)$ its identee, where $\kappa$ is the 2-universal 2-cell making $f\kappa$ an identity. We denote with $(W,\omega)$ the invertee of $f$, where $\omega$ is the 2-universal 2-cell making $f\omega$ an isomorphism.

Later on, we will take advantage of the following results concerning isofibrations.
Recall that a morphism is called $f$-vertical if its image under $f$ is an identity.

\begin{Lemma} \label{lemma:psvert=vert}
Let $f$ be an isofibration and $\alpha$ a 2-cell such that $f\alpha$ is an isomorphism. Then $\alpha$ factorizes as $\alpha=\sigma\cdot\tau$, where $\tau$ is $f$-vertical and $\sigma$ is an isomorphism.
\end{Lemma}

\begin{proof}
Since $f$ is an isofibration, the isomorphism $f\alpha$ admits a cartesian lifting $\sigma$, which is an isomorphism, at the codomain of $\alpha$. $\tau$ is then the unique $f$-vertical factorization of $\alpha$ through $\sigma$.
\end{proof}

\begin{Corollary} \label{cor:isofib.cons=gpd(isofib)}
Let $f$ be an isofibration, then:
\begin{enumerate}
 \item $f$ is conservative if and only if its fibres are groupoids;
 \item the coinverter of the identee of $f$ coincides with the coinverter of its invertee.
\end{enumerate}
\end{Corollary}

\begin{proof}
Let $(W,\omega)$ and $(K,\omega c)$ be the invertee and the identee, respectively, of $f$:
\[
\xymatrix@C=6ex{
K \ar[r]^-c & W \ar@/_2ex/[r]_{w_1}^{}="1" \ar@/^2ex/[r]^{w_0}_{}="2" \ar@{=>}"2";"1"^{\omega} & A \ar[r]^-f & B\,.
}
\]
Since $f\omega$ is an isomorphism by definition, as in Lemma \ref{lemma:psvert=vert}, we can factorize $\omega$ as a composite $\omega=\sigma\cdot\tau$, where $\sigma$ is a cartesian lifting of $f\omega$, and $\tau$ the unique $f$-vertical comparison 2-cell. $\tau$ being vertical, there is a unique $c'\colon W\to I$ such that $\omega cc'=\tau$. So we have factorized $\omega$ as in the following diagram:
\[
\xymatrix@!=7ex{
W \ar[r]^-{c'} \ar@/_/[drr]_{w_1}^{}="3" & K \ar[r]^-c_{}="4"  \ar@{=>}"4";"3"^{\sigma} & W \ar@/_2ex/[d]_{w_1}^{}="1" \ar@/^2ex/[d]^{w_0}_{}="2" \ar@{=>}"2";"1"^{\omega} \\
& & A\,.
}
\]
It is now easy to see that $f$ is conservative, i.e.\ its invertee is an isomorphism, if and only if its fibres are groupoids, i.e.\ its identee is an isomorphism.

As for the second statement, it suffices to observe that the coinverter of the identee $\omega c$ coinverts also $\omega=\sigma\cdot\omega cc'$.
\end{proof}

Obviously the last result does not hold in general if $f$ is not an isofibration, as it is whitnessed by the non-constant functor from the arrow-category $\mathbf{2}$ to the groupoid $\mathbf{I}$ with two objects and two non-trivial arrows.

It is easy to check that identees and coidentifiers in $\ensuremath{\mathcal{C}}/B$ are computed in \ensuremath{\mathcal{C}}, and the same property holds for coinverters of identees.

\begin{Lemma} \label{lemma:inv/B}
The forgetful 2-functor $U\colon \ensuremath{\mathcal{C}}/B \to \ensuremath{\mathcal{C}}$ creates coinverters of identees.
\end{Lemma}

We are going to explore the behaviour of the monad $R$ with respect to these limits and colimits. Analogous results can be proved for the monad $L$.

\begin{Remark} \label{rem:R}
It is worth observing that the functor $R$ can be described by means of the following construction:
\[
\xymatrix@!=5ex{
B/f \ar[r]_{d_1^*f} \ar@/^3ex/[rr]^{Rf} \ar[d]_{d_1} & B/B \ar[r]_-{d_0} \ar[d]_{d_1} & B \ar@{=}[d] \\
A \ar[r]_-{f} & B \ar@{=}[r] \ar@{}[ur]|(.3){}="1" \ar@{}[ur]|(.7){}="2" \ar@{=>}"2";"1"^{\mu_B} & B.
}
\]
That is, $R=(d_0)_{!}d_1^*$, i.e.\ the composite of the change-of-base 2-functor along $d_1$ with the composition 2-functor with $d_0$, which is left 2-adjoint to $d_0^*$.
\end{Remark}

\begin{Lemma} \label{lemma:R.pres.id}
The monad $R\colon\ensuremath{\mathcal{C}}/B\to\ensuremath{\mathcal{C}}/B$ preserves identees.
\end{Lemma}

\begin{proof}
By rk \ref{rem:R}, the thesis follows from the fact that $d_1^*$ preserves limits, being a right adjoint, and $(d_0)_{!}$ preserves identees.
\end{proof}

\begin{Lemma}
The identee of a morphism $p\colon(A,f)\to(C,g)$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ can be computed as in \ensuremath{\mathcal{C}}.
\end{Lemma}

\begin{proof}
Let
\begin{equation} \label{diag:identee}
\xymatrix@!=6ex{
K \ar@/_2ex/[dr]_{h} \ar@/^2ex/[r]^{k_0}_{}="1" \ar@/_2ex/[r]_{k_1}^{}="2" \ar@{=>}"1";"2"^{\kappa} & A \ar[d]_{f} \ar[r]^{p} & C \ar[dl]^{g} \\
& B
}
\end{equation}
be an identee diagram in $\ensuremath{\mathcal{C}}/B$ (which means that $\kappa$ is also the identee of $p$ in \ensuremath{\mathcal{C}}). Since $p$ is a morphism in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ and $R$ preserves identees by Lemma \ref{lemma:R.pres.id}, it is straightforward to prove that the adjunctions $v_f\dashv r_f$ and $v_g\dashv r_g$ in the diagram
\[
\xymatrix@!=6ex{
B/h \ar@{-->}@/^/[d]^{r_h} \ar@/^2ex/[r]^{Rk_0}_{}="1" \ar@/_2ex/[r]_{Rk_1}^{}="2" \ar@{=>}"1";"2"^{R\kappa} & B/f \ar@/^/[d]^{r_f} \ar[r]_-{Rp} & B/g \ar@/^/[d]^{r_g} \\
K \ar@{-->}@/^/[u]^{v_h}_{\dashv} \ar@/^2ex/[r]^{k_0}_{}="1" \ar@/_2ex/[r]_{k_1}^{}="2" \ar@{=>}"1";"2"^{\kappa} & A \ar[r]^-{p} \ar@/^/[u]_{\dashv}^{v_f} & C \ar@/^/[u]_{\dashv}^{v_g}
}
\]
induce an adjunction $v_h\dashv r_h$ by the universal property of the identees.
\end{proof}

Coinverters and coidentifiers may not be preserved by the monad $R$, however this happens to be true in some cases of interest which we will explore later on. So, for a given object $B$ in \ensuremath{\mathcal{C}}, we shall consider the property
\begin{itemize}
\item[($\dagger$)] The monad $R\colon \ensuremath{\mathcal{C}}/B\to \ensuremath{\mathcal{C}}/B$ preserves coinverters and coidentifiers of identees.
\end{itemize}

\begin{Proposition} \label{prop:coinv}
Let $B$ be an object in \ensuremath{\mathcal{C}}\ satisfying $(\dagger)$, $p\colon(A,f)\to(C,g)$ a morphism in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ and $\kappa$ its identee in \ensuremath{\mathcal{C}}. Then the coinverter $q\colon A\to Q$ of $\kappa$ in \ensuremath{\mathcal{C}}\ induces a factorization
\[
\xymatrix@!=6ex{
A \ar@/^3ex/[rr]^{p} \ar[dr]_{f} \ar[r]_-{q} & Q \ar[d]_(.4){gs} \ar[r]_-{s} & C \ar[dl]^{g} \\
& B
}
\]
of $p$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$, and $q\colon(A,f)\to(Q,gs)$ is the coinverter of $\kappa$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$.
\end{Proposition}

\begin{proof}
Recall that $\kappa$ is also the identee of $p$ in $\ensuremath{\mathcal{C}}/B$ and consider the corresponding identee diagram (\ref{diag:identee}) in $\ensuremath{\mathcal{C}}/B$. Since $p$ (and then $f$) coinverts $\kappa$, the morphisms $s$ and $gs$ in the factorization above are uniquely determined by the universal property of $q$, and this explains why $q$ is a coinverter of $\kappa$ in $\ensuremath{\mathcal{C}}/B$.

