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Aston Martin Virage coupe (2011 – 2012) review
Review Aston Martin Virage coupe (2011 – 2012) review
Read the Aston Martin Virage coupe (2011 - 2012) car review by Auto Trader's motoring experts, covering price, specification, running costs, practicality, safety and how it drives.
Last updated: 7th November 2014
Aston Martin spots a gap between its DB9 and DBS and fills it with the Virage. In doing so it has effectively replaced both with this hugely talented all-rounder.
Sharper looks
Performance and specification gains over DB9
More drivable than its DBS relative
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If the DB9 is a head turning work of automotive art then the Aston Martin Virage is an absolute masterpiece. The changes are subtle, but details like the re-profiled grille, revised wings and the new headlamps all combine to create an edgier, more contemporary take on the DB9's elegant lines. Not hugely adventurous stuff from Aston Martin admittedly, but enough to allow the Virage fulfil its role as the middle-ground between the DB9 and DBS. Sat alongside a DB9, the Virage is tauter and more chiselled, though it manages to eschew the DBS's rather extrovert, somewhat aftermarket styling.
Like the exterior there are few real surprises inside the Virage for those used to the cabin of a DB9. That does mean there is plenty of beautifully finished leather, high gloss wood and brushed metal. Unsurprisingly, it's styled particularly neatly, Aston's mix of materials particularly nicely judged – even the Volvo-sourced air vents. It might be easy on the eye, but aside for the obvious big push buttons for the automatic transmission – the Virage not being offered with a manual – the smaller controls are a bit fiddly.
The specification says 2+2, but you'll need to be flexible and small to fit in the rear of the Virage. Best then not to bother trying and instead specify the optional hard-backed sports seats. This means losing the rear seats altogether and using the shelves for additional luggage space. The boot will take a golf bag or two, but getting them in won't be particularly easy. Soft bags are best for the boot then. In the cabin there's decent oddment storage, though the glovebox isn't huge – think sunglasses and iPod only.
Riding on the same basic suspension of the DB9 but with a few revisions to make it sharper, the Virage is supple but exceptionally well controlled. It rides on an adaptive damping system that manages this tricky balance, though in Sport mode the ride goes from taut to too-stiff for our crumbling roads. Reduced unsprung mass thanks to lighter standard carbon ceramic brake discs help the Virage ride so well – they also stop it very effectively. The steering is quick and nicely weighted and the electronic stability control systems intervention levels are high enough to enjoy the Virage's ample power.
The Virage's 6.0-litre V12 engine's output is a healthy 490bhp, which allows it to reach 62mph in 4.6 seconds and a top speed of 186mph. It's only offered with Aston Martin's TouchTronic gearbox, which can be either left alone or shifted via steering column mounted paddles. It's a quick, smooth shifter, while the V12's ample mid and low range torque is backed up with real high-rev fireworks when you're in a hurry. It sounds great, though it's better from the side of the road than within. Fast, yet not as fearsome as the DBS, the Virage is a consummate all-rounder.
Aston Martins aren't cheap to buy, so don't expect them to be cheap to run. The 6.0-litre V12 engine in the Virage develops more power than the DB9's, which suggests it'll consume a bit more fuel. Aston hasn't revealed official figures as yet, but expect everyday driving to result in consumption of around 12-15mpg. Fuel aside, servicing, insurance and road tax won't be inexpensive either.
The DB9, which this Virage is heavily based on, has been around for a while now, and although it's not got a blemish-free reliability record Aston Martin has improved in recent years. That should mean the Virage offers relatively trouble-free motoring. The company's recent endurance racing success underlines the durability of the mechanicals.
Like its DB9 relative the Virage comes with electronic stability and traction control systems, anti-lock brakes, front and side driver and passenger airbags and seatbelt pretensioners among its standard safety equipment. If you opt for the sports bucket seats you lose not only the rear seats, but the side airbags, too. The Virage hasn't been independently crash tested, but its long nose, front engined layout and stiff extruded aluminium structure should mean it's safe should the worst happen.
Equipment is where the Virage really aces its DB9 relative. Along with the additional power and poise the Virage gains powerful and lightweight carbon ceramic brakes as standard, while the hopeless Volvo-sourced sat-nav is replaced by an all-new system developed by Aston Martin and Garmin. Add in the usual luxury refinements you'd expect for this money – leather, electrically operated seats etc. – and the Virage actually represents decent value in its class.
If you like the DB9 you'll love the Virage. It looks better, drives better, has more equipment and yet it's only around £25,000 more. To specify the carbon ceramic brakes alone in a DB9 you'd need £15,000, which makes the DB9 look a bit defunct. What's impressive though is how it nibbles away at the DBS making a case for itself too. It retains its impressive GT credentials, but gains a degree of more sporting appeal, making it the only sensible choice in the DB9-based family line-up.
Performance, prestige and classic cars
Aston Martin Virage Coupe (2011 - 2012) image gallery


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Listen: imaJJine's "Your Voice"
by W. Andrew Powell August 14, 2019
written by W. Andrew Powell August 14, 2019
Award-winning producer, songwriter, and author Keith "Keef" Whiting has an ambitious project on his hands with the new all-star 37-member mega-ensemble, imaJJine. Blending contemporary jazz, gospel, and R&B for the creative project, the group will launch their debut album, Love Songs & Lullabies, this fall.
imaJJine's first single, "Your Voice", features Michael Stuart and Brittany Miranda with a lot of sway between the jazz, gospel, and R&B influences.
"I've always been a big fan of Joni Mitchell," Whiting said about his inspirations. "I was trying to capture the nuances of her voice–like 'a wolf's mournful howl on a Canadian night' and 'a loon's plaintive calls as it rises in flight'."
"The song started out as random musings, evolved into a poem, and finally into a song."
"If you listen carefully, you'll hear echoes of her work with Jaco Pastorius," he added. "But, ultimately, all you'll find is a writer at a loss for words to describe her voice!"
So who are imaJJine? The mega group features 13-year-old Rishita Barua, Guido Basso, as well as Lou Pomanti, Bill King, Russ Little, John Capek, Carlos Del Junco, Rich Brown, Bruce Cassidy, Donald Quan, Tom Szczesniak, Kobenna Harrison, Vito Reza, John Roby, Scott Alexander, Darcy Hepner, Kevin Barrett, Kieran Overs, John Johnson, Bob Cohen, John Alcorn, Gavin Hope, Jackie Richardson, Arlene Duncan, Alisha Oliver, Alex Samaris, Jeremy Tozer, Gabriella Rodgers, Tara Wink, Keith Whiting, Joe Sealy, Pat Sullivan, Jeremy Darby, Matt Snell, and Michael Jack.
Whiting originally came to Canada in 1975 and his first production was the Gold Certified Black Noise by FM. He later worked with numerous artists, including Marc Jordan, Ken Tobias, Aaron Davis, and produced classical music albums, and worked at CBC Radio as the senior music producer.
imaJJine's Love Songs & Lullabies will be released this September.
Watch the lyric video for imaJJine's "Your Voice" below and follow them on Twitter, Facebook, and Instagram.
Brittany MirandaimajjineMichael Stuart
W. Andrew Powell
W. Andrew Powell lives, sleeps, eats, and breaths movies and entertainment. Since launching The GATE, Andrew has enjoyed being a pest to any publicist who would return his calls.In his "spare time," Andrew is also an avid photographer, and writes about leisure travel and hotels around the world.
Christee Palace's new single, "What Have You Done...

redpajama_set_name: RedPajamaCommonCrawl


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\section{Introduction}

The importance of formal semantics should not be underestimated, especially when aimning to design \emph{reliable} dynamical systems in heterogeneous environments. Therefore, a variety of state based modelling frameworks at different levels of abstraction have been proposed; to quote Goguen \cite{Goguen92sheafsemantics}: \emph{one person's syntax is another person's semantics}.
Diversity in
algorithms can be desirable; however, as
argued in \cite{Goguen92sheafsemantics,abramsky:critic-2006,presheaves-as-transitionsys:1997}, the proliferation of semantic theories indicates our scattered understanding of concurrent systems.
Thus, we seek a framework that provides semantic structure describing the behaviour of a dynamical system and its refinement independently of syntax.

This goal is shared to an extent by the theory of coalgebras \cite{Rut03:universal}.
In \cite{bk2017}, we abandoned state-based modelling in favour of describing behaviour as the set of executions (inspired by \cite{Cuijpers:2013:DCM,control-theory-book}) because the branching structure of a state in the presence of invisible actions is described by the set of executions (not states).
This situation further escalates when one is interested in
infinite executions (e.g. fairness properties \cite{hennessy:futureperfect,Kwiatkowska1989:fairness-survey}) or dense executions which are omnipresent in hybrid systems (e.g. \cite{Cuijpers:lost-in-translation}). The point is not that executions are inexpressible in a coalgebra, but rather that we need a semantic framework where they are treated as first-class citizens just as states are in a coalgebraic framework. Thus, our hypothesis is that behaviour of a system is given solely by its executions.

We anticipate presheaves to be the ``right'' semantic structure to study executions without fixing a kind of dynamical system. Note that we are \emph{not} the first in proposing presheaves as the mathematical universe to studying behaviour. Winskel and his colleagues \cite{presheaves-as-transitionsys:1997,cattani_winskel_2005,hildebrandt:fairness,jnw96:bisimopenmaps,Fiore:wbisim_open-maps}
have already employed presheaves (among other things) by giving a denotational semantics of process-algebraic terms supported by characterisations of strong bisimulation and weak bisimulation relations using open maps in the context of transition systems. For a more modern treatment, Hirschowitz and his colleagues \cite{Hirschowitz:LICS19,Hirschowitz:sheaf-pi,hirschowitz:inncocent} advocated game semantics using (pre)sheaves.

The novelty of our work lies in refining the notion of open maps (which we christened \emph{bisimulation maps}) in a presheaf category and using it to characterise $\forall$-fair bisimulation \cite{Kupferman2003:fair_equiv_rel,Henzinger:2002} and branching bisimulation \cite{bbisim:1996} relations. In addition, the prospect of having to specify notions of time and observation (which was absent in the earlier works on presheaf semantics) leads to a clearer modelling, so explicitly highlighting  these two dimensions of system modelling is at the core of our contribution (cf. Section~\ref{sec:prelim}). This distinction was in turn essential to capture branching bisimulation in the presence of invisible actions $\tau$.

Interestingly, unlike open maps, bisimulation maps are always retracts in the category of presheaves (in turn, they are surjective at the level of executions). As a slogan, presheaf maps are refinement maps, while bisimulation maps (which are special presheaf maps) are complete refinement maps (Section~\ref{sec:bisimmaps}). By moving to a finer notion, we are still able to capture functional bisimulations without fairness. However, in the context of fairness, we can show (Theorem~\ref{thm:existsfairisfairbisimfct}) that the behavioural equivalence induced by a bisimulation map coincides with $\forall$-fair bisimulation relation. Note that our $\forall$-fair bisimulations are equivalence relations by definition in contrast to the existing definition  \cite{Kupferman2003:fair_equiv_rel,Henzinger:2002} (see the dicussion after Theorem~\ref{thm:existsfairisfairbisimfct} on Page~\pageref{thm:existsfairisfairbisimfct}). This is an improvement with respect to the previous characterisation of $\exists$-fair bisimulation \cite{Kupferman2003:fair_equiv_rel,Henzinger:2002} (called \emph{extended bisimulation} in \cite{hennessy:futureperfect}) obtained by Hildebrandt \cite{hildebrandt:fairness} using open maps, since any $\forall$-fair bisimulation relation is strictly finer than an $\exists$-fair bisimulation relation \cite{Henzinger:2002} and our correspondence does not impose any restrictions on the fairness predicates. These restrictions, originally from \cite{hennessy:futureperfect}, asserted that fairness predicates on infinite executions are closed under the removal and the addition of finite prefixes.