Since $f$ is a fibration, the unit component $v_f\colon (A,f) \to (B/f,Rf)$ admits a right adjoint $r_f$ in $\ensuremath{\mathcal{C}}/B$. We call $\eta_f$ and $\epsilon_f$ the corresponding unit and counit. Likewise, $v_g$ has a right adjoint $r_g$, and $pr_f=r_g(Rp)$ since $p$ is a morphism in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$. Let us consider the following diagram:
\[
\xymatrix@!=6ex{
B/h \ar@/^/[d]^{r_h} \ar@/^2ex/[r]^{Rk_0}_{}="1" \ar@/_2ex/[r]_{Rk_1}^{}="2" \ar@{=>}"1";"2"^{R\kappa} & B/f \ar@/^/[d]^{r_f} \ar@/^3ex/[rr]^{Rp} \ar[r]_-{Rq} & B/(gs) \ar@{-->}@/^/[d]^{r_{gs}} \ar[r]_-{Rs} & B/g \ar@/^/[d]^{r_g} \\
K \ar@/^/[u]_{\dashv}^{v_h} \ar@/^2ex/[r]^{k_0}_{}="1" \ar@/_2ex/[r]_{k_1}^{}="2" \ar@{=>}"1";"2"^{\kappa} & A \ar@/_3ex/[rr]_-{p} \ar[r]^-{q} \ar@/^/[u]_{\dashv}^{v_f} & Q \ar@/^/[u]_{\dashv}^{v_{gs}} \ar[r]^-{s} & C \ar@/^/[u]_{\dashv}^{v_g}
}
\]
Now, $pr_f(R\kappa)=r_g(Rp)(R\kappa)=1$, hence $r_f(R\kappa)$ factors through $\kappa$ and $qr_f(R\kappa)$ is an isomorphism. By the assumption $(\dagger)$, $Rq$ is the coinverter of $R\kappa$, so there exists a unique $r_{gs}\colon B/(gs)\to Q$ such that $r_{gs}(Rq)=qr_f$. By the 2-dimensional universal property of the coinverters $q$ and $Rq$, one can prove that a unit $\eta_{gs}$ and a counit $\epsilon_{gs}$ are induced by $\eta_f$ and $\epsilon_f$ respectively, making $v_{gs}\dashv r_{gs}$ an adjoint pair in $\ensuremath{\mathcal{C}}/B$, so that $gs$ is a fibration. As a consequence of this construction, $q$ turns out to be a morphism of fibrations over $B$.

It remains to show that for each $c\colon (A,f) \to (Y,y)$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ such that $c\kappa$ is an isomorphism, the unique comparison morphism $t$ in $\ensuremath{\mathcal{C}}/B$, induced by the coinverter $q$ and such that $tq=c$, is actually a morphism in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$. Let us denote by $r_{y}$ the $R$-pseudo-algebra structure on $y$, i.e.\ the right adjoint to $v_y$, and observe that the diagram
\[
\xymatrix@C=6ex{
B/(gs) \ar[d]_{r_{gs}} \ar[r]^-{Rt} & B/y \ar[d]^{r_{y}} \\
Q \ar[r]_-{t} & Y
}
\]
commutes since $Rq$ is a coinverter, then epimorphic, and precomposition with $Rq$ gives the commutative square presenting $c$ as a morphism of $R$-pseudo-algebras.

Finally, the fact that $s$ is in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ follows from the last argument, taking $c=p$.
\end{proof}

\begin{Corollary} \label{cor:fib.to.frac}
Let $B$ be an object in \ensuremath{\mathcal{C}}\ satisfying $(\dagger)$, $f\colon A \to B$ a fibration in \ensuremath{\mathcal{C}}, $\kappa$ its identee in \ensuremath{\mathcal{C}}, and $q\colon A \to Q$ its coinverter in \ensuremath{\mathcal{C}}. Then the unique comparison morphism $s\colon Q \to B$, such that $sq=f$, is a fibration and $q$ is the coinverter of $\kappa$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$.
\end{Corollary}

\begin{proof}
Apply Proposition \ref{prop:coinv} to the morphism $f\colon(A,f)\to(B,1_B)$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$.
\end{proof}

\begin{Proposition} \label{prop:coid}
Let $B$ be an object in \ensuremath{\mathcal{C}}\ satisfying $(\dagger)$, $p\colon(A,f)\to(C,g)$ a morphism in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ and $\kappa$ its identee in \ensuremath{\mathcal{C}}. Then the coidentifier $q'\colon A\to Q'$ of $\kappa$ in \ensuremath{\mathcal{C}}\ induces a factorization
\[
\xymatrix@!=6ex{
A \ar@/^3ex/[rr]^{p} \ar[dr]_{f} \ar[r]_-{q'} & Q' \ar[d]_(.4){gs'} \ar[r]_-{s'} & C \ar[dl]^{g} \\
& B
}
\]
of $p$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$, and $q'\colon(A,f)\to(Q',g's')$ is the coidentifier of $\kappa$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$.
\end{Proposition}

\begin{proof}
The proof is the same as for Proposition \ref{prop:coinv}, replacing coinverters by coidentifiers.
\end{proof}

\begin{Corollary} \label{cor:fib.to.sfrac}
Let $B$ be an object in \ensuremath{\mathcal{C}}\ satisfying $(\dagger)$, $f\colon A \to B$ a fibration in \ensuremath{\mathcal{C}}, $\kappa$ its identee in \ensuremath{\mathcal{C}}, and $q\colon A \to Q$ its coidentifier in \ensuremath{\mathcal{C}}. Then the unique comparison morphism $s\colon Q \to B$, such that $sq=f$, is a fibration and $q$ is the coidentifier of $\kappa$ in $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$.
\end{Corollary}

In the cases we are interested in, which will be studied in the next section \ref{sec:casestudy}, the property $(\dagger)$ relies upon the exponentiability of split opfibrations in \ensuremath{\mathcal{C}}. In this context, exponentiability is to be understood in a 2-categorical sense: a 1-cell $f$ is exponentiable if the change-of-base 2-functor along $f$ has a right 2-adjoint.

\begin{Lemma} \label{lem:exp.to.dag}
If for an object $B$ in \ensuremath{\mathcal{C}}, the comma projection $d_1$ in the diagram
\[
\xymatrix{
B/B \ar[r]^-{d_0} \ar[d]_{d_1} & B \ar@{=}[d] \\
B \ar@{=}[r] \ar@{}[ur]|(.3){}="1" \ar@{}[ur]|(.7){}="2" \ar@{=>}"2";"1"^{\mu_B} & B
}
\]
is exponentiable, then the functor
\[
R\colon \ensuremath{\mathcal{C}}/B \to \ensuremath{\mathcal{C}}/B
\]
has a right 2-adjoint. As a consequence, $B$ satisfies the condition $(\dagger)$. In particular, this holds for any $B$ when split opfibrations in \ensuremath{\mathcal{C}}\ are exponentiable.
\end{Lemma}

\begin{proof}
It is easy to see that the functor $R$ can be described by means of the following construction:
\[
\xymatrix@!=5ex{
B/f \ar[r]_{d_1^*f} \ar@/^3ex/[rr]^{Rf} \ar[d]_{d_1} & B/B \ar[r]_-{d_0} \ar[d]_{d_1} & B \ar@{=}[d] \\
A \ar[r]_-{f} & B \ar@{=}[r] \ar@{}[ur]|(.3){}="1" \ar@{}[ur]|(.7){}="2" \ar@{=>}"2";"1"^{\mu_B} & B.
}
\]
That is, $R=(d_0)_{!}d_1^*$, i.e.\ the composite of the change-of-base 2-functor along $d_1$ with the composition 2-functor with $d_0$, which is left 2-adjoint to $d_0^*$. Hence $R$ is left 2-adjoint to $\Pi_{d_1}d_0^*$, where $\Pi_{d_1}$ denotes the right 2-adjoint to $d_1^*$, which exists by assumption.
\end{proof}

\begin{Remark} \label{rem:dagger'}
{\rm
If instead of $(\dagger)$ we ask for
\begin{itemize}
 \item[($\dagger'$)] The monad $L\colon \ensuremath{\mathcal{C}}/B\to \ensuremath{\mathcal{C}}/B$ preserves coinverters and coidentifiers of identees,
\end{itemize}
then the results of \ref{prop:coinv} and \ref{prop:coid} hold with $\ensuremath{\mathsf{Fib}_\mathcal{C}}(B)$ replaced by $\ensuremath{\mathsf{OpFib}}_{\ensuremath{\mathcal{C}}}(B)$. Accordingly, if $d_0$ is exponentiable, and in particular when split fibrations are exponentiable in \ensuremath{\mathcal{C}}, then $L$ admits a right 2-adjoint and $(\dagger')$ holds for $B$.
}
\end{Remark}