Another practical aspect of the theory of presheaves is that it guides us in finding the right semantic categories once a notion of time and observation is fixed.
Moreover, we can apply concepts (like, e.g., essential geometric morphism \cite{sheafbook}) that transform a dynamical system from one observation space $\obs$ to another space $\obs'$.
This way we can transform (see Section~\ref{sec:invsible}) a presheaf of executions (induced by a given transition system) into a presheaf of minimal executions (i.e., executions in which trailing $\tau$-transitions are chopped off). This property is specific to branching bisimulation, which may be the reason why this construction was not discussed in \cite{Fiore:wbisim_open-maps} (their objective was to capture weak bisimulation).

\paragraph{Organisation of the paper.}
In Section~\ref{sec:prelim}, we introduce our mathematical framework to model behaviour of a dynamical system with a special focus on the aspects of time and observation. Then, we introduce the notion of bisimulation maps in presheaves on an arbitrary (small) category in Section~\ref{sec:bisimmaps}. Turning our attention towards the first major example, we characterise $\forall$-fair bisimulation relations in Section~\ref{sec:fairness}. The case of invisible actions in Section~\ref{sec:invsible} is based on a change of observation space. We first outline an obvious (but ultimately failed) attempt to capture branching bisimulation, before giving the correct (yet intuitive) construction that characterises branching bisimulation.

\section{Our universe of discourse}\label{sec:prelim}
The objective of this section is to describe our semantic framework in which one can model behaviour of a dynamical system. By \emph{behaviour} of a dynamical system, we understand some \emph{phenomena that evolve over time}. Our aim is to formalise this intuition. We begin by modelling time as a small category $\timeaxis$, whose objects are points in time and arrows describe passing of time.

\begin{notation}
  An object $C$ (an arrow $f$) of a category $\cat C$ will be denoted by the predicate $C\in \cat C$ ($f\in\cat C$). Moreover, the codomain and domain of an arrow $f\in\cat C$ are denoted as $\cod f$ and $\dom f$, respectively.
\end{notation}

Invariably, dynamical systems come with a notion of observation.
For instance, a letter from a fixed alphabet may denote the assignment of model variables in a computer program/controller. We assume that a system under study has a display unit together with the existence of a hypothetical `observer' $\obs$ who is watching/measuring behaviour of the system using this display unit over time. In addition, our observer $\obs$ can remember its observations over time, i.e., earlier observations can be deduced from the later observations. Mathematically, this amounts to saying that $\obs$ is a contravariant functor $\timeaxis \rTo \set$.

\begin{proposition}
  Let $\cat C$ be a small category. Then, the collection of functors of type $\op{\cat C} \rTo \set$ (i.e., presheaves on a category $\cat C$) and natural transformations between them form a category $\presheaves{\cat C}$.
\end{proposition}

\begin{notation}
  Given a presheaf $F\in\presheaves{\cat C}$, we follow \cite{sheafbook} in writing $x\cdot f$ to denote the restriction of $x\in FC$ along $C'\rTo^f C$, i.e., $Ff(x)=x\cdot f$. In case $\cat C$ is a poset (viewed as a category) $\cat C$, we write $x\cdot C'$ to denote the restriction of $x\in FC$ along $C'\preceq C$.
  Note that we use calligraphic letters for specific presheaves, whereas arbitrary ones are denoted by capital letters as above.
\end{notation}

\begin{example}\label{ex:observation-lts}
  In this example, we fix the notion of time $\timeaxis$ and observation $\obs$ associated with a (labelled) transition system.
  For time $\timeaxis$ we take the set of natural numbers $\nat$ viewed as a category (arrows are the less-than-equal-to relations). For the given alphabet $\act$, we now define a presheaf $\A\in\presheaves{\nat}$:
  \[
  \A(n) =
  \{\sigma\in\fseq\act \mid |\sigma| = n\} \qquad
    \text{(for every $n\in\nat$),}
  \]
  together with the action on $\A$ given by $\sigma\cdot n =\sigma|_n$ (for every $\sigma\in \A(n')$ and $n\leq n'$).

  In other words, $\A(n)$ is the set of those finite words $\sigma\in \fseq\act$ whose length is $n$ (denoted by $|\sigma|=n$), while the action $\_\cdot n$ simply maps a word $\sigma$ of length $n'$ to its unique prefix of length $n$ (denoted by $\sigma|_n$). Note that $\A(0)$ is a singleton set containing the empty word which we denote by $\eseq$.
\end{example}
\begin{remark}\label{remark:sheaves}
  In modelling some dynamical systems, like, e.g., those arising from control theory \cite{control-theory-book}, $\obs$ may have even more structure in that global observations can be constructed by gluing  the local observations (smaller neighbourhoods). In such situations, the category of sheaves $\sheaves{\cat C,J}$ equipped with a Grothendieck topology $J$ on $\cat C$ is more suitable (cf. \cite{spivak_hybrid-systems}) for semantic purposes. Moreover, sheaves equipped with discrete Grothendieck topology are exactly presheaves (cf.\thinspace\cite{sheafbook}), so our mathematical universe is actually the category of sheaves (rather than presheaves). But due to the discrete nature of dynamical systems considered in this paper, we restrict ourselves to presheaves. Nevertheless, we will state our definitions so that they are applicable on sheaves (see, e.g., Remark~\ref{remark:sheaves-bisim}).
\end{remark}
Once a notion of time $\timeaxis$ and an observation $\obs\in\presheaves{\timeaxis}$ is fixed, then a system essentially describes the \emph{runs} (also known as \emph{trajectories} or \emph{executions}) of the system and the observation associated with each run.
To answer both, we envisage that a dynamical system is nothing but an object in the slice category $\presheaves{\cat C}/\obs$. In other words, a dynamical system corresponds to a presheaf $F$ modelling the runs of the system and a natural transformation $F \rTo^\alpha \obs$ modelling the observation associated with each run of the system. More importantly, a \emph{system homomorphism} $\phi$ between two systems $(F,\alpha)$ and $(G,\beta)$, denoted $(F,\alpha) \rTo^\phi (G,\beta)$, is a natural transformation $F \rTo^\phi G$ preserving the observations, i.e., $\beta \circ \phi = \alpha$. Intuitively, a system homomorphism $(F,\alpha) \rTo^\phi (G,\beta)$ says that the system $(F,\alpha)$ is a \emph{refinement} of $(G,\beta)$ (i.e., every observable behaviour of $F$ is also part of the observable behaviour of $G$).

\subsection{Refining our framework by unifying time and observation}\label{subsec:guide}
Although the slice category $\presheaves{\timeaxis}/\obs$ is close to our system theoretic intuition, its presentation can be further simplified.
Recall the \emph{category of elements} of a presheaf $F\in\presheaves{\cat C}$, denoted $\mathbb E_{\cat C}(F)$ (we drop the subscript $\cat C$ whenever clear from the context), has as objects the tuples $(x,C)$ with $C\in\cat C,x\in FC$ and as arrows $(x,C) \rTo^f (x',C')$ the morphism $C \rTo^f C'\in\cat C$ such that $x'\cdot f=x$.

\begin{theorem}[\!\!\protect{\cite[Exercise~III.8(a)]{sheafbook}}]\label{thm:elements-equivalence}
  For a presheaf $F$ over a small category $\cat C$, there is an equivalence of categories
  $\presheaves{\cat C}/F \cong \presheaves{\elements{F}}$.
\end{theorem}
\noindent
Note that a similar result also holds in the setting of sheaves (cf.\thinspace \cite[Exercise~III.8(b)]{sheafbook}).

In other words, time can be made inherent with observation and, thus, we can work in a simpler setting without worrying about the bookkeeping associated with slice categories. To see this, recall Example~\ref{ex:observation-lts} and the poset of finite words $\fseq\act$ (a.k.a.\thinspace free monoid) generated by a set $\act$, which is ordered by the prefix relation $\preceq\ \subseteq\fseq\act\times\fseq\act$. Notice that the categories $\elements{\A}$ and $\fseq\act$ are isomorphic: since the length of a word is redundant in the objects of $\elements{\A}$ dropping the length results in the elements of $\fseq\act$. Thus, we obtain
\begin{corollary}
There is an equivalence of categories $\presheaves{\nat}/\A \cong \presheaves{\fseq\act}$.
\end{corollary}

As a result, the category of presheaves on $\fseq\act$ can serve as the semantic universe to study behaviour of a transition system (cf. Example~\ref{ex:semantics-lts}). More generally, by giving the semantics to a `syntactic' category of a computational model $\cat M$, we mean identifying the notion of time $\cat T$ and observation $\obs\in\presheaves{\cat T}$ together with a faithful functor $\cat M \rTo^{\semantics{\_}} \presheaves{\mathbb E_{\cat T}(\obs)}$, called the \emph{semantics} functor. By interpreting an arrow $M \rTo^f M'$ in $\cat M$ as \emph{$M$ is an implementation of $M'$ witnessed by $f$}, then faithfulness of $\semantics{\_}$ asserts: if an implementation is witnessed by two semantically same morphisms $\semantics{f}=\semantics{g}$, then $f=g$ must be the same syntactically.

\begin{example}\label{ex:semantics-lts}
  Consider a transition system $\tuple{X,\act,\rightarrow}$ where $X$ is the set of states, $\act$ is the set of actions, and $\rightarrow \subseteq X \times \act \times X$ is the transition relation\footnote{Transition systems without initial states are standard in process algebraic literature (see \cite{process_algebra_reference}).}. Then the collection of transition systems together with simulation functions form a category denoted $\lts$. Note that a \emph{simulation function}
  is a function $X \rTo^f Y$ satisfying:
  \begin{equation}\label{eq:trans-preserve}
    \forall_{x,x'\in X,a\in\act}\ x\step a x' \implies f(x) \step a f(x').
  \end{equation}
  As usual, we write $x \step a x'$ to denote $(x,a,x')\in\rightarrow$. Let $\history{\sigma}=\{\sigma'\in\fseq\act\mid \sigma' \preceq \sigma \}$ be the prefixes of $\sigma$.
  Next, define a presheaf $\semantics{X}\in\presheaves{\fseq\act}$ which records all the executions whose trace is $\sigma$ at $\semantics{X}(\sigma)$:
  \begin{align*}
    \semantics{X}(\sigma) =&\ \left\{\history{\sigma} \rTo^p X \mid \forall_{\sigma',a}\ \big(\sigma'a\preceq\sigma \implies p(\sigma') \step a p(\sigma' a)\big) \right\},\\
    p\cdot \sigma' =&\ p_{\history{\sigma'}} \quad \text{\big(for any $\sigma'\preceq \sigma $ and $p\in\semantics{X}(\sigma)$\big)}.
  \end{align*}

  Moreover, any function $X \rTo^f Y$ induces a family of maps $\semantics{f}_\sigma (p) = f\circ p$ (for each $\sigma\in\fseq\act$).
\end{example}
Thus, we obtain the following result; wherein, the result that presheaf maps encode simulation maps is well known from the early work of Joyal {et al.} in \cite{jnw96:bisimopenmaps}.
\begin{proposition}\label{prop:lts-semantics}
  The map $\semantics{\_}$ defined in Example~\ref{ex:semantics-lts} is a faithful functor. Moreover,
  \begin{itemize}
    \item if $\semantics{f}$ is a presheaf map for any $f\in\lts$, then $f$ is a simulation function.
    \item if a function $f$ between the underlying state spaces induces a presheaf map $\semantics f$, then $f$ is a simulation map.
  \end{itemize}
\end{proposition}
As a result, the category of transition systems $\lts$ has presheaf semantics on $\fseq\act$. In the subsequent sections, we will show the applicability of our semantic framework by giving presheaf semantics to different computational models; namely, transition systems with fairness predicates and invisible transitions.
\section{Bisimulation maps: towards complete refinement}\label{sec:bisimmaps}