\subsection{Case study: \ensuremath{\mathsf{Cat}}\ and $\ensuremath{\mathsf{Fib}}(B)$} \label{sec:casestudy}

It is well-known that fibrations in \ensuremath{\mathsf{Cat}}\ are exponentiable \cite{Giraud} in the classical 1-categorical sense. As observed by Johnstone in \cite{Johnstone}, this property holds also in the 2-categorical sense recalled above. As a consequence, by Lemma \ref{lem:exp.to.dag} and Remark \ref{rem:dagger'}, we have:

\begin{Corollary} \label{cor:cat.dag}
In the 2-category \ensuremath{\mathsf{Cat}}, each object $B$ satisfies the conditions $(\dagger)$ and $(\dagger')$.
\end{Corollary}

One can extend the last property from \ensuremath{\mathsf{Cat}}\ to $\ensuremath{\mathsf{Fib}}(B)$ for each $B$, by means of the pseudo-functorial interpretation of fibrations in \ensuremath{\mathsf{Cat}}.

\begin{Remark} \label{rem:coinv.pseudofunct}
{\rm
As far as coinverters and coidentifiers of identees in $\ensuremath{\mathsf{Fib}}(B)$ are concerned, whose existence is guaranteed by Proposition \ref{prop:coinv} and Corollary \ref{cor:cat.dag}, let us observe that their construction can be performed fibrewise. In fact, given an identee
\[
\xymatrix{
K \ar@/^2ex/[r]_{}="1" \ar@/_2ex/[r]^{}="2" \ar@{=>}"1";"2"^{\kappa} & A
}
\]
in $\ensuremath{\mathsf{Fib}}(B)$, the collection of the coinverters (respectively coidentifiers) $q_b$ of its restrictions $\kappa_b$ to the fibres
\[
\xymatrix@!=6ex{
K_b \ar@/_2ex/[r]^{}="1" \ar@/^2ex/[r]_{}="2" \ar@{=>}"2";"1"^{\kappa_b} \ar[d]_{\beta^*} & A_b \ar[r]^{q_b} \ar[d]^{\beta^*} & Q_b \ar@{-->}[d]^{\beta^*} \\
K_{b'} \ar@/_2ex/[r]^{}="1" \ar@/^2ex/[r]_{}="2" \ar@{=>}"2";"1"^{\kappa_{b'}} & A_{b'} \ar[r]_{q_{b'}} & Q_{b'}
}
\]
gives rise to a natural transformation between pseudo-functors (see Section \ref{sec:pseudo} for details). It is easy to check that the corresponding morphism in $\ensuremath{\mathsf{Fib}}(B)$ is the coinverter (resp. coidentifier) of $\kappa$.
}
\end{Remark}

\begin{Lemma} \label{lem:internal.fib.pres}
For each internal fibration $p\colon(E,e)\to(A,a)$ in $\ensuremath{\mathsf{Fib}}(B)$, the change of base 2-functor $p^*\colon\ensuremath{\mathsf{Fib}}(B)/(A,a)\to\ensuremath{\mathsf{Fib}}(B)/(E,e)$ preserves coinverters and coidentifiers of identees.
\end{Lemma}

\begin{proof}
We will prove the result concerning internal fibrations and coinverters, the variations involving opfibrations and coidentifiers are obtained analogously.

Let the arrow $q\colon((C,c),f)\to((D,d),g)$ in the diagram
\[
\xymatrix@!=1.5ex{
K \ar@/_2ex/[ddrr]_{k} \ar@/^2.5ex/[rr]^{u}_{}="1" \ar@/_2.5ex/[rr]_{v}^{}="2" \ar@{=>}"1";"2"^{\kappa} & & C \ar[dd]_{c} \ar[rrr]^{f} \ar[dr]^{q} & & & A \ar@/^3ex/[ddlll]^{a} \\
& & & D \ar[urr]_{g} \ar[dl]_{d} \\
& & B
}
\]
be the coinverter in $\ensuremath{\mathsf{Fib}}(B)/(A,a)$ of an identee $\kappa$, and consider its image under the change of base 2-functor $p^*$, i.e.\ the upper part of the next diagram (we omit all arrows over $B$, all pullbacks provide in fact fibrations over $B$):
\[
\xymatrix@R=3ex@C=5ex{
K \times_A E \ar[dd] \ar@/^2.5ex/[rr]^{p^*u}_{}="1" \ar@/_2.5ex/[rr]_{p^*v}^{}="2" \ar@{=>}"1";"2"^{p^*\kappa} & & C \times_A E \ar[dd] \ar[rr]^{p^*f} \ar[dr]_{p^*q} & & E \ar[dd]^{p} \\
& & & D \times_A E \ar[ur]_{p^*g} \ar[dd] \\
K \ar@/^2.5ex/[rr]^{u}_{}="1" \ar@/_2.5ex/[rr]_{v}^{}="2" \ar@{=>}"1";"2"^{\kappa} & & C \ar[rr]^(.35){f} \ar[dr]_{q} & & A. \\
& & & D \ar[ur]_{g}
}
\]
We would like to show that $p^*q$ is the coinverter of the identee $p^*\kappa$ in $\ensuremath{\mathsf{Fib}}(B)/(E,e)$. To this end, we consider the restriction of the above diagram to the fibres over any object $b$ in $B$. By limit commutation, the latter is the same as the corresponding change of base diagram in the fibres over $b$:
\[
\xymatrix@R=3ex@C=5ex{
K_b \times_{A_b} E_b \ar[dd] \ar@/^2.5ex/[rr]^{p_b^*u_b}_{}="1" \ar@/_2.5ex/[rr]_{p_b^*v_b}^{}="2" \ar@{=>}"1";"2"^{p_b^*\kappa_b} & & C_b \times_{A_b} E_b \ar[dd] \ar[rr]^{p_b^*f_b} \ar[dr]_{p_b^*q_b} & & E_b \ar[dd]^{p_b} \\
& & & D_b \times_{A_b} E_b \ar[ur]_{p_b^*g_b} \ar[dd] \\
K_b \ar@/^2.5ex/[rr]^{u_b}_{}="1" \ar@/_2.5ex/[rr]_{v_b}^{}="2" \ar@{=>}"1";"2"^{\kappa_b} & & C_b \ar[rr]^(.35){f_b} \ar[dr]_{q_b} & & A_b. \\
& & & D_b \ar[ur]_{g_b}
}
\]
Now observe that, by Remark \ref{rem:coinv.pseudofunct}, $q_b$ is the coinverter of $\kappa_b$. Moreover, since $p_b$ is a fibration in \ensuremath{\mathsf{Cat}}\ by assumption, it is exponentiable, hence $p_b^*$ is a left 2-adjoint and $p_b^*q_b$ is the coinverter of $p_b^*\kappa_b$. Finally, again by Remark \ref{rem:coinv.pseudofunct}, $p^*q$ is the coinverter of $p^*\kappa$ in $\ensuremath{\mathsf{Fib}}(B)$, and hence in $\ensuremath{\mathsf{Fib}}(B)/(A,a)$.
\end{proof}

\begin{Proposition} \label{prop:dag.to.fib}
In the 2-category $\ensuremath{\mathsf{Fib}}(B)$, each object satisfies the conditions $(\dagger)$ and $(\dagger')$.
\end{Proposition}

\begin{proof}
Let $a\colon A \to B$ be a fibration of categories, then by Corollary \ref{cor:p0p1} the projections $d_0$ and $d_1$ of the comma square in $\ensuremath{\mathsf{Fib}}(B)$
\[
\xymatrix{
(A,a)/(A,a) \ar[r]^-{d_0} \ar[d]_{d_1} & (A,a) \ar@{=}[d] \\
(A,a) \ar@{=}[r] \ar@{}[ur]|(.3){}="1" \ar@{}[ur]|(.7){}="2" \ar@{=>}"2";"1"^{\mu_{(A,a)}} & (A,a)
}
\]
are an internal fibration and opfibration respectively. As a consequence, by Lemma \ref{lem:internal.fib.pres}, the corresponding change of base 2-functors $d_0^*$ and $d_1^*$ preserve coinverters and coidentifiers of identees. Now likewise in the proof of Lemma \ref{lem:exp.to.dag}, the thesis follows from the fact that $R=(d_0)_{!}d_1^*$ and $(d_0)_{!}$ is a left adjoint (and similarly for $L$).
\end{proof}

\begin{Proposition}
Let $p\colon(A,f)\to(C,g)$ be an internal fibration in $\ensuremath{\mathsf{Fib}}(B)$. Then the morphism $s$ in the factorization of Proposition \ref{prop:coinv} is an internal fibration in $\ensuremath{\mathsf{Fib}}(B)$.
\end{Proposition}

\begin{proof}
By Proposition \ref{prop:dag.to.fib}, we can apply Corollary \ref{cor:fib.to.frac} to the fibration $p$ in $\ensuremath{\mathsf{Fib}}(B)$.
\end{proof}

\section{Two factorization systems for (fibrewise) opfibrations in $\ensuremath{\mathsf{Fib}}(B)$}

\subsection{Two factorization systems in \ensuremath{\mathsf{Cat}}}

Let us consider the diagram
\[
\xymatrix@!=6ex{
W \ar@/^2ex/[r]_{}="1" \ar@/_2ex/[r]^{}="2" \ar@{=>}"1";"2"^{\omega} & A \ar[d]_{f} \ar[r]^{q} & Q \ar[dl]^{s} \\
& B
}
\]
in \ensuremath{\mathsf{Cat}}, where $q$ is the coinverter of the invertee $\omega$ of $f$.

The comparison functor $s$ is not conservative in general. One has to repeat this ``invertee-coinverter'' procedure possibly infinitely many times in order to get a conservative comparison, and an actual factorization system in \ensuremath{\mathsf{Cat}}\ \cite[Theorem C.0.31]{Joyal}. A similar phenomenon occurs when taking the ``identee-coidentifier'' analogue of the previous procedure, which allows to factor any functor as a (possibly infinite) sequence of coidentifiers followed by a discrete functor, i.e.~a functor whose fibres are discrete, yielding another factorization system in \ensuremath{\mathsf{Cat}}.

We are going to show that if we restrict ourselves to fibrations, both factorization procedures above simplify to a single-step factorization. Let us start with the second one showing that, for fibrations, it coincides with the comprehensive factorization system introduced in \cite{StW}. This actually holds not only in \ensuremath{\mathsf{Cat}}, but in any 2-category $\ensuremath{\mathsf{Cat}}(\ensuremath{\mathcal{E}})$ of internal categories where the construction of the comprehensive factorization of any functor provided in \cite{StV} is still valid, as, for example, when \ensuremath{\mathcal{E}}\ is a finitely cocomplete locally cartesian closed category, like any topos \ensuremath{\mathcal{E}}.

\begin{Proposition} \label{prop:comprehensive.cat}
Let $f\colon A \to B$ be a fibration in $\ensuremath{\mathsf{Cat}}(\ensuremath{\mathcal{E}})$ as above. The coidentifier of the identee of $f$, together with the comparison functor, factorizes $f$ into a final functor followed by a discrete fibration, giving then the comprehensive factorization of $f$. Starting with $f$ opfibration, the same procedure yields the dual comprehensive factorization of $f$ given by an initial functor followed by a discrete opfibration.
\end{Proposition}

\begin{proof}
We consider just the case of fibrations.
Following the approach of Section 3 in \cite{StV}, we perform the comprehensive factorization of $f$ by taking the free $R$-algebra $Rf$, which is a split fibration, and then reflecting it into a discrete fibration $p$:
\[
\xymatrix@!=7ex{
K_f \ar@/_/[r]_{\bar v}^{\top} \ar@/_2ex/[d]^{}="1" \ar@/^2ex/[d]_{}="2" \ar@{=>}"1";"2"^{\kappa_f} & K_{Rf} \ar@/_/[l]_{\bar r} \ar@/_2ex/[d]^{}="1" \ar@/^2ex/[d]_{}="2" \ar@{=>}"1";"2"_{\kappa_{Rf}} & Q \\
A \ar@/_/[r]_{v}^{\top} \ar@/^/[rru]^(.75){q} \ar[d]_{f} & B/f \ar[d]_{Rf} \ar@/_/[l]_{r} \ar[r]_-d & \pi_{0B}(B/f) \ar[u]^{t} \ar[d]_{p} \\
B \ar@{=}[r] & B \ar@{=}[r] & B.
}
\]
By construction of the above reflection, $d$ is the coidentifier of the identee $\kappa_{Rf}$ of $Rf$. Considering the adjunction $v \dashv r$ provided by the fact that $f$ is a fibration, we get $dv\kappa_f=d\kappa_{Rf}\bar{v}=1$, where $\kappa_f$ is the identee of $f$. Let now $q$ be a functor such that $q\kappa_f=1$, and consider the unit $\eta\colon 1 \to rv$ of the adjunction $v\dashv r$ in $\ensuremath{\mathsf{Cat}}(\ensuremath{\mathcal{E}})/B$. Then $f\eta=1$ and $\eta$ is contained in $\kappa_f$, so that $q\eta=1$ as well, and $qrv=q$. On the other hand, the counit $\epsilon\colon vr\to 1$ is such that $(Rf)\epsilon=1$, so it is contained in $\kappa_{Rf}$ and hence $d\epsilon=1$ and $dvr=d$.

Now, $qr\kappa_{Rf}=q\kappa_f\bar{r}=1$, so by the universal property of the coidentifier $d$ there exists a unique $t$ such that $td=qr$. Hence $q=qrv=tdv$, and $t$ is unique with this last property. Indeed, if $t'dv=q$ for some $t'$, then $t'd=t'dvr=qr=td$ and hence $t'=t$ since $d$ is cancellable. This proves that $dv$ is the coidentifier of $\kappa_f$, and then it is final \cite{StV}.
\end{proof}

\begin{Remark}{\rm
The last result also shows that, when $f$ is a fibration, the factorization of $f$ given by (sequence of coidentifiers, discrete functor) reduces to a single coidentifier and coincides with the comprehensive factorization.
}
\end{Remark}

In the special case of \ensuremath{\mathsf{Cat}}, Proposition \ref{prop:comprehensive.cat} can be proved directly by means of the pseudo-functorial interpretation of fibrations. This indeed is what we are going to do in order to obtain the analogous result, where coidentifiers are replaced by coinverters and discrete fibrations are replaced by fibrations in groupoids (i.e.~fibrations whose identee is an isomorphism).

\begin{Proposition} \label{prop:isocompr.cat}
Each fibration (respectively opfibration) $f\colon A \to B$ in \ensuremath{\mathsf{Cat}}\ admits a factorization given by the coinverter of the identee of $f$ followed by a fibration (resp.~opfibration) in groupoids. This factorization of $f$ coincides with the one given by (sequence of coinverters, conservative functor).
\end{Proposition}

\begin{proof}
Let us denote by
\[
\xymatrix{
\ensuremath{\mathsf{Cat}} \ar@<1.2ex>[r]^{\pi}_{\bot} & \ensuremath{\mathsf{Gpd}} \ar@{_(->}@<1.2ex>[l]^{i}
}
\]
the reflection of categories in groupoids, where the left 2-adjoint $\pi$ can be obtained by taking as unit component, for each category $A$, the coinverter $\eta_A$ of the 2-cell $\mu_A$ associated with the comma category $A/A$:
\[
\xymatrix@C=6ex{
A/A \ar@/_2ex/[r]^{}="1" \ar@/^2ex/[r]_{}="2" \ar@{=>}"2";"1"^{\mu_A} & A \ar[r]^-{\eta_A} & \pi(A)\,.
}
\]
Consider now a fibration $f\colon A\to B$ and denote by $[f]\colon B^{\ensuremath{\mathrm{op}}}\to \ensuremath{\mathsf{Cat}}$ the corresponding pseudo-functor. The composite $\pi[f]\colon B^{\ensuremath{\mathrm{op}}}\to \ensuremath{\mathsf{Gpd}}$ gives rise to a fibration in groupoids $\overline{f}\colon\overline{A}\to B$. On the other hand, $\eta[f]$ corresponds to a morphism $q\colon (A,f) \to (\overline{A},\overline{f})$ in $\ensuremath{\mathsf{Fib}}(B)$:
\[
\xymatrix{
A \ar[dr]_f \ar[rr]^{q} & & \overline{A} \ar[dl]^{\overline{f}} \\
& B.
}
\]
The component $q_b$ of $\eta[f]$ at an object $b$ of $B$ is actually the coinverter $\eta_{A_b}$ of $\mu_{A_b}$:
\[
\xymatrix@C=6ex{
A_b/A_b \ar@/_2ex/[r]^{}="1" \ar@/^2ex/[r]_{}="2" \ar@{=>}"2";"1"^{\mu_{A_b}} & A_b \ar[r]^-{\eta_{A_b}} & \pi(A_b).
}
\]
It is not difficult to see that the pair $(A_b/A_b,\mu_{A_b})$ coincides with the restriction $(K_b,\kappa_b)$ of the identee $(K,\kappa)$ of $f$ to the fibre over $b$. Hence, as explained in Remark \ref{rem:coinv.pseudofunct}, $q$ turns out to be the coinverter of $\kappa$ in $\ensuremath{\mathsf{Fib}}(B)$. Thanks to Corollary \ref{cor:isofib.cons=gpd(isofib)}, $\overline{f}$ is conservative and we get the desired factorization of $f$.
\end{proof}