Consider a category of computational models $\cat M$ together with its semantics over a presheaf category $\presheaves {\cat C}$, i.e., $\cat M \rTo^{\semantics{\_}} \presheaves{\cat C}$. As mentioned earlier, an arrow between any two images of the semantics functor as a refinement map from the modelling point of view. In particular, we interpret $\semantics{M} \rTo^{\semantics{f}} \semantics{M'}$ (induced by an arrow $M \rTo^f M'\in\cat M$) as the information that $\semantics{M}$ is an implementation of $\semantics{M'}$ witnessed by the refinement map $\semantics{f}$. Note that this is a straightforward generalisation of Proposition~\ref{prop:lts-semantics} since simulation maps encode the refinement of behaviour in the case of labelled transition systems.
Therefore, to study bisimulation maps at the level of presheaves (executions), we will restrict ourselves to those presheaf morphisms that represent `complete' refinement of behaviour in the following sense. Theorem~\ref{thm:bisim-epi} expresses this in a more formal manner.
\begin{itemize}
  \item Every observable behaviour in $\semantics{M'}$ is also observable in $\semantics{M}$ (thus, $\semantics{M}$ is a refinement of $\semantics{M'}$).
  \item Moreover, every observable behaviour in $\semantics{M}$ can be retracted onto an observable behaviour in $\semantics {M'}$.
\end{itemize}

One possibility is to use the open maps of Joyal et al. \cite{jnw96:bisimopenmaps}, which gave a unified definition of functional bisimulations over the range of computational models. In particular, when invoked in a presheaf category, open maps correspond to natural transformations whose naturality squares are the weak-pullback squares in $\set$ (see \cite[Proposition~2.3]{cattani_winskel_2005}). Nevertheless, an open map falls short in capturing the complete refinement point of view since an arbitrary open map may not even be a surjective map at the level of executions; i.e., our implementation may not even implement or cover all the behaviour present in the specification.

In \cite{openmaps-concrete:2015}, open maps \cite{jnw96:bisimopenmaps} were refined to embedding-open maps in the setting of concrete categories. An important difference to the classical definition of open maps is that the parametric notion of path extensions can be replaced by embeddings (see \cite[Definition~8.6(2)]{book:con-cat} for a formal definition).
Moreover, embedding-open maps are always retracts under some mild restrictions on concrete categories (cf. \cite{openmaps-concrete:2015}). Now, if a category is concrete over itself, then an embedding corresponds to simply a monomorphism. Consequently, we propose the following definition of an embedding-open map, which we simply call a \emph{bisimulation} map.
\begin{definition}\label{def:bisim-presheaf}
  A map $F \rTo^f G \in \presheaves{\cat C}$ is a  \emph{bisimulation} if, and only if, for every commutative square depicted in \eqref{eq:bisim-presheaf} with a mono $P \rMono^g Q$ and maps $m,n$ in $\presheaves {\cat C}$, there is a map $Q \rTo^k F\in\presheaves{\cat C}$ such that the two triangles commute, i.e., $k\circ g=m$ and $f\circ k=n$.
  \begin{diagram}[LaTeXeqno]\label{eq:bisim-presheaf}
    Q & \rTo^{n} & G \\
    \uMono^{g} & \rdExistMap^k & \uTo_{f} \\
    P & \rTo_{m} & F
  \end{diagram}
\end{definition}
Intuitively, \eqref{eq:bisim-presheaf} states that every extension of behaviour observable in $G$ can be reflected as an extension observable in $F$ through the witnesser $f$.
\begin{remark}\label{remark:sheaves-bisim}
  It is interesting to note that a similar definition for sheaves can be derived from the definition of embedding-open maps as given in \cite{openmaps-concrete:2015}. First, note that $\sheaves{\cat C,J}$ is concrete over $\presheaves{\cat C}$ due to the forgetful functor $\isheaf$, which is fully faithful. Moreover, embeddings in this concrete category are actually monomorphisms. This is because any mono is a regular mono in $\sheaves{\cat C, J}$ and the faithful functor $\isheaf$ preserves regular monos (since $\isheaf$ is right adjoint to the associated sheaf functor $\asheaf$ \cite{sheafbook}). Lastly, every regular mono is an embedding whenever the faithful functor preserves regular monos \cite[Proposition~8.7.3]{book:con-cat}. Thus, we have the exact same definition of bisimulation maps in $\sheaves{\cat C,J}$.
\end{remark}
\begin{theorem}\label{thm:bisim-epi}
  Every bisimulation map in a (pre)sheaf category is a retract
\end{theorem}
We end this section by capturing functional bisimulations in terms of bisimulation maps whose proof can be extracted from the proof of Theorem~\ref{thm:fair-bisimulation-map}. Note that a similar theorem was proven earlier in the seminal paper \cite{jnw96:bisimopenmaps} for functional bisimulation; however, the difference is that we use bisimulation maps (not open maps) in our characterisation.
\begin{theorem}
Given a simulation function $X \rTo^f Y$, then $\semantics{f}$ is a bisimulation map in $\presheaves{\fseq\act}$ if, and only if, the function $f$ is a surjection satisfying:
\begin{equation}\label{eq:bisim-reflect}
  \forall_{x\in X,y\in Y}\ \big( f(x) \step a y \implies \exists_{x'\in X}\ (x\step a x' \land f(x')=y) \big).
\end{equation}
\end{theorem}

\section{The case of fairness}\label{sec:fairness}

When considering infinite, rather than finite, behaviour of systems, it is common to take fairness  into account. There are various notions of fairness \cite{Henzinger:2002} but the general idea for fair (bi)simulation is to demand that fair executions are matched by fair executions, in addition to classical (bi)simulation properties. In this section, we give a presheaf semantics to fair transition systems and outline how $\forall$-fair bisimulation can be captured via bisimulation maps.

Let $\iseq\act=\{\sigma \mid \nat \rTo^\sigma \act\}$ be the set of infinite words generated from $\act$ and fix $\aseq\act=\fseq\act\cup\iseq\act$, which is ordered by the prefix relation $\preceq$. The object of study are \emph{fair transition system}s $\tuple{X,\act,\rightarrow, \fairness {X}}$, where $\tuple{X,\act,\rightarrow}$ is a transition system and $\fairness {X}$ is a fairness predicate on infinite executions. An \emph{infinite execution} is a function $\history{\sigma} \rTo^p X \cup \{\chaos\}$ whose domain is the history of an infinite word $\sigma\in\iseq\act$ such that
$p(\sigma) = \chaos$ and $\forall_{\sigma',a}\ (\sigma' a\prec \sigma \implies p(\sigma') \step a p(\sigma'a))$.
Here, $\Omega$ is the \emph{chaos} state which the system enters once it has executed an infinite execution. Let $\fexec{X}=\fairness{X}\cup\exec{X}$ be the set of all fair executions ordered by the prefix relation $\preceq$, i.e., $p\preceq p' \iff p'|_{\dom p} = p$ (for $p,p'\in\fexec{X}$). The set of (finite) executions $\exec{X}$ is defined as earlier, i.e., $\exec{X}=\{\history\sigma\rTo^p X\mid \sigma\in\fseq A\wedge\forall_{\sigma'a\in \history\sigma}\ p(\sigma')\step ap(\sigma'a)\}$.

\begin{definition}
\label{def:fairbisim}
 A chaos preserving extension $X\cup\{\Omega\} \rTo^{f_\Omega} Y \cup\{\Omega\}$ (i.e., $f_\Omega(x)=f(x)$ for $x\in X$ and $f_\Omega(\Omega)=\Omega$) of a function $X\rTo^f Y$ is a \emph{fair simulation} between $\tuple{X,\act,\rightarrow, \fairness {X}},\tuple{Y,\act,\rightarrow, \fairness {Y}}$ if, and only if, $f$ satisfies \eqref{eq:trans-preserve} and
  $
  \forall_{p\in\fairness{X}}\ f_\Omega\circ p\in \fairness {Y}
  $.
  Henceforth, we do not distinguish between $f_\Omega$ and $f$.
\end{definition}

Note that an infinite, but fair, execution can be seen as the limit of a monotonically increasing sequence of finite executions in $\fexec X$. Since this limiting sequence is part of behaviour, we should reflect it. Thus, we say a \emph{fair bisimulation} $X\cup \{\Omega\} \rTo^f Y \cup \{\Omega\}$ is a surjective fair simulation $f$ satisfying \eqref{eq:bisim-reflect} and the following condition for any increasing sequence of finite executions $(p_i)_{i\in\nat}$ in $X$:
  \begin{equation}\label{eq:fbisim-reflect}
  \bigsqcup_{i\in \nat} f\circ p_i \approx f\circ \bigsqcup_{i\in\nat} p_i.

  \end{equation}
Here, $\approx$ is the Kleene equality used to equate the  partially defined terms above.

To the best of our knowledge, the above notion of fair bisimulation is novel; however, below we will establish its connection with the literature after characterising it in terms of presheaf morphisms.
So let $\fts$ be the category of fair transition systems and fair simulation between them. In addition,
\begin{itemize}
  \item Time: Since infinite executions are allowed in a fair transition system, we take $\timeaxis=\nat\cup\{\infty\}$ to be the category of natural numbers extended by a number representing infinity (i.e., $\forall_{n\in\nat}\ n\leq \infty$).
  \item Observation: Using the definition of $\A$ in Example~\ref{ex:observation-lts}, we define: $\obs(\infty)=\iseq\act$ and $\obs(n)=\A(n)$ (for $n\in\nat$).
\end{itemize}
Clearly, the categories $\elements{\obs}$ and $\aseq\act$  are isomorphic and by applying Theorem~\ref{thm:elements-equivalence} we obtain $\presheaves{\elements{\obs}} \cong \presheaves{\aseq\act}$. So, we take the category $\presheaves{\aseq\act}$ as the semantic universe to study fair transition systems. Just as in Example~\ref{ex:semantics-lts}, for a given fair transition system $\tuple{X,\act,\rightarrow,\fairness{X}}$, we define a presheaf
\[
\semantics{X}(\sigma) =  \left\{p\in\fexec{X} \mid \max \dom p =\sigma\right\} \quad (\text{for each $\sigma\in\aseq\act$}),
\]
and the action is given by restricting the domain of an execution. Moreover, for any fair simulation function $f$, we let $\semantics{f}_\sigma = f\circ \_$ (for each $\sigma\in\aseq\act$); thus, resulting in a presheaf semantics to fair transition systems.
\begin{proposition}\label{prop:fair-semantics}
  The above map $\fts \rTo^{\semantics{\_}} \presheaves{\aseq\act}$ is a faithful functor.
\end{proposition}
Invoking the bisimulation map definition between any two presheaves generated by fair transition systems results in the characterisation of fair bisimulation.
\begin{theorem}\label{thm:fair-bisimulation-map}
  A fair simulation function $f$ is a fair bisimulation if, and only if, the underlying map $\semantics{f}$ is a bisimulation map in $\presheaves{\aseq\act}$.
\end{theorem}

\noindent\stepcounter{thm}\textit{Remark~\thethm.}
  It should be noted that fair bisimulation maps are stronger than the open maps studied by Hildebrandt \cite{hildebrandt:fairness} in $\presheaves{\aseq\act}$ to characterise the extended bisimulation of Hennessy and Stirling \cite{hennessy:futureperfect} for pointed systems, i.e. systems with explicitly initial states. To demonstrate this, recall that an $\text{Inf}_\bot$-open map of Hildebrandt \cite[Proposition~25]{hildebrandt:fairness} is a fair simulation function (not necessarily surjective) $X\cup\{\Omega\} \rTo^f Y \cup \{\Omega\}$ satisfying \eqref{eq:bisim-reflect} and the property:
  \begin{equation}\label{eq:hildebrandt-fair}
    \forall_{x\in X,q\in\fexec Y}\ \big( q(\eseq)=f(x)  \implies \exists_{p\in\fexec{X}}\ (p(\eseq)=x \land f\circ p=q) \big).
  \end{equation}

Consider the two systems drawn on the right with a function $f$ between the states depicted by dashed lines.
\[\scalebox{1}{
  \begin{tikzpicture}
    \node (x1) {$x$};
    \node (x2) at ($(x1.center)+(1,0)$) {$x'$};
    \node (x12) at ($(x2.center)+(1.5,0)$) {$y$};
    \path[->]
        (x1) edge node[above]{$a$} (x2)
        (x1) edge[loop left] node[left] {$a$} (x1)
        (x2) edge[loop above] node[right] {$a$} (x2)
        (x12) edge[loop right] node[right] {$a$} (x12);
    \path[->,dashed]
        (x1) edge[bend right] (x12)
        (x2) edge (x12);
  \end{tikzpicture}}
  \]
In the left system, the infinite executions visiting $x'$ infinitely often are considered fair, whereas the only infinite execution in the right system is fair. Clearly, $f$ is a fair simulation satisfying \eqref{eq:hildebrandt-fair}.
  But $f$ is \emph{not} a fair bisimulation because the sequence of finite executions $(p_i)_{i\in\nat}$ formed by unfolding the self-loop on $x$ has a fair execution $\bigsqcup_{i\in\nat} f\circ p_i$ (looping on $y$) as the limit in the right system. Yet, $(p_i)_{i\in\nat}$ has no limit, thus, violating \eqref{eq:fbisim-reflect}.

\begin{remark}
In \cite{hildebrandt:fairness}, (separated) presheaves with sup topology are used because an increasing sequence of finite executions induced by a fair transition system has at most one limit point. Unlike \cite{hildebrandt:fairness}, we are not interested in one-to-one semantic representation of our syntactical models, so we work with arbitrary presheaves. This is because the choice whether the semantic universe should be (separated) presheaves or sheaves depends on observations, i.e., how $\obs$ is modelled. This is also why we do \emph{not} require our semantic functor $\semantics{\_}$ to be full.
\end{remark}

Next, we relate fair bisimulation maps with $\forall$-fair bisimulation relations.
\begin{definition}\label{def:forall-fairbisim}
A \emph{$\forall$-fair bisimulation} on $\tuple{X,\act,\rightarrow,\fairness {X}}$ is an equivalence relation $\mcal R\subseteq X\times X$ satisfying the following transfer properties:
\begin{enumerate}
  \item\label{def:forall-fairbisim1} $\forall_{x,y,x',a}\ \big( (x\step a x' \land x \mcal R y ) \implies \exists_{y'}\ (y \step a y' \land x' \mcal R y') \big)$, and
  \item\label{def:forall-fairbisim2} $\forall_{p,q}\
    \big( (p=_{\mcal R} q \land p\in\fairness X ) \implies q\in\fairness{X}\big)
    $.

\end{enumerate}
Here, $p=_{\mcal R} q$ is an abbreviation for $\dom q=\dom p \land \forall_{\sigma\in\dom p\cap\fseq\act}\ p(\sigma) \mcal R q(\sigma)$.
\end{definition}

\begin{theorem}\label{thm:existsfairisfairbisimfct}
Two states $x$ and $x'$ are related by a $\forall$-fair bisimulation relation if, and only if, there is a fair bisimulation function $f$ such that $f(x)=f(x')$.
\end{theorem}
The requirement of equivalence relations in the above definition may look superfluous at first glance. This is because, traditionally (i.e., when fairness predicates are empty sets), a strong bisimulation relation by definition is not necessarily an equivalence relation on the set of states. Moreover, \emph{bisimilarity} which is defined as the union of all strong bisimulation relations turns out to be both an equivalence relation and a strong bisimulation relation. However, such closure results do not hold in general for $\forall$-fair bisimulation relations.
$\forall$-fair bisimulation relations are not closed under union and the relational composition (even if we relax Definition~\ref{def:forall-fairbisim} by replacing `an equivalence relation' for `a symmetric relation')\footnote{This is how $\forall$-fair bisimulation relations are defined in \cite{Kupferman2003:fair_equiv_rel} on the states of Kripke structures. In regards to the above closure properties, we are only aware of \cite{Hojati:96-phdthesis}
who showed that $\forall$-fair simulation relations are not closed under union.}. Thus, $\forall$-fair bisimilarity may, in general, neither be an equivalence relation nor a $\forall$-fair bisimulation; in other words, $\forall$-fair bisimilarity is not a coinductive definition (like how strong bisimilarity is). Nevertheless, we deemed all $\forall$-fair bisimulation relations to be equivalences because: first, the main use of a $\forall$-fair bisimulation relation is to equate two systems that have the same behaviour sensitive to fair executions; second, the mathematics tells us that the kernel of a (fair bisimulation) function is an equivalence relation.

\section{The case of invisible actions}\label{sec:invsible}

The behaviour of a specification and its implementation is often spread over different levels of abstraction. The standard process algebraic way to relate the behaviour of an implementation with its specification is by delineating the effect of actions in lower levels of abstraction as invisible. For instance, removing a message from a buffer is considered unobservable in a rendezvous between communicating processes. This is made formal by reinterpreting the notion of (bi)simulation functions in the presence of the invisible action $\tau$.

\subsection{Branching bisimulation}
\begin{notation}
  Henceforth, $\tau\not\in\act$ will denote the invisible action and $\tact=\act\cup\{\tau\}$. Furthermore, $\exec{X,\sigma}$ is the set of executions $p$ having trace $\sigma$, i.e., $\max \dom p=\sigma$ and $\steps{} \subseteq X \times \fseq{\act} \times X$ is the \emph{weak reachability} relation on $\tuple{X,\tact,\rightarrow}$
  given as the smallest relation satisfying the following conditions:
  \[
  \frac{ }{x\steps \eseq x} \qquad \frac{x\step\tau x'}{x\steps \eseq x'} \qquad \frac{x\steps\eseq x'\wedge x'\steps\eseq x''}{x\steps \eseq x''} \qquad \frac{x \steps\sigma x' \land x'\step a x''}{ x \steps{\sigma a}x''}.
  \]
\end{notation}
\begin{definition}\label{def:bsimulation}
  A \emph{branching simulation} function $f$ between systems $\tuple{X,\tact,\rightarrow}$ and $\tuple{Y,\tact,\rightarrow}$ is a function $X \rTo^f Y$ satisfying the following properties:
  \begin{enumerate}
    \item Simulation of observable transitions, i.e., $\forall_{x,x'\in X,a\in\act}\ x \step a x' \implies f(x) \step{a} f(x')$, and
    \item (Possible) simulation of invisible transitions, i.e., $\forall_{x,x'\in X}\ x \step \tau x' \implies  \big(f(x)=f(x') \lor f(x) \step{\tau} f(x')\big)$.
    \item Stuttering of $\tau$-transitions, i.e., for any $x_1,x_2,x_3$ we have
    \begin{equation}\label{eq:b-stutter}
      \big(x_1 \steps\eseq x_2 \steps\eseq x_3 \land f(x_1)=f(x_3)\big) \implies f(x_1)=f(x_2).
    \end{equation}
  \end{enumerate}
  A \emph{branching bisimulation} $f$ is a branching simulation surjection $f$ satisfying the `weak' reflection of transitions, i.e., for any $x\in X,y\in Y$, and $a\in\tact$ we have
  \begin{equation}
  \label{eq:trans-breflect}
  f(x) \step a y \implies \exists_{x',x''\in X}\ (x \steps {\eseq} x' \step a x'' \land f(x')=f(x) \land f(x'')=y).
  \end{equation}
\end{definition}
\begin{definition}
  Given a labelled transition system $(X,\tact,\rightarrow)$, then a symmetric relation $\mcal R\subseteq X \times X$ is a \emph{branching bisimulation} \cite{bbisim:1996} if, and only if, the following transfer property is satisfied
  \[
  \forall_{x_1,x_2,y_1,a\in\tact} \Big( (x_1 \step a x_2 \land x_1 \mcal R y_1) \implies (a=\tau \land x_2 \mcal R y) \lor \exists_{y,y_2}\ \big( y_1 \steps{\eseq} y \step a y_2 \land x_1 \mcal R y \land x_2 \mcal R y_2\big) \Big).
  \]
  Two states $x,x'\in X$ are branching bisimilar if there exists a branching bisimulation $\mcal R$ such that $x\mcal R x'$.
\end{definition}
We work with branching bisimulation functions (not relations) because of the following result (Theorem~\ref{thm:bbisim-kernel}), which is similar in spirit to Lemma~{2.7} proved by Caucal in \cite{caucal:1992}. The difference is that Caucal's branching bisimulation functions (which he calls \emph{reduction} in \cite{caucal:1992}) do not respect the stuttering of invisible steps \eqref{eq:b-stutter}. Nevertheless, we are still able to obtain the following correspondence since the largest branching bisimulation relation satisfies the so-called \emph{stuttering lemma} of \cite{bbisim:1996}. In particular, any reduction $X \rTo^f Y$ in the sense of Caucal can be extended to a branching bisimulation function by composing it with the quotient map $Y \rTo^q Y/\mcal R$, where $\mcal R\subseteq Y \times Y$ is the largest branching bisimulation relation on $Y$.
\begin{theorem}\label{thm:bbisim-kernel}
  Two states $x,x'\in X$ of a transition system $\tuple{X,\tact,\rightarrow}$ are branching bisimilar if, and only if, there are a transition system $\tuple{Y,\tact,\rightarrow}$ and a branching bisimulation function $X \rTo^f Y$ such that $fx=fx'$.
\end{theorem}
\subsection{The setup}
Our first step towards the characterisation of branching bisimulation is the presheaf representation of executions induced by a transition system with invisible actions. Since a transition system evolves in a step-based manner, it is sufficient to model the time by the set of natural numbers $\nat$. The only difference, when compared to the strong case (cf.\thinspace Example~\ref{ex:observation-lts}), is in the notion of observation. Instead of recording words from $\fseq\act$, our hypothetical `observer' now records words from $\ftact$. So, consider a presheaf $\op\nat \rTo^{\A_\tau} \set$ in the spirit of $\A$ as defined in Example~\ref{ex:observation-lts}.
Recall that the slice category $\presheaves{\nat}/\A_\tau$ is equivalent to the category of presheaves on the category of elements $\elements{\A_\tau}$, which is isomorphic to the category $\ftact$.
Thus, for a given transition  system $(X,\tact,\rightarrow)$, we define a presheaf $F_{X}\in\presheaves{\ftact}$ as follows:
\begin{equation}\label{eq:presheaf-tact}
  F_X(\sigma) = \exec{X,\sigma},\quad \text{(for every $\sigma\in\ftact$)}.
\end{equation}
The action on $F_{X}$ is given by the restriction of the domain of an execution.