\subsection{From \ensuremath{\mathsf{Cat}}\ to $\ensuremath{\mathsf{Fib}}(B)$} \label{sec:pseudo}

We are going to use now the results of the previous section to produce analogous factorizations for (fibrewise) opfibrations in $\ensuremath{\mathsf{Fib}}(B)$. Our focus on this case is motivated by the study of cohomology theories provided in \cite{CMMV17}. Analogous results are still valid while considering (fibrewise) fibrations in $\ensuremath{\mathsf{Fib}}(B)$ or (fibrewise) opfibrations in $\ensuremath{\mathsf{OpFib}}(B)$. It is a well known result, due to B\'enabou, that fibrations in $\ensuremath{\mathsf{Fib}}(B)$ (opfibrations in $\ensuremath{\mathsf{OpFib}}(B)$) are just fibrations (opfibrations) in \ensuremath{\mathsf{Cat}}, so that this case reduces to Propositions \ref{prop:comprehensive.cat} and \ref{prop:isocompr.cat}, thanks to Propositions \ref{prop:coinv} and \ref{prop:coid}.

Let us recall from \cite{CMMV17} the following definitions and results.

\begin{Definition} \label{def:fw} {\rm(see \cite[Definition 2.1]{CMMV17})}
We say that a morphism $p\colon (A,f)\to (C,g)$ in $\ensuremath{\mathsf{Fib}}(B)$ is a \emph{fibrewise (discrete) opfibration} if, for every object $b$ of $B$, the restriction $p_b\colon A_b\to C_b$ of $p$ to the $b$-fibres is a (discrete) opfibration.
\end{Definition}

From Theorem 2.8 in \cite{CMMV17} it follows that every internal opfibration in $\ensuremath{\mathsf{Fib}}(B)$ is a fibrewise opfibration, while the latter is exactly a morphism in $\ensuremath{\mathsf{Fib}}(B)$ which is an internal opfibration in $\ensuremath{\mathsf{Cat}}/B$ (see Proposition 2.5 and Proposition 2.7 in \cite{CMMV17}). By Corollary 2.9 in [\emph{loc.~cit.}] in the discrete case the two notions coincide. Recall also from [\emph{loc.~cit.}] that Yoneda's \emph{regular spans} and \emph{two-sided fibrations} are instances, respectively, of fibrewise opfibrations and internal opfibrations in $\ensuremath{\mathsf{Fib}}(B)$.
\medskip

Let us consider a fibrewise opfibration $p\colon(A,f)\to(C,g)$ in $\ensuremath{\mathsf{Fib}}(B)$ and focus our attention on its restriction $p_b$ to a single fibre over some $b$ in $B$. By the dual of Proposition \ref{prop:comprehensive.cat}, we can perform the comprehensive factorization of the opfibration $p_b\colon A_b \to C_b$ by means of the coidentifier $q_b$ of its identee:
\[
\xymatrix{
A_b \ar@/_3ex/[rr]_{p_b} \ar[r]^{q_b} & Q_b \ar[r]^{s_b} & C_b.
}
\]
Since $f$ and $g$ are fibrations, the assignments $b\mapsto A_b$ and $b\mapsto C_b$ are pseudo-functorial and the collection of the functors $p_b$ gives rise to a natural transformation of pseudo-functors from $B^{op}$ to \ensuremath{\mathsf{Cat}}. By the universal property of the coidentifiers $q_b$ for each $b$, the $Q_b$'s are also pseudo-functorial and the $q_b$'s and $s_b$'s organize in two natural transformations. Let us briefly show how this can be proved.

For each $b$ in $B$, we denote by $(K_b,\kappa_b)$ the identee of $p_b$. Let us observe that also the assignment $b\mapsto K_b$ is pseudo-functorial and together with the collection of the $\kappa_b$'s, it determines the identee $(K,\kappa)$ of the cartesian functor $p$. Given an arrow $\beta\colon b'\to b$ in $B$, we always denote by $\beta^*$ its associated change of base functor for any chosen fibration over $B$. Since $q_{b'}\beta^*\kappa_b=q_{b'}\kappa_{b'}\beta^*=1$, by the universal property of the coidentifier $q_b$ there is a unique functor $\beta^*\colon Q_b\to Q_b'$ such that $\beta^*q_b=q_{b'}\beta^*$:
\[
\xymatrix@!=6ex{
K_b \ar@/_2ex/[r]^{}="1" \ar@/^2ex/[r]_{}="2" \ar@{=>}"2";"1"^{\kappa_b} \ar[d]_{\beta^*} & A_b \ar[r]^{q_b} \ar[d]^{\beta^*} & Q_b \ar[r]^{s_b} \ar@{-->}[d]^{\beta^*} & C_b \ar[d]^{\beta^*} \\
K_{b'} \ar@/_2ex/[r]^{}="1" \ar@/^2ex/[r]_{}="2" \ar@{=>}"2";"1"^{\kappa_{b'}} & A_{b'} \ar[r]_{q_{b'}} & Q_{b'} \ar[r]_{s_{b'}} & C_{b'}.
}
\]
Given a composable pair of arrows
\[
\xymatrix{
b'' \ar[r]^{\beta'} & b' \ar[r]^{\beta} & b
}
\]
in $B$, let $\phi_{\beta,\beta'}\colon (\beta')^*\beta^*\to (\beta\beta')^*$ and $\gamma_{\beta,\beta'}\colon (\beta')^*\beta^*\to (\beta\beta')^*$ be the corresponding coherence isomorphisms induced by the fibrations $f$ and $g$ respectively. Since $q_{b''}\phi_{\beta,\beta'}$ is a 2-cell between $q_{b''}(\beta')^*\beta^*=(\beta')^*\beta^*q_b$ and $q_{b''}(\beta\beta')^*=(\beta\beta')^*q_b$, then by the universal property of the coidentifier $q_b$ there exists a unique invertible 2-cell $\psi_{\beta,\beta'}$ such that $\psi_{\beta,\beta'}q_b=q_{b''}\phi_{\beta,\beta'}$.
\[
\xymatrix@C=10ex@R=8ex{
A_b \ar[r]^{q_b} \ar[d]^{\beta^*}_(.8){}="1" \ar@/_6.5ex/[dd]_{(\beta\beta')^*}^(.4){}="2" \ar@{=>}"1";"2"_{\phi_{\beta,\beta'}}^\sim & Q_b \ar[r]^{s_b} \ar[d]_{\beta^*}^(.8){}="3" \ar@/^6.5ex/[dd]^(.25){(\beta\beta')^*}^(.4){}="4" \ar@{=>}"3";"4"^{\psi_{\beta,\beta'}}_\sim & C_b \ar[d]_{\beta^*}^(.8){}="5" \ar@/^6.5ex/[dd]^{(\beta\beta')^*}^(.4){}="6" \ar@{=>}"5";"6"^{\gamma_{\beta,\beta'}}_\sim \\
A_{b'} \ar[r]^{q_{b'}} \ar[d]^{(\beta')^*} & Q_{b'} \ar[r]|(.42)\hole^(.7){s_{b'}} \ar[d]_{(\beta')^*} & C_{b'} \ar[d]_{(\beta')^*} \\
A_{b''} \ar[r]_{q_{b''}} & Q_{b''} \ar[r]_{s_{b''}} & C_{b''}.
}
\]
Moreover, the equality $\gamma_{\beta,\beta'}s_bq_b=s_{b''}q_{b''}\phi_{\beta,\beta'}=s_{b''}\psi_{\beta,\beta'}q_b$ implies that $\gamma_{\beta,\beta'}s_b=s_{b''}\psi_{\beta,\beta'}$ since $q_b$ is right cancellable. Finally, the coherence conditions on the $\psi$'s making the assignment $b\mapsto Q_b$ into a pseudo-functor can be deduced once again by the universal property of the $q_b$'s. If $(K,\kappa)$ is the identee of $p$, since, for a morphism $t\colon(A,f)\to(Y,y)$ in $\ensuremath{\mathsf{Fib}}(B)$, $t\kappa=1$ if and only if $t_b\kappa_b=1$ for each $b$ in $B$, $q$ is actually the coidentifier of $\kappa$ in $\ensuremath{\mathsf{Fib}}(B)$. Recalling that, since each $s_b$ is a discrete opfibration, $s$ is an internal discrete opfibration, we have just proved the following result.