Incidentally, a branching simulation function $X \rTo^f Y$ between $(X,\tact,\rightarrow)$ and $(Y,\tact,\rightarrow)$ does \emph{not} induce a system homomorphism between the underlying dynamical systems (presheaves) $F_X$ and $F_Y$ because a branching simulation function does not necessarily preserve the length of the executions. For example, a sequence of transitions $\bullet \step \tau \bullet \step a \bullet$ may get mapped to a transition $\bullet \step a \bullet$.

Thus, we need a procedure that transforms a given presheaf on $\ftact$ to a presheaf on $\fseq\act$. It turns out that there already is a general result in category theory for this purpose, which we explain next.
In particular, recall the cocompletion of a category $\cat C$ through the Yoneda embedding $\cat C \rTo^{\yoneda{\cat C}} \presheaves{\cat C}$.
\begin{theorem}[see \cite{sheafbook}]\label{thm:cocompletion}
  For any functor $\cat C \rTo^h \cat D$, when $\cat C$ is small and $\cat D$ is cocomplete, there is a colimit preserving functor $\presheaves{\cat C} \rTo^{L_h} \cat D$ satisfying $L_h\circ \yoneda{\cat C} \cong h$. Moreover, $L_h$ has a right adjoint $R_h$ given by: $R_hD(C)=\cat D(hC,D)$, for each $C\in\cat C,D\in\cat D$.

\end{theorem}
Using the language of Kan extensions, $L_h$ is the left Kan extension of $h$ along the Yoneda embedding $\yoneda{\cat C}$. Moreover, using the notion of a coend \cite{categories-working-mat}, we have:
\[L_h F \cong \int^{C\in\cat C} FC \odot hC,\]
where $\cat D \rTo^{S \odot\_ } \cat D$ is the copower functor given by $S\odot D =\coprod_{s\in S} D$ (taking $S$ disjoint copies of $D$). If we replace $\cat C \rTo^h \cat D$ in the above diagram by a map $\cat C \rTo^h \cat D \rTo^{\yoneda{\cat D}} \presheaves{\cat D}$, then we obtain the so-called \emph{essential geometric morphism} \cite{sheafbook} between $\presheaves{\cat C} \rTo \presheaves{\cat D}$. In full, this means that the composition functor $h^*$ not only has a right adjoint $\Pi_h$, but also a left adjoint $\Sigma_h$ (see, for instance, \cite{sheafbook} for a formal definition of a geometric morphism).
Below, the composition functor $h^*$, its left adjoint $\Sigma_h$, and its right adjoint $\Pi_h$ are given by (up to isomorphism) $R_{\yoneda{\cat D}h}, L_{\yoneda{\cat D}h}$, and $R_{h^*\yoneda{\cat D}}$, respectively.
\begin{corollary}[see \cite{sheafbook}]\label{cor:ess_geometric}
  Given a functor $\cat C \rTo^h \cat D$ between small categories, then the inverse image functor $\presheaves {\cat D} \rTo^{h^*} \presheaves{\cat C}$ given by $h^*G=G\circ \op{h}$ (for each $G\in\presheaves{\cat D}$) has both left and right adjoints:
  \[\Sigma_h \dashv h^* \dashv \Pi_h.\]
\end{corollary}
So, in principle, there are two ways to land in $\presheaves{\fseq\act}$ from $\presheaves{\ftact}$ whenever there is a functor between $\ftact\rTo\fseq\act$. Nevertheless, since our aim is to characterise branching bisimulation functions, we will choose the left adjoint (over the right adjoint) of the composition functor for this task. This is because to reflect (recall \eqref{eq:trans-breflect}) an observable transition -- say, of the form $\bullet \step a \bullet$ --, we only need \emph{minimal} executions which are of the form $\bullet \steps{} \bullet \step a \bullet$ (cf.\thinspace Definition~\ref{def:mexecution}). In particular, we will show (cf.\thinspace Theorem~\ref{thm:adjoint-h}) that the left adjoint $\Sigma_h$ (induced by a suitable $h$) transforms a presheaf of executions into a presheaf of minimal executions (or those executions in which trailing $\tau$-transitions are chopped off).
\subsection{An obvious, but failed attempt}\label{subsec:failed}
There is an evident functor $\ftact \rTo^h \fseq\act$ which treats the letter $\tau$ as an empty word. It is then natural to investigate whether the bisimulation maps in $\presheaves{\fseq\act}$ between any two induced presheaves $\Sigma_h F_X,\Sigma_h F_Y$ characterise the branching bisimulation map. Unfortunately, the answer is no! We explain this extensively, as this lays the formal foundation for the desired characterisation given in the next subsection.

So, consider the hiding function $\tact \rTo^h \fseq\act$ which treats $\tau$ as the unit of $\fseq\act$ (i.e., $h(\tau)=\eseq$) and treats an action $a\in\act$ as observable (i.e., $h(a)=a$ when $a\in\act$).  This lifts to an order-preserving function $\ftact \rTo^h \fseq\act$ (denoted again as $h$ by abuse of notation). Thus, we have a functor $\ftact \rTo^h \fseq\act$.
\begin{proposition}\label{prop:A*-binary-products}
  The categories $\ftact,\fseq\act$ have binary products
\end{proposition}
\begin{remark}
  It is worthwhile noting that $h$ does not generally preserve finite limits (or infimum $\sqcap$ in this case). Consider the words $a\tau b,ab\in\ftact$. Then, $h(a\tau b \sqcap ab) = h(a)=a$; however, $h(a\tau b) \sqcap h(ab) = ab \sqcap ab =ab$.
\end{remark}
\noindent
Furthermore, $S\odot X \cong S \times X$ (for any two sets $S,X$) and the (co)limit in any presheaf category is computed point-wise. Thus, using these observations, we calculate $\Sigma_h F$ (for $F\in\presheaves{\ftact}$) at $\varrho\in\fseq\act$ as follows:
\begin{equation}\label{eq:coend-simplification}
  \Sigma_h F (\varrho) \cong
  \int^{\sigma\in \ftact} F(\sigma)\times \yoneda {\fseq\act} (h (\sigma))(\varrho) \ \cong\
  \colimit{\sigma\in \op{\ftact},\varrho \preceq h(\sigma)} F(\sigma).
\end{equation}
Note that the reason for the last isomorphism is that every colimit can be encoded as a coend (see \cite[p.~224--225]{categories-working-mat} for a dual statement).
In addition, \eqref{eq:coend-simplification} is the colimit of $\op{\ftact} \rTo^{F} \set$ when $\varrho=\eseq$.
Notice that $\op{\ftact}$ is directed; thus, the colimit of the filtered diagram is $\coprod_{\eseq\preceq h(\sigma)} F\sigma/{\sim}$, where $\sim\ \subseteq \coprod_{\eseq\preceq h(\sigma)} F(\sigma)\times \coprod_{\eseq\preceq h(\sigma)} F(\sigma)$ is the equivalence relation defined as follows:
\begin{equation}\label{eq:filtered-colimit}
  (\sigma, p) \sim (\sigma',p') \iff \exists_{\sigma''}\ \left( \sigma''\preceq \sigma \land \sigma''\preceq \sigma' \land p\cdot \sigma'' =p'\cdot \sigma''\right).
\end{equation}
It turns out that from a system theoretic viewpoint, the set $\colimit{\eseq\preceq h(\sigma)}F_{X}(\sigma)$ is nothing but the set of minimal executions whose observable trace is an empty observation. Theorem~\ref{thm:adjoint-h} states this result in full generality.
\begin{definition}\label{def:mexecution}
  Given a transition system $\tuple{X,\tact,\rightarrow}$ and a word $\varrho\in\fseq\act$, we say an execution $p$ has an \emph{observable trace} $\varrho$ if $h(\max \dom p)=\varrho$. Furthermore, $\mexec{X,\varrho}$ is the set of all \emph{minimal executions} w.r.t.\thinspace the prefix order $\preceq$ on executions whose observable trace is $\varrho$. Formally, $p\preceq p' \iff p'|_{\dom p} = p$ and
  \[\mexec{X,\varrho} = \Big\{p \mid \ \history{p} \cap \{q \in\exec X \mid h(\max\dom{q})=\varrho\} = \{p\}\Big\}.\]
\end{definition}
Before we prove Theorem~\ref{thm:adjoint-h}, we need a category theoretic result (which is probably folklore; see \cite[Exercise~IV.2.7]{categories-working-mat} for a dual statement), namely, that the colimit of a diagram can be decomposed into the coproducts of the colimits of diagrams with smaller shapes under certain restrictions.
\begin{lemma}\label{lem:colimit-decompose}
  Given a set $J$ and a small category $\cat C=\coprod_{j\in J}\cat C_j$ with injections $\iota_j$, then for any functor $F$ from $\cat C$ to a cocomplete category $\cat D$, we have
  $\colimit{\cat C} F  \cong \coprod_{j\in J} \colimit{\cat C_j} F \circ \iota_j.$
\end{lemma}
\begin{theorem}\label{thm:adjoint-h}
  For a given transition system $\tuple{X,\tact,\rightarrow}$ and $\varrho\in\fseq\act$, we have
  $\Sigma_hF_X(\varrho) \cong \mexec{X,\varrho}.$
\end{theorem}
Next, we explore the action of the presheaf $\Sigma_hF_X$ from a system theoretic viewpoint. Firstly, it is defined by the universal property of the colimit. Let $\varrho'\preceq \varrho$ and let $\sigma$ be a minimal word such that $h(\sigma)=\varrho$. Then, there is a unique minimal word $\sigma_{\varrho'}=\bigsqcap \{\sigma'\preceq\sigma \mid h(\sigma')=\varrho'\}$. Note that this infimum exists since the history of a word in $\ftact$ is a finite totally ordered set of words. Therefore, the family of arrows depicted by the dotted arrows in \eqref{eq:sigma_h-action} forms a cone, where $\act_{\tau,\varrho}^\star=\{\sigma\in\ftact\mid \varrho \preceq h(\sigma)\}$ is a sub-forest of $\ftact$. Thus, the universal property of the colimit gives a map $\Sigma_hF_{X}\varrho \rTo \Sigma_hF_{Y}\varrho'$, which we denote by $\Sigma_h(\varrho,\varrho')$.