\begin{Proposition} \label{prop:comprehensive.int}
Every fibrewise opfibration $p\colon(A,f)\to(C,g)$ in $\ensuremath{\mathsf{Fib}}(B)$ admits a \emph{comprehensive factorization}
\[
\xymatrix@!=6ex{
A \ar@/^3ex/[rr]^{p} \ar[dr]_{f} \ar[r]_-{q} & Q \ar[d]_(.4){h} \ar[r]_-{s} & C \ar[dl]^{g} \\
& B
}
\]
where $q$ is the coidentifier of the identee of $p$ and $s$ is a discrete opfibration in $\ensuremath{\mathsf{Fib}}(B)$.
\end{Proposition}

The same result (with $g$ a split fibration) is obtained in Section 3.3 of \cite{CMMV17}, by providing an explicit construction of the discrete opfibration $s$ together with an ad hoc definition of $q$, which is later on proved to be the coidentifier of the identee of $p$.
\smallskip

As a corollary of Proposition \ref{prop:comprehensive.int}, we get an extension of Proposition \ref{prop:comprehensive.cat} in the case of \ensuremath{\mathsf{Cat}}.

\begin{Corollary}
For every fibrewise opfibration $p\colon(A,f)\to(C,g)$ in $\ensuremath{\mathsf{Fib}}(B)$, the factorization of Proposition \ref{prop:comprehensive.int} coincides with the one given by (sequence of coidentifiers, discrete functor) in \ensuremath{\mathsf{Cat}}.
\end{Corollary}

The present approach allows us to obtain an analogous result concerning the factorization given by (coinverter, opfibration in groupoids).

\begin{Proposition} \label{prop:refl.gpd.fib}
Every fibrewise (resp.~internal) opfibration $p\colon(A,f)\to(C,g)$ in $\ensuremath{\mathsf{Fib}}(B)$ admits a factorization
\[
\xymatrix@!=6ex{
A \ar@/^3ex/[rr]^{p} \ar[dr]_{f} \ar[r]_-{q'} & Q' \ar[d]_(.4){h'} \ar[r]_-{s'} & C \ar[dl]^{g} \\
& B
}
\]
where $q'$ is the coinverter of the identee of $p$ and $s'$ is a fibrewise (resp.~internal) opfibration in groupoids in $\ensuremath{\mathsf{Fib}}(B)$.
\end{Proposition}

\begin{proof}
Let us start with a fibrewise opfibration $p$. The coinverter $q'$ of the identee of $p$ in $\ensuremath{\mathsf{Fib}}(B)$ and the comparison arrow $s'$ can be constructed following the lines of the previous paragraph, by means of coinverters taken fibrewise. This means that for each $b$ in $B$, $(q'_b,s'_b)$ gives the factorization of $p_b$ into a coinverter followed by an opfibration in groupoids, thanks to Proposition \ref{prop:isocompr.cat}. Hence $s'$ is a fibrewise opfibration in groupoids.

If moreover $p$ is an internal opfibration, then $s'$ is also an internal opfibration by Corollary \ref{cor:fib.to.frac} applied to $\ensuremath{\mathsf{Fib}}(B)$, thanks to Proposition \ref{prop:dag.to.fib}.
\end{proof}

As a consequence, we get an extension of Proposition \ref{prop:isocompr.cat}.

\begin{Proposition} \label{prop:ultima}
For every fibrewise opfibration $p\colon(A,f)\to(C,g)$ in $\ensuremath{\mathsf{Fib}}(B)$, the factorization of Proposition \ref{prop:refl.gpd.fib} coincides with the one given by (sequence of coinverters, conservative functor) in \ensuremath{\mathsf{Cat}}.
\end{Proposition}

\begin{proof}
By Proposition \ref{prop:coinv}, the arrow $q'$ in the above factorization can be obtained as the coinverter of the identee $\kappa$ of $p$ in \ensuremath{\mathsf{Cat}}\ (which is also the identee in $\ensuremath{\mathsf{Cat}}/B$). Actually, $q'$ is also the coinverter of the invertee $\omega$ of $p$ in \ensuremath{\mathsf{Cat}}. Indeed, since $f\omega$ is an isomorphism, thanks to Lemma \ref{lemma:psvert=vert}, it factorizes as $\sigma\cdot\tau$, with $\sigma$ an isomorphism and $\tau$ an $f$-vertical 2-cell. Then $p\tau=p(\sigma^{-1}\cdot\omega)$ is an isomorphism, hence $\tau$ factorizes through the invertee $\omega'$ of $p$ in $\ensuremath{\mathsf{Cat}}/B$.
Since $p$ is an opfibration in $\ensuremath{\mathsf{Cat}}/B$, by Corollary \ref{cor:isofib.cons=gpd(isofib)}, the coinverter $q'$ of $\kappa$ in $\ensuremath{\mathsf{Cat}}/B$ is also the coinverter of $\omega'$ in $\ensuremath{\mathsf{Cat}}/B$, hence $q'\tau$ is an isomorphism. As a consequence $q'\omega$ is an isomorphism, and the universal property in \ensuremath{\mathsf{Cat}}~follows easily.

Using the same technique as before, considering now the invertee $\omega$ of $s'$ in \ensuremath{\mathsf{Cat}}, we get that $\omega=\sigma\cdot\tau$ with $\sigma$ an isomorphism and $\tau$ an $h'$-vertical 2-cell. As above, $\tau$ factorizes through the invertee $\omega'$ of $s'$ in $\ensuremath{\mathsf{Cat}}/B$, which is an isomorphism by Corollary \ref{cor:isofib.cons=gpd(isofib)}, because $s'$ is an opfibration in groupoids in $\ensuremath{\mathsf{Cat}}/B$. In conclusion, the factorization of $p$ in \ensuremath{\mathsf{Cat}}\ through a conservative functor is obtained by means of just one coinverter $q'$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


The long history of Qigong has led to a wide variety of forms, some more meditational, others more physical. Zhineng Qigong focuses on both mind and body but sees the mind as paramount.
More than 5000 years ago, the ancient Chinese Masters came to the understanding that everything is composed of the same energetic substance, which they called Qi. They observed that there is a oneness and wholeness in all existence and that everything is energetically interconnected as one. The ancient Chinese believed that all transformations happened due to the influence of Qi. All things (birth, death, etc.) happening between heaven and earth are caused by and formed out of Qi. Qi is vibrating in constant energetic motion within all things.
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WASHINGTON – The Senate Judiciary Committee today released another batch of records from Judge Brett Kavanaugh's service as a lawyer in the George W. Bush White House. Following today's release more than 103,000 pages of material from Judge Kavanaugh's Executive Branch service is now public. That's in addition to nearly 17,500 pages of material Judge Kavanaugh submitted to the committee in response to its bipartisan questionnaire and more than 10,000 pages from his service on the D.C. Circuit.
Nomination material is being posted HERE as it becomes available.
The Committee expects to continue receiving future productions on a rolling basis from both the Office of George W. Bush and NARA.


From shipping inquiries to ingredient questions, we've got your back… er, tail.
We hand blend every recipe specific to your dog's needs. We feel that every dog is different, so instead of trying to fit your dog into pre-made dog food, we custom formulate a blend of food that works for them.
We looked at all varieties of dog food and ultimately decided there is a reason a vast majority of dog food sold is kibble. It is easy to serve, requires no refrigeration or freezing and is full of nutrients.
We offer both grain inclusive and grain free diets.
Is your dog food suitable for puppies and senior dogs?
Yep, our dog food is made to meet the needs of all dog life stages. Simply enter the age of your dog in the wizard and we do the rest.
Are your ingredients sourced from the USA?
Yep, we only use ingredients from within the United States.
Who came up with the questions for your wizard?
We worked with a top pet food nutritionist to come up with a suitable list of questions to help determine the perfect blend of food for your dog.
What types of meat & proteins do you offer?
Most of our recipes are multi-protein, but we allow the ability to exclude certain proteins. We offer recipes with pork, chicken, beef, and fish.
What do you mean "blend"?
We use a wide variety of proprietary recipes that are then carefully selected and blended together to create a recipe unique to your dog. After completing the wizard, you will see your custom blend.
Can I feed one blend to both of my dogs?
Each of our blends meets the AAFCO guidelines for dog food, so will be suitable for any dog. We do recommend though you fill out the wizard for each dog though so it can be suited to their needs.
What are the protein and fat contents of each blend?
After you complete the wizard, the results page will provide detailed information on the nutritional and ingredient contents of each recipe.
The food is sourced, much of it from local farms, all within the USA. It's manufactured in Texas.
At the very least, we recommend the bag be rolled up and clipped shut. If you have an airtight container available for storage, that is the most suitable option.
Written on the bag will be the correct amount of food to feed your dog a day. This should be broken up into 2 meals a day, but you can feed as to what works for your dog.
We offer a satisfaction guarantee on all of our food. If you find that your dog doesn't like it we will refund your money and cancel your subscription.
Nope! Your package will be left for you unless you explicitly choose for it to be held at a pickup location.
Will I get notified when my package is on its way each month?
Yep! You'll get an automated email with a tracking number each shipment.
Depending on where you are, we use USPS or Fedex, with the occasional dash of UPS.
Yep, simply log in to your account, click "Subscription" and then under actions "Cancel".
Can I skip a subscription shipment?
Yep! To do so, login to your account, click "Delivery Schedule" and find the upcoming shipment you'd like to skip and click "skip".
Log in to your account and click "Billing Information" and then "Update Card".
Log in to your account, click "Subscriptions" and then under your current shipping address at the top, click "Edit".


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News - TransfersManchester United Target Sean Longstaff Set To Be Subject of Steve Bruce Charm Offensive
Rangers All But Done For Serie A Centre-Back Filip Helander
Rangers are set to complete the signing of Sweden centre-back Filip Helander after agreeing a fee for his services.
Gers boss Steven Gerrard wants to bring in another defender and the Glasgow giants have been linked with a host of potential signings.
He has though zeroed in on Serie A and is set to snap up Helander, according to Swedish daily Aftonbladet.
Helander currently turns out for Bologna and it is claimed Rangers have agreed a fee with the Italian club, with the player costing the Gers a fee of around the £4m mark to lure to Scotland.
The centre-back is now expected to complete the fomalities of the switch and become Rangers' latest signing this summer.
Helander has another three years left to run on his contract at Bologna, but the Swede has had his head turned by the prospect of turning out for Rangers.
He made 20 appearances in the Italian top flight for Bologna last season, being booked on six occasions.
Experienced in Italian football, Helander has turned out in Serie A on 84 occasions, but is now set for a new challenge in Scotland.
He has been capped by Sweden on eight occasions since making his international debut in 2017.


Finding Henry Hunan's Restaurant in Chinatown may be a little bit difficult because of its secluded location, yet everything is worth the effort. Home to the spiciest Chinese dishes, you can delight your taste buds with their impressive selection of authentic Sichuan dishes. Dumplings are served hot and spicy, so are stir fried noodles and vegetables. Make sure to order their extra special meat pie. Freshly baked and served hot straight from the oven, it is filled with a generous amount of beef seasoned with authentic Chinese spices and herbs. Of course, it is spicy just the way you want it.


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Home » Magnolia Bakery arrives in Washington, D.C. in September
Magnolia Bakery arrives in Washington, D.C. in September
By Brian Amick
Image courtesy of Magnolia Bakery
One of the biggest names in baking will soon make its mark in on the biggest cities in the country. Magnolia Bakery is set to open its first location in Washington, D.C. in September.
Magnolia Bakery's new D.C. store will be located at transit hub Union Station, where another big name – French macaron specialist Ladurée – opened last year. It's located just blocks from the U.S. Capitol Building, and has become an iconic destination for visitors looking to shop and dine.
There is no official launch date for the bakery yet, but it is scheduled to operate from Monday through Saturday from 10 a.m. to 9 p.m. EST. and Sunday from 12 p.m. to 6 p.m.
Customers will be able to enjoy Magnolia's signature cupcakes, as well as favorites such as banana pudding, cheesecake, pies, muffins, cakes, cookies, brownies, and much more. Other dining options that have arrived at Union Station recently include pizza chain Neapolitan Express, San Francisco's Blue Bottle Coffee, and chicken sandwich chain Chik-fil-A.
Magnolia Bakery debuts in Washington, D.C.
Magnolia Bakery expected to open in D.C. in 2018
Levain Bakery to open first non-New York City location in Washington, D.C.
New D.C. bakery becomes first in city with in-house mill
Washington, D.C. gets brick-and-mortar cookie dough bakery


Celebrity moms are just like us, they say. When we see Beyoncé performing pregnant in front of thousands of people, or basically any photo of Gwyneth, that cliché is hard to believe. But when it comes to feeding their babies, starlet moms might actually be more relatable than we thought. So, in honor of World Breastfeeding Week, here is a compilation of memorable breastfeeding moments in Hollywood.


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We invite you to contact us today to discuss your care. Please don't hesitate to call us at (507) 876-9997.
Welcome to Urbain Family Chiropractic, PLLC!
Urbain Family Chiropractic, PLLC believes that you can reach maximum improvement in your health using an approach that evaluates your condition in three ways.
Most people think of Dr. Colleen Urbain Hess DC as a wellness doctor because the focus is on helping people get and stay well, naturally.
Here at Urbain Family Chiropractic, PLLC, we have many different services to fit your specific needs. Whether you are experiencing back pain, feeling out of alignment, or just want to get healthier, we can help.


"You see, my second sun already appears to you to be more than a hollow dream! But this is only a rough and imperfect work. The light does not reproduce itself; the gases are exhausted; the refractory body loses its properties. It requires a man versed in science to watch over the apparatus incessantly, to prevent an explosion. But I shall create a sun that will be, in approximate proportion, as bright and as durable as God's sun!"
S. Henry Berthoud (1804-1891) was a writer of considerable ability, remembered today mostly for his often reprinted collections of folkloric legends from Northern France. Yet, he was a writer who came close to inventing science fiction as early as the 1840s.
This unique collection assembled, translated and introduced by Brian Stableford, gathers 32 of Berthoud's best works, displaying his often pioneering surges of imagination. It includes samples of his remarkable supernatural stories, his eccentric scientific fantasies, featuring real or imaginary scientists, his ground-breaking visions of the prehistoric past of Paris, and his futuristic novella about the Year 2865.
"Berthoud's work would have been just as difficult for contemporary readers to assess as for modern ones, but its esotericism should not be allowed to detract from his achievement, which is as remarkable for its fervor as for its uniqueness." Brian Stableford.