\vspace{-0.3cm}
\begin{minipage}[t]{0.4\textwidth}
  \begin{diagram}[width=1.6cm]
  F_X\sigma & \rAllMap & F_X\sigma_{\varrho'}\\
  \dTo & \eqnum\label{eq:sigma_h-action} & \dAllMap\\
  \colimit{\sigma\in\act_{\tau,\varrho}^\star} F_{X}\iota_{\varrho} \sigma & \rExistMap &\colimit{\sigma\in\act_{\tau,\varrho'}^\star} F_{X}\iota_{\varrho'} \sigma
 \end{diagram}
\end{minipage}
\hspace{0.8cm}
\begin{minipage}[t]{0.45\textwidth}
  \begin{diagram}[width=1.8cm]
    \Sigma_h F_{X} \varrho & \rTo~{\cong} & \mexec{X,\varrho}\\
    \dTo~{\Sigma_h(\varrho,\varrho')} & \eqnum\label{eq:naturality-mexec} & \dTo~{\mpast(\varrho,\varrho')}\\
    \Sigma_h F_{X} \varrho' & \rTo~{\cong} & \mexec{X,\varrho'}
  \end{diagram}
\end{minipage}
\vspace{0.1cm}

\noindent
Secondly, for any $\varrho' \preceq \varrho$ with $\varrho,\varrho'\in\fseq\act$, we define a map:
\[
\mexec{X,\varrho}\rTo^{\mpast(\varrho,\varrho')} \mexec{X,\varrho'} \quad \text{given by}\quad p \mapsto p|_{\history{\sigma_{\varrho'}}}\ ,
\]
where $\sigma=\max \dom p$.
Now we can establish that the isomorphism in Theorem~\ref{thm:adjoint-h} is natural in $\varrho\in\fseq\act$.
\begin{theorem}\label{thm:square8commutes}
  For any $\varrho,\varrho'\in\fseq\act$ with $\varrho'\preceq\varrho$, the square in \eqref{eq:naturality-mexec} commutes.
\end{theorem}
Now that we know what $\Sigma_h$ does to a presheaf, we use the category $\presheaves{\fseq\act}$ as our semantic universe to handle invisible actions. Thus, for a given $\tuple{X,\tact,\rightarrow}$, we let $\semantics{X}=\Sigma_h F_X$. In addition, for a branching simulation function $X\rTo^f Y$, we use the isomorphic view of minimal executions and let $\semantics{f}_{\varrho}(p)=p_f$ (for each $\varrho\in\fseq\act$), where $p_f$ is an execution defined inductively using the following rules.
\begin{itemize}
  \item If $\dom p=\eseq$ then $\dom {p_f} = \eseq$ and $p_f(\eseq)=f(p(\eseq))$.
  \item If $p \step a p'$, $p_f \step a q$, $\last q=f(\last {p'})$, and $a\in\act$ then $p'_f =q$.
  \item If $p \step \tau p'$ and $f(\last p) = f(\last {p'})$ then $p'_f=p_f$.
  \item If $p \step \tau p'$, $f(\last p) \neq f(\last {p'})$, $p_f \step \tau q,\last q =f(\last {p'})$ then $p'_f = q$.
\end{itemize}
Here, $p \step a p'\iff \dom {p'}= \dom p a \ \land \ p\prec p'$; the function $\last{p}$ returns the last visited state by the execution $p$, i.e., $\last p = p(\max \dom p)$.
\begin{lemma}\label{lem:mexec-preservation}
  For a given branching simulation function $X \rTo^f Y$ and a minimal execution $p\in\mexec{X,\varrho}$, we have $p_f\in\mexec{Y,\varrho}$.
\end{lemma}
Next let $\lts_\tau$ be the category of transition systems with possible invisible steps, which comprises of transition systems as objects and morphisms are the collection of identity functions and branching simulation functions. Thus, we can now give a presheaf semantics for the category $\lts_\tau$.
\begin{theorem}\label{thm:lts-tau_semantic1}
  The mapping $\lts_\tau \rTo^{\semantics{\_}} \presheaves{\fseq\act}$ is a faithful functor.
\end{theorem}

Finally, we can invoke the bisimulation maps in $\presheaves{\fseq\act}$ but the characterisation of branching bisimulation functions fails to hold. The mismatch is in the reflection \eqref{eq:trans-breflect} of invisible steps (not with the reflection of observable transitions), which we explain next in the following example.
\begin{example}
  Consider the two transition systems and the branching simulation $f$  depicted below with dashed lines. Note that $f$ is \emph{not} a branching bisimulation function (even though it is surjective) because it fails to reflect the transition $y_1 \step \tau y_3$.
  \begin{center}
  \begin{tikzpicture}
    \node (x1) {$x_1$};
    \node (x2) at ($(x1.center)+(0,-1.5)$) {$x_2$};
    \node (x3) at ($(x2.center)+(0.8,0)$) {$x_3$};
    \node (y1) at ($(x1.center)+(3,0)$) {$y_1$};
    \node (y3) at ($(y1.center)+(0.8,-1.5)$) {$y_3$};
    \node (y2) at ($(y1.center)+(-0.8,-1.5)$) {$y_2$};
    \path[->]
        (x1) edge node[left]{$a$} (x2)
        (y1) edge node[above right]{$\tau$} (y3)
        (y1) edge node[above left]{$a$} (y2);
    \path[->,dashed]
        (x1) edge (y1)
        (x2) edge [bend right] (y2)
        (x3) edge [bend left] (y3);
  \end{tikzpicture}
  \end{center}
  Yet $\semantics{f}$ is a bisimulation map in $\presheaves{\fseq\act}$. For this consider a given commutative square \eqref{eq:bisim-presheaf} in $\presheaves{\fseq\act}$. The crucial case in defining the functions
	$k_\varrho$ is when $\varrho\in\{\eseq,a\}$.
  \begin{itemize}
    \item Let $q\in Q(\eseq)$. If $n_\eseq (q)=\eseq_{y_i}$ then we let $k_\eseq(q)=\eseq_{x_i}$ (for $i\in\{1,2,3\}$).
    \item Let $q\in Q(a)$. Then, $n_a(q)$ is the minimal execution witnessing $y_1 \step a y_2$ since there are no other minimal executions in $\mexec{Y,a}$. Thus, we let $k_a(q)$ to be the minimal execution witnessing the transition $x_1 \step a x_2$.
  \end{itemize}
\end{example}

\subsection{Characterisation of branching bisimulation}
Unfortunately, all the information related to silent transitions is lost by transforming a presheaf using the functor $\Sigma_h$ because all the executions with empty observations get fused into their corresponding empty (minimal) executions. In retrospect, the problem lies in our specification of observation since our hypothetical `observer' $\obs$ is unable to differentiate between \emph{empty observation due to no system move} and \emph{empty observation due to zero or more system moves}. Therefore, to model this latter observation, we introduce a constant $\bar\tau$ which can be observed anytime after $0$. So for this section, we define $\obs$ in the next page as follows:

\begin{equation*}\begin{aligned}
   \obs(0) = \A_\tau(0) \quad \text{and} \quad \obs(n)=\A_\tau(n) \cup  \{\bar\tau\}\quad (\text{for every $n>0$});\\
   \sigma\cdot m=
\begin{cases}
  \sigma|_{m}, & \text{if}\ \sigma\neq\bar\tau\\
  \eseq, & \text{if}\ m=0 \land \sigma=\bar\tau  \\
  \bar\tau, & \text{if}\ m>0 \land \sigma=\bar\tau.
\end{cases}
\qquad \Big(\text{for any $m\leq n$ and $\sigma\in\obs(n)$}\Big).
\end{aligned}\end{equation*}

\noindent
Put differently, $\bar\tau$ is `really' a constant observation over time except at $0$. The introduction of $\bar\tau$ is new and inspired from the constant $\eta$ \cite{eta-abstraction} which can be viewed as an \emph{empty observation due to at least one system move}.

\begin{proposition}\label{prop:Oispresheaf}
  The above mapping $\op{\nat} \rTo^{\obs} \set$ is a presheaf.
\end{proposition}
\noindent
Moreover, the categories $\presheaves{\nat}/\obs$ and $\presheaves{\elements{\obs}}$ are equivalent (Theorem~\ref{thm:elements-equivalence}). So, consider the category of elements $\elements{\obs}$ whose objects are tuples $\tuple{n,\sigma}$ (for $n\in\nat,\sigma\in\obs(n)$) and whose arrows are given by the rule:
\[\tuple{n,\sigma} \step{} \tuple{n',\sigma'} \iff n\leq n' \land \sigma'\cdot n=\sigma.\]
Just as the category $\elements{\A_\tau}$ has a simpler description in terms of $\ftact$, we simplify $\elements{\obs}$ by defining a set $\ftbtact = \ftact \cup \{(n,\bar\tau) \mid n>0\} $ ordered by the smallest relation $\preceq\subseteq \ftbtact \times \ftbtact$ satisfying the following rules:
\[
\frac{(|\sigma|,\sigma) \step{} (|\sigma'|,\sigma') \in\elements{\A_\tau}}{\sigma \preceq \sigma'} \quad
\frac{(m,\bar\tau) \step{} (n,\bar\tau)\in\elements{\obs}}{(m,\bar\tau) \preceq (n,\bar\tau)}\quad
\frac{n>0}{\eseq \preceq (n,\bar\tau)}.
\]
Note that the leftmost rule is concerned with the elements of $\ftact$. Thus, the slice category $\presheaves{\nat}/\obs$ is equivalent to the category $\presheaves{\ftbtact}$ since $\elements{\obs}\cong\ftbtact$. Using the presheaf $F_X$ (cf. \eqref{eq:presheaf-tact}) induced by a given transition system $\tuple{X,\tact,\rightarrow}$, define a presheaf $\bar F_X\in\presheaves{\ftbtact}$ as follows:
\[\bar F_X\tuple{\sigma}=
\begin{cases}
  F_X(\sigma), & \text{if}\ \sigma\in\ftact\\
  \coprod_{n\in\nat} F_X(\tau^n), & \text{otherwise}
\end{cases},
\ \text{where}\
\begin{array}{l}
  \tau^0=\{\eseq\} \\
  \tau^{n+1}=\tau^n \cup \prod_{n+1}\{\tau\}
\end{array}.
\]
The executions based on invisible steps are seen \emph{stretchable in time} by $\obs$, i.e., a system may perform invisible executions of length independent of time instants. This is encoded in the second clause of $\bar F_X$. To complete the definition of $\bar F_X$ as a presheaf, we define, for any $p\in\bar F_X(\sigma)$ and $\sigma'\preceq \sigma$ in $\ftbtact$, the action of $\bar F_X$ as:
\[
p\cdot \sigma' =
\begin{cases}
  p|_{\history{\sigma'}}, & \text{if}\ \sigma\in\ftact \land \sigma'\in\ftact\\
  p|_{\{\eseq\}}, & \text{if}\ \sigma\not\in\ftact \land \sigma'=\eseq\\
  p, & \text{if}\ \sigma\not\in\ftact \land \sigma'\not\in\ftact.
\end{cases}
\]
Notice how the above formulation closely follows the three defining clauses of $\obs$.
\begin{proposition}\label{prop:barfxcfunctor}
  The mapping $\bar F_X$ defined above is a contravariant functor.
\end{proposition}
To characterise branching bisimulation,
define a structure $\ftbact$ similar to $\fseq\act$ and a structure preserving map $\ftbtact \rTo \ftbact$ similar to $h$ (Section~\ref{subsec:failed}). To this end, our semantic category for branching bisimulation will be $\ftbact=\fseq\act\cup\{\bar\tau\}$ ordered by the relation $\preceq\subseteq\ftbact\times\ftbact$ consisting of prefix relation and the pairs $\tuple{\eseq,\bar\tau},\tuple{\bar\tau,\bar\tau}$. We consider $\ftbtact \rTo^{\bar h} \ftbact$ given by: $\bar h(\sigma)=h(\sigma)$, if $\sigma\in\ftact$; and $\bar h(n,\bar\tau)=\bar\tau$ (for all $n\in\nat$).

Next, we aim to compute the presheaf $\Sigma_{\bar h}\bar F_X$ at $\varrho\in\ftbact$. At this stage, we can take the advantage of similarity between the structures $\ftact,\ftbtact$ and $\fseq\act,\ftbact$ to compute the colimits, similar to  \eqref{eq:coend-simplification} where $\op{\ftact}$ is replaced by $\op{\ftbtact}$ as the indexing category. In lieu of Theorems~\ref{thm:adjoint-h} and \ref{thm:square8commutes}, we obtain
\begin{theorem}\label{thm:adjoint-barh}
  For a given transition system $\tuple{X,\tact,\rightarrow}$, we find that $\Sigma_{\bar h}\bar F_X(\bar\tau) \cong \coprod_{n\in\nat} \exec{X,\tau^n}$ and $\Sigma_{\bar h}\bar F_X(\varrho) \cong \mexec{X,\varrho}$ (for any $\varrho\neq\bar\tau$).
  Moreover, the above isomorphisms are natural in $\varrho\in\ftbact$. I.e., for any $\varrho'\preceq\varrho$ with $\varrho,\varrho'\in\fseq\act$, the square in \eqref{eq:naturality-mexec} and the following square commute.
  \vspace{-0.3cm}
  \begin{diagram}[size=3em,width=2cm,midshaft]
			\Sigma_{\bar h} F_{X} (\bar\tau) & \rTo~{\cong} & \coprod_{n\in\nat} \exec{X,\tau^n}\\
			\dTo_{\Sigma_{\bar h}(\bar\tau,\eseq)} & & \dTo_{\_|_{\eseq}}\\
			\Sigma_h F_{X} (\eseq) & \rTo~{\cong} & \mexec{X,\eseq}
		\end{diagram}
\end{theorem}

Note that the definition of the maps $\Sigma_{\bar h}(\varrho',\varrho)$ (for $\varrho'\preceq \varrho$) is similar to the maps $\Sigma_{h}(\varrho',\varrho)$ defined by the universal property of colimits (see \eqref{eq:sigma_h-action}). Just like in the previous subsection, we utilise the isomorphic view of minimal executions to define our semantic map $\lts_\tau \rTo^{\semantics{\_}} \presheaves{\ftbact}$:
\begin{itemize}
  \item For a given transition system, we let $\semantics{X} = \Sigma_{\bar h}\bar F_X$.
  \item For a given branching simulation function $X \rTo^f Y$, we let $\semantics{f}_\varrho (p)=p_f$.
\end{itemize}
\begin{lemma}\label{lem:semanticsfunctor}
  The above mapping $\semantics{\_}$ is a faithful functor.
\end{lemma}
Finally, we have obtained the desired result of this section.
\begin{theorem}\label{thm:bbisim-char}
  A branching simulation function $f$ is a branching bisimulation function iff $\semantics f$ is a bisimulation map in $\presheaves{\ftbact}$.
\end{theorem}

\section{Related work and Conclusion}

The core idea of \textbf{Goguen's sheaf semantics} \cite{Goguen92sheafsemantics} is: systems are diagrams of sheaves, behaviour (interconnection) of systems is their limit (colimit). In retrospect, Goguen gave a sheaf semantics of nondeterministic automata by constructing presheaves on the Alexandroff topology induced by the downward closed subsets of $\fseq\act$ (in contrast to presheaves over $\fseq\act$). However, it is well known that the category of sheaves on Alexandroff spaces induced by a poset $(X,\preceq)$ is equivalent to the category of presheaves on $X$ (a consequence of the so-called comparison lemma in topos theory; see \cite[Corollary~3 on Page~590]{sheafbook}). In short, we use simpler structures to represent executions and focused on defining bisimulations abstractly which were absent in \cite{Goguen92sheafsemantics}.

\textbf{Relational presheaves} \cite{rel-presheaves:2015} generalise transition systems that are labelled by words from the free monoid $\fseq\act$. The idea was to accommodate the earlier presheaf approaches \cite{presheaves-as-transitionsys:1997,jnw96:bisimopenmaps} with algebraic structure on labels. The most insightful observation of \cite{rel-presheaves:2015} was the well-known `saturation' construction on transition systems can be captured using a 2-adjunction induced by a homomorphism between
$\ftact,\fseq\act$.
Unlike \cite{rel-presheaves:2015}, our left adjoint $\Sigma_h$ records the minimal executions induced by a transition system with silent steps.
We expect this to be relevant for probabilistic systems, where minimal executions are used to define a probability measure (cf. \cite{bk2017}).

\textbf{Open maps} between presheaves as defined in \cite{jnw96:bisimopenmaps} are instances of the open maps in a topos as introduced in \cite{openmaps-completness:1994}. This is because open maps between presheaves (as in \cite{jnw96:bisimopenmaps}) are natural transformations whose naturality square is a weak pullback in $\set$ (cf. \cite[Example~1.1]{openmaps-completness:1994}). We discarded the open maps between presheaves because they are incapable of establishing complete refinement between an implementation and its specficiation; though it is still interesting to assert whether the bisimulation maps (Def.~\ref{def:bisim-presheaf}) satisfy the axioms given in \cite{openmaps-completness:1994}.

\textbf{Prefix orders} are generalisations of trees proposed in \cite{Cuijpers:2013:DCM} to study executions of dynamical systems in an order theoretic manner. In \cite{openmaps-concrete:2015}, the authors defined functional bisimulation between prefix orders by reinterpreting the definition of open maps in concrete categories. Our bisimulation maps are an instance of this general definition (Section~\ref{sec:bisimmaps}). It is unclear, though, how to enrich prefix orders with observations so that we can model labelled executions in a uniformly. This  question lead us to model observations as presheaves.

To sum up, bisimulation maps between presheaves are versatile enough to capture
different notions of behavioural equivalence.
We
demonstrated
this by characterising $\forall$-fair bisimulation and branching bisimulation, two notions that are notoriously difficult to capture with a coalgebraic approach. The clear distinction between time and observation proved fruitful in dealing with silent actions, but we also expect our framework to lend itself well to modelling
hybrid systems. For instance, sheaves over the translation-invariant interval domain $\mathbb {IR}_{/\vartriangleright}$ were introduced in \cite{spivak_hybrid-systems} to model hybrid systems. It will be interesting to explore whether bisimulation maps between such sheaves (Remark~\ref{remark:sheaves-bisim}) coincides with stateless bisimulation \cite{hcif,Cuijpers-hypa,vanBeek:2006-chi}.

\begin{ack}
  We thank the anonymous reviewers of FOSSACS'19 for their feedback on an earlier draft that greatly improved this manuscript. In particular, Reviewer 1 not only identified a mistake in our earlier characterisation of branching bisimulation, but also provided a detailed feedback including the proofs in the appendix for which we are grateful.
  We thank Barbara K\"onig and Christina Mika for valuable comments on earlier drafts of this paper. We also thank Pieter Cuijpers for asking us about the importance of presheaf categories in system modelling, which formed the basis of Section~\ref{sec:prelim}. Moreover, we are grateful to Alex Simpson for discussing early ideas and in particular, for guiding our attention from sheaves on Alexandroff spaces induced by the poset $\fseq\act$ to just presheaves on $\fseq\act$. Finally, we acknowledge
Paul Taylor's diagram package for commutative diagrams.
\end{ack}
\bibliographystyle{entcs}

redpajama_set_name: RedPajamaArXiv


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13 July 2015 (London and Hereford) – Boultbee Brooks Real Estate (BBRE), the UK-based residential and commercial property investor and developer, has further strengthened its presence in Manchester City Centre's growing commercial property through the acquisition of a three-property portfolio from M20 Property Group for a total consideration of £16M.
BBRE will undertake a comprehensive refurbishment of the two properties on Mosley Street and transform them into new work space, for which there is high demand. The area will soon benefit from extensive regeneration and will become one of Manchester's key locations, attracting an extra 4,500 professionals upon completion of No.1 and No.2 St Peter's Square.
The Chapel, located in Castlefield's Urban Heritage Park and less than 100 metres away from Deansgate Railway Station and Metrolink, is a Grade II listed building providing 12,132 sq ft of space over lower ground, ground, first and second floors. The property was originally opened as a Congregational chapel in 1858 and was designed by the local architect, Edward Walters. In more recent years it has undergone significant refurbishment and now provides contemporary, high specification workspace in one of Manchester's most desired locations.
This acquisition is the latest in a string of recent purchases in Manchester by BBRE and takes the firm's total investment in Manchester this year to £28M, a sign of the firm's growing confidence in the strength of the Manchester market and further evidence of its strategy to invest in high-quality properties in prime locations, which offer attractive financial returns. BBRE, which is a major landlord in the Shoreditch and Clerkenwell areas of London where many of its tenants are fast growing digital and creative firms, also recently purchased 30 Brown Street and Croxley House in Manchester, for £5.5M and £6.5M respectively.
BBRE was advised by Rob Cannock at Aurum Real Estate. Legal advice was provided by Darren Hamer of DWF who advised M20 and BBRE were represented by Michelmores.
9 July 2015 (London and Hereford) – Boultbee Brooks Real Estate (BBRE), the UK-based property investor and developer, is pleased to announce the acquisition of Venture House, a prominent office building located opposite Reading mainline railway station.
Venture House, 37- 43 Blagrave Street, Reading,RG1 1PZ, was constructed in the early 1980s behind an attractive retained façade and now provides office accommodation over six floors. The building totals 20,640 sq ft NIA. The property is partly vacant and will be refurbished to provide modern Grade A office suites of between 2,700 sq ft to 3,600 sq ft.
The commercial property, situated on the corner of Blagrave Street and Forbury Road, is exceptionally well positioned. Opposite Reading Railway Station, Apex Plaza and R+, the property is located within the main office core that is set to improve further as the prime office area around the railway station is strengthened by the completion of a number office developments and the arrival of CrossRail.
The purchase of Venture House is BBRE's second acquisition in Reading since 2014 and demonstrates its confidence in the strength of this market for property investment.
Reading is considered to be the capital and the commercial centre of the Thames Valley region, one of the economic powerhouses of the British economy. With a highly skilled labour pool to draw on, Reading has attracted a number of world leading companies in the fields of finance, IT and global communications, as well as a high concentration of FTSE100. The area boasts the UK's highest employment rate of 78%.
Aurum Real Estate and Lambert Smith Hampton acted on behalf of Boultbee Brooks Real Estate.
3 July 2015 (London and Hereford) – Boultbee Brooks Real Estate (BBRE), the UK based real estate investor/developer who recently purchased 24 – 30 Brown Street, Manchester for £6.5 million, is pleased to announce that OBI Property has been appointed to lead the comprehensive refurbishment of the building 24 Brown Street.
BBRE intends to completely remodel the building including new open services, a new façade and an additional floor to increase the net internal area to 48,000 sq. ft. A planning application will be submitted by the end of the summer. Once complete the building will comprise retail/leisure units on the ground floor, with offices on the floors above.
BBRE is a privately owned property company with a development and investment portfolio valued at approximately £350 million. The company, which is a major landlord in the Shoreditch and Clerkenwell areas of London where many of their its tenants are fast growing digital and creative firms, also recently purchased Croxley House in Manchester.
OBI Property has assembled the project team which includes PRP Architects, Tier Consult Structural Engineers and REDS as M&E consultants.
OBI Property was set up in 2010 and is a commercial real estate firm which provides innovativeproviding innovative solutions to its landlord and occupier clients.


MANILA, Philippines — Patients confined in hospitals hit by the ongoing water shortage would only be allowed one visitor or watcher to minimize water use, the Department of Health (DOH) said on Thursday.
DOH Secretary Francisco Duque appealed to patients to bear with hospital management water use is being prioritized for critical care.
"Given the situation sana po maintindihan ng mga magulang na hindi po taghirap sa tubig ay amin pong papahintulutan ang mas marami sa isa, pero ngayon po mapipilitan po tayo na isa lang po talaga dapat," Duque explained in a press briefing at the National Kidney Transplant Institute (NKTI).
"That's to be strictly enforced. Hindi naman pwedeng hayaan natin, kasi kung hayaan natin na lahat tatlo, apat (ang bisita) eh mauubos ang tubig natin. Are we part of the solution or are we part of the problem?" he asked.
According to Duque, they have identified five hospitals being served by Manila Water that are affected by the water supply crisis. These are the NKTI, the Rizal Medical Center in Pasig, the National Center for Mental Health in Mandaluyong, the Philippine Children's Medical Center and the Quirino Memorial Medical Center in Quezon City.
Representatives from the East Avenue Medical Center, Philippine Heart Center, and the Philippine Lung Center were also present at the press briefing to discuss their contingency measures.
Duque said hospitals can still function on just 50 percent of their daily consumption.
"We have come to an agreement that amid the water shortage that we are facing, 50 percent of their average daily requirement should be enough for them to sustain their current level of operations," he said.
"Manila Water has been tasked to make sure this temporary solution be implemented in the soonest possible time, that means today, which means that via the tankering system, they're going to provide the 50 percent water requirement of these five hospitals," he added.
Water is important for hospitals such as the NKTI, which needs around 1,500 cubic meters of water daily for its hemodialysis units and hospital operations.
According to NKTI Executive Director Rosemarie Liquete, each hemodialysis unit needs 100 to 150 liters of clean water in a single dialysis cycle which lasts for four hours. Currently, there are 70 hemodialysis units operating in four shifts.
Earlier, the NKTI management said they have started shutting water supply to hospital rooms, to ensure that hemodialysis units will have water to use. However, should Manila Water fails to bring tankers to refill its cisterns, there will be no afternoon dialysis sessions, Manila Water said.
Liquete added that the NKTI is also shifting dialysis methods, from hemodialysis to peritoneal dialysis which needs just around six liters of water per session.