At a time when global equity strategies are maturing and the search for returns are ever more prevalent after a relatively strong extended period of positive returns, there are two arguments that can be made for making a discrete investment allocation to global small- and mid-cap (SMID) companies.
The first argument considers Global SMID from an asset allocation perspective. If you believe that growth in sales and earnings are a scarce fundamental commodity that ultimately dictates a company's share price performance over time, together with stock price inefficiencies of the asset class, then an allocation to Global SMID may add value going forward. Not only does Global SMID hold superior risk-adjusted return characteristics to broad global benchmarks, it also offer a strong growth option without a lot of the shortcomings of other global growth asset classes and its underfollowing by sell-side analysts provides the active investment manager with a greater opportunity to produce a differentiated view to help generate alpha.
The second argument considers Global SMID as a style diversifier. In an environment where style diversification is becoming less effective, and where Global portfolios are becoming more subject to unintended consequences, diversifying by size, and not style, is becoming an increasingly important driver behind maximising risk-adjusted returns.
Since 2000, an allocation to Global SMID would have added 3.3% per annum over and above the MSCI ACWI Index(1). Additionally the asset class has performed particularly well in recent periods, especially when compared to Emerging markets, one of the more popular Global 'growth' asset classes(2), as evidenced in Chart 1 and Table 1 below.
Source: eVestment Alliance. Quarterly returns in A$, annualised. As at 30 December 2016.
Since 2000, Global SMID has exhibited the lowest level of risk out of the Growth asset classes, and with risk-adjusted returns second behind Small cap. Table 3 below further shows Global SMID provided for better risk-adjusted returns versus Emerging markets and significantly better versus Global World indices, that it outperformed MSCI ACWI 68% of the time and held the lowest tracking error amongst Global growth asset classes.
Performance metrics are calculated against MSCI ACWI.
Historically, as an asset class, Global SMID has provided growth shown to outstrip that found in other common global equity buckets such as Global Large caps and Emerging Market equities. Table 2 below shows this has been true for fundamentals such as sales and earnings in particular, and where both SMID and Small caps have led dividend growth.
It is entirely reasonable to observe SMID cap, and Small cap, return on capital (ROC) at the lower end of the group, leverage slightly higher and free cash flow weaker, as they are typically in their 'growth' phases where the focus is on growing capital expenditures and investing in their franchises. As such, returns as measured by ROC in table above, can take time to grow.
Interesting to further point out over the last 8 years, that despite actual ROCs not being overly impressive, all Global indices have shown an improvement, with the exception of Emerging markets, and that the greatest improvement generated has come from SMID.
This points to the fact that Global SMID, as a global asset class, is less directly exposed to more macro issues (such as weak economics and weak commodity prices) and heightened political risks within key Emerging markets that can sometimes influence poor capital allocation decisions made by management teams of associated companies. Markets such as Brazil, Russia and China, that tend to be driven by one or two Energy and Resource focused 'heavy-weights', also come with their own political ties. For example (1) Petrobras, an integrated energy company and Brazil's top publicly traded company by revenues and which is controlled by the Brazilian government ; (2) Gazprom, another oil and gas company, one of Russia's largest company and one of the world's largest publicly listed gas producer, is majority-owned by the Russian government and accounts for approximately 25% of Russia's tax revenues with its various stakes in local financial institutions, electricity assets and telecom networks; and (3) over 150 state-owned enterprises (SOEs) in China which in themselves have an abundance of financial resources and power to effectively act like independent 'empires' or oligopolies influencing political and economic decisions of the country.
Are in the sweet spot of their business life cycle - Once companies move past the start-up phase they enter the SMID cap stage – the sweet spot – where their business model has been proven, they have achieved some product and customer diversification, are on their way to building a strong franchise with an improving balance sheet and cash flows to support them, yet still with plenty of growth opportunities available to capture and exploit. This can lead to consistently above average revenue and earnings growth, and over time a strong return profile.
Face minimal issues with the law of large numbers - Many large caps can find it difficult to sustain good growth rates due to their sheer size, making it tough to create new products or services that move the needle in terms of revenue or earnings, and where incremental R&D spending for these large caps often has a diminishing effect on profitability and returns. Large caps have often already expanded into all corners of the globe and therefore have less opportunity to penetrate new markets through geographic diversification. SMID caps do not face these same issues and usually have plenty of organic growth opportunities available to them.
Offer better liquidity - Better trading liquidity (buyers) can help to reduce transaction costs and lower risks associated with investing in companies with poor liquidity either due to size or market exposure. This is an important consideration, especially during bear markets, where the ability to sell a small cap position if required may be nearly impossible; notwithstanding the overall longer waiting period for selling in normal market conditions.
Offer better diversification, where at the company level it can sometimes be similar to large caps but on a smaller scale. Small caps tend to sell a limited number of products and/or services into a small number of geographies/regions. Global SMID caps generally have more diversified revenue streams, creating more stability while still benefitting from ample growth opportunities. Somewhat reflected in Global SMID's historically higher ROC versus Small caps and comparable ROC to Large caps (Table 2).
Command stronger balance sheets, often generate better cashflow and have better access to capital markets during times of 'stress' which can help them hold up better in market downturns (Table 4).
Not a forced seller due to market cap - Investment Managers of small caps often have to sell an investment once it reaches a certain market cap level. This sell decision could come at a time when the company still has plenty of growth opportunities ahead. By having exposure to Global SMID caps, an active manager is able to hold the position for longer and capture more of the potential upside. Whilst a SMID cap manager may still be forced to sell when the company enters the large cap space, at least they may have been able to benefit from more of the market cap growth over time.
Offer superior outperformance in different environments – where small caps are often the strongest performing asset class in strong up markets and large caps typically hold up better in down markets, SMID caps often perform well in both types of markets (see Table 4 below). Actually SMID performed the strongest within the Global growth asset classes in down markets, which perhaps help support the case for SMID to be seen as a possible lower risk 'growth' option. Whilst small caps may present the strongest upside capture and returns, this comes with the risk of less favourable liquidity characteristics and arguably being at the relatively riskier early stage of their business life cycle.
More micro focused – the type of risk taken within a SMID company is less volatile or more controllable given it is potentially more micro-driven versus that within Emerging markets, where company growth and stock market performance can be more dependent, if not sometimes entirely reliant, upon macro and political issues.
Are underfollowed by sell-side analysts, when compared to large caps. This can help create stock price inefficiencies and a greater opportunity for active investment managers to produce a differentiated view to consensus and greater ability to generate alpha. For the purpose of illustration, assume a market cap cut off at A$31bn, under which is regarded as SMID cap down to A$1bn, at which point and below is Small cap. Table 5 highlights there are 4-5x the amount of overall recommendations and estimates on Large caps' sales and earnings vs SMID caps despite just over two-thirds of the Global Market consisting of companies with market caps in the SMID cap range.
Can be M&A targets, given their attractiveness for larger companies looking for growth. Investing in companies that become take-out targets can be an excellent way to generate alpha as many large companies tend to overpay in their desperate search for growth.
Active and passive exposures for global equity portfolios are looking increasingly similar and are arguably going through, what we at BAM are referring to a 'portfolio eclipse'. A result of this is a build-up of stock specific risk causing asset owners to end up holding a handful of stocks, mostly and unintentionally large caps, that are seemingly driven by momentum as a dominant factor. Additionally, this effect can lead to a disconnect between valuation and fundamentals, where the diversification benefits of an active / passive split dissipates and downside protection is compromised.
Arguably the true diversification 'test' with any portfolio comes during periods of major weakness. Table 6 below compares the traditional style buckets of Value and Growth to how the three key market cap groups i.e. Global all cap, Global SMID and Global Small, performed during two major market corrections since 2000, namely the Tech Crash and the GFC and the subsequent two year periods. Specifically, both Global SMID and Global Small cap stocks on average pulled back less than the World and on average provided more upside, pointing to size acting as a more effective portfolio construction tool. Additionally, a similar picture is concluded when compared to Value, Growth and Quality styles, suggesting that style diversification has not saved portfolios from meaningful losses in drawdowns, nor allowed for participation in maximum upside in subsequent years.
Is the only style bucket with the optimal blend of high Upside market capture and low Downside market capture - Analysing the performance characteristics of the major style and size buckets in all market conditions over the last 20 years in Chart 2 below shows Global Quality and Global SMID to have preserved capital well and offered the best downside protection, whilst Global Growth has not done well. Particularly interesting is the outcome for Global SMID which is somewhat counter-intuitive but points to many of the advantages it may hold as an asset class made earlier.
Source: eVestment Alliance: +MSCI Indices is the median of the five indices shown in the chart. Period is for 20 years ending June 2017, run on a monthly basis. All results are in AUD terms and measured against MSCI World-ND.
Offers meaningfully positive actual return and risk outcomes. Chart 3 below shows over a 20 year period, MSCI World, Global Value and Global Growth have all produced a similar level of return, with an increasing level of risk respectively. And whilst Global growth and Global SMID have virtually identical risk levels, Global SMID provided higher returns.
Proven to act as a solid preserver of capital. Chart 4 below shows since 2000, an allocation to Global SMID would have generated 0.8x outperformance against Global Value, 1.4x against Global Growth and 0.7x against Global Quality. Notwithstanding the fact that Global SMID has also proven to act as a solid preserver of capital during this period.
To conclude, a discrete investment allocation to Global SMID companies could be seen as a compliment to Global small cap exposure with better liquidity, with the potential growth option that historically has been available within emerging markets but without the macro and political risks associated with these markets, and with superior risk adjusted return characteristics to the broader, and particularly larger, global benchmarks.
Additionally, as traditional style bucketing is becoming increasingly challenged, greater diversification at a market cap level also provides for positive risk / return characteristics and can further help create great investment opportunities for fundamental / active investors.
Important information:This has been produced by Bell Asset Management Limited (BAM) ABN 84 092 278 647, AFSL 231091. This has been prepared by BAM for information purposes only and does not take into consideration the investment objectives, financial circumstances or needs of any particular recipient – it contains general information only. Before making any investment decision, you should consider your needs and objectives, consult with a licensed financial adviser and obtain a copy of the relevant offering document. No representation or warranty, express or implied, is made as to the accuracy, completeness or reasonableness of any assumption contained herein. To the maximum extent permitted by law, none of BAM and its directors, employees or agents accepts any liability for any loss arising, including from negligence, from the use of this document or its contents. This document shall not constitute an offer to sell or a solicitation of an offer to purchase or advice in relation to any securities within or of units in any investment fund or other investment product described herein. Any such offer shall only be made pursuant to an appropriate offer document. Past performance is not necessarily indicative of expected future performance.


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