Motherhood. It's not so glamorous. Especially that whole, "becoming a mother" act of birthing a child. Sure, making the baby was fun but who thinks about what motherhood is going to be during that, um, private time? I didn't! Did you? I thought about all the cute baby clothes, the baby shoes, the baby bottles and the baby bibs and the softest baby blankets.
And then there's a real baby. In your arms. And your body is blissed out and you just love that bundle because you just do.
And the DIAPERS. All the diapers. There were so many diapers that smelled like one million miles of cow manure emanating from the bum of your sweet little one.
And how pretty did you feel? Did you even wear deodorant the first few weeks? I recall a shower here and there. I remember many long nights, awake. And I'm not even a single mom. Where was the father of my children you ask? Snoring, peacefully through all the night time feedings, changings, and scary-first-time-fevers and stuffy noses.
And I still got the kids to day care, preschool and now school. And I still show up to work. Note. I show up. Sometimes I'm on time. Mostly I'm not. And my appearance now is much how I looked like during the first few months of brand new motherhood. I catch a glimpse of my reflection in the windows to my office building and pray security doesn't stop me for trespassing. I see how my coworkers look at me: first my hair; then my outfit; and always my shoes. I rock the pumps, no matter what. But I still look like hell when I get to work. But my kids? Cute as a GAP ad.
I know. I know. This only happens to me. Your kids do no wrong (at someone else's house). But at my house, I can let kids be kids. Because my house is designed to be lived in. By kids. By us. If it's breakable, it will be broken. If it's soft and bouncy, it will be bounced on and if it's vertical a ball will be thrown at it and probably break a framed 2nd hand piece of art or a much adored school picture.
And we are not always gentle people. We move quickly. Determined. Ok, we're late for stuff. Whatever. Often, my front door is found open, my kids and the neighbor kids streaming through the front door to the back door. At the back door, this also includes frantic screams for help because our three big dogs (really, it's two big dogs and one little dog with a size issue) gate rush the back door, plow over whatever size human is unfortunate enough to open that back door gate to "hello dogs!" and I'm the one running around the house, yard or neighborhood screaming out ridiculous dog names, which brings out the neighbors, looking, watching….judging. And then, there is a pack of tiny neighborhood humans, searching for a wild pack of domesticated dogs enjoying their freedom.
Yes, Motherhood is chaos. It's crazy and full of unexpected sights and smells. An adventure that no one can tell you about because you have to live it to believe it. And that's a pretty glamorous life to live.


text:
Library and Research
Home Feature news Apple's Music and TV apps for Windows are now available in preview
Apple's Music and TV apps for Windows are now available in preview
Plus an app for managing your iPods and iPads, since you can't use iTunes with the apps installed.
Apple's Music and TV apps for Windows have made an appearance, with preview versions of the apps being spotted on the Microsoft Store by The Verifier (via MacRumors). Last October, Microsoft announced they were coming to the platform, letting users finally ditch the old iTunes app and Apple TV web player for native software that is closer to the experience embedded in macOS.
There's also a third preview of an app called Apple Devices, which is meant to let you manage and sync things like iPods and iPads — functionality that's currently handled by iTunes on Windows and Finder on the Mac. It also, apparently, contains some references to Reality OS and xrOS, two codenames that have reportedly been associated with Apple's virtual reality headset.
The TV app is similar to the version on macOS, though you access different sections via a list to the side rather than tabs on the top. Using it, I was able to watch Apple TV Plus shows, as well as movies I'd purchased on iTunes; it even had a miniplayer mode. The Music app is a similar to the macOS version, though the iTunes store is ever so slightly harder to access.
Two Verge staffers running Windows 11 were able to install and run the apps using The Verifier's links. You can download them yourself here: Apple Music, Apple TV, Apple Devices.
When you open the apps for the first time, they warn that iTunes will stop working if you use them and that you'll have to uninstall the previews to bring it back. It does seem like the Music app brought over my music and settings from iTunes, but if you absolutely have to sync an iPod, it may be worth holding off on trying these apps out until the Apple Devices app is out of preview. MacRumors also points out that you may want to hold off if you use iTunes for Windows to manage or listen to podcasts and audiobooks.
So far, the previews seem to be mostly stable with a lot of the basic functionality I'd expect, though the apps do say that "not all features may work as expected."
This article was originally published by Theverge.com. Read the original article here.
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It is Berger's theory that the United States Supreme Court has embarked on "a continuing revision of the Constitution, under the guise of interpretation," thereby subverting America's democratic institutions and wreaking havoc upon Americans' social and political lives.
Raoul Berger (1901–2000) was Charles Warren Senior Fellow in American Legal History, Harvard University.
Foreword, preface to the second edition, abbreviations, introduction, appendixes, bibliography, bibliography of Raoul Berger's writings, index.


We'd love your help. Let us know what's wrong with this preview of Harambee by Karen Jones.
Profits of 50% of the sale of this children's book go to the Machao Orphanage.
"MACHAO" is an acronym for Makueni Christian Home for Assisting Orphans. MACHAO is a faith-based organization formed by seven Christians of the Africa Inland Church (AIC) within Wote Division in Makueni District. MACHAO aims at covering all the seventeen divisions of the Makueni District, with a Profits of 50% of the sale of this children's book go to the Machao Orphanage.
"MACHAO" is an acronym for Makueni Christian Home for Assisting Orphans. MACHAO is a faith-based organization formed by seven Christians of the Africa Inland Church (AIC) within Wote Division in Makueni District. MACHAO aims at covering all the seventeen divisions of the Makueni District, with a population of about 900,000 people. It is estimated that over 5% of this population die and leave many children orphaned.
From Author Karen Jones: I met Carolyn while my family was relocating to Los Angeles, California. She and her family embraced me and mine as if we'd known each other our whole lives. I learned of Carolyn's involvement with the Machao Orphanage and was deeply moved. My hope is that this book will change lives - mine, perhaps; Carolyn's, certainly; but most of all the children in need of food, clothing, care and education.
Harambee is the Kenyan tradition of community self-help events, often used to describe fundraising or developmental projects and has become the official motto of Kenya appearing on its coat of arms. Literally, it means "all pull together" in Swahili. I think it is the perfect title for a book meant to encourage those who've been blessed to "pull together" to help those less fortunate.
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Karen Jones grew up in rural Indiana on a small family farm. Her father worked construction all his life and her mother was a writer. The last born in a family of seven, Karen wore too many hand-me-downs and heard "grown-up" conversations long before she should have. But that large, semi-dysfunctional family taught her a lot about life, love, and sticking together. Each special moment, each family Karen Jones grew up in rural Indiana on a small family farm. Her father worked construction all his life and her mother was a writer. The last born in a family of seven, Karen wore too many hand-me-downs and heard "grown-up" conversations long before she should have. But that large, semi-dysfunctional family taught her a lot about life, love, and sticking together. Each special moment, each family crisis, and each unique family personality from her childhood molded her into the crazy, quirky, and creative person she is. And she wouldn't trade a single triumph or trauma.
She married her complete opposite. Her husband, Rob, is rational, logical and analytical. They make a great pair. She shows him the creative side of life and he keeps her from floating away into oblivion on the winds of inspiration. They live in Indiana but long to move to the beach. They're daughter, Allyson, is a complex mixture of them both. Using both sides of her brain equally, she is both surprising and sensible.


The funding is part of a new £2 million Business Basics Fund launched by the Department for Business, Energy and Industrial Strategy (BEIS) under the Government's Industrial Strategy to boost productivity and back business. Newcastle Business School is among just 15 successful bids across the UK to win funding on the projects designed to help small businesses make better use of technology and modern management practices. It will now develop a Digitally Enabled Business Clinic (DEBC) model – based on the success of its own Business Clinic where final year students work on a range projects with real clients.
The concept of the DEBC is that it can be scaled up and offered to SME's as a free consultancy service by other universities across the UK.
Newcastle Business School established its Business Clinic in 2013. Since then the total value of the students' free consultancy advice and reports has been estimated by more than 200 clients to benefit, to exceed £1.2m.
Nigel Coates, Director of the Business Clinic, said: "Winning this funding is a significant achievement that reflects the positive impact our business clinic has - both for our clients and in the learning experience it offers to our students. Thanks to quality and commitment of colleagues and our students we were able to make a compelling case to the BEIS.


Epson WorkForce Pro WF-C869R Drivers Setup Download For Windows and Mac - The new WorkForce Pro WF-C869R signs up with Epson's line of inkjet printers and also MFPs aimed at small to medium sized businesses. It includes printing, copying, scanning and faxing as common functions. The WF-C869R is similar in regards to rate to the older WorkForce Pro WF-6590 that was released in 2015.
The WorkForce Pro WF C869R has an ISO speed of as much as 24ppm in both black and shade. The maximum draft rate depends on 35ppm as well as the duplex speed is 16ipm. This is thanks to Epson's inkjet modern technology, PrecisionCore. PrecsionCore utilizes a high-density print chip that could produce approximately 40 million precise dots each secondly. This modern technology was initially located in Epson's large-format printers and also is currently used in all of its printers and MFPs.
Along with Epson's exclusive printing language, the Epson WorkForce Pro WF-C869R sustains both PCL and PostScript. USB, Ethernet, as well as Wi-Fi connectivity, is common. Mobile printing is additionally supported by means of the Epson Link solution. Within the Epson Link are the adhering to functions: Email Print, iPrint Mobile Application, and also Remote Print. Customers can also print using the Apple AirPrint, Kindle Fire as well as Google Cloud Print.
WorkForce Pro WF-C869R is also a shade touchscreen so users could access typical printing, copying, scanning and faxing process. The tool likewise integrates with various other web-based applications, such as PaperCut MF, through the Epson Open System.
One innovative feature of this tool is in the return of the replacement print cartridges. Epson uses both high-capacity and extra-high-capacity substitute cartridges for each of the 4 inks (black, cyan, yellow as well as magenta). The returns for the high-capacity black cartridge is 22, 500 pages and also 22,000 web pages for every shade. Nevertheless, the extra-high-capacity web page yields are even higher: 86,000 pages per black as well as 84,000 pages for every shade.
The WF-C869R includes a large range of protection functions, such as PIN number accreditation in order to release jobs, individual control accessibility, printer and network setups using Internet Configuration, SSL/TLS and also IPsec.
The WF-C869R is sold through business Modern Technology Organization (BTA) reseller channel, so regrettably, we don't have a market price. Firms can contact their local reseller for a price quote.
Rapid outcome-- prints at speeds of 24 ISO ppm (black) and also 24 ISO ppm (shade).
Functional media handling-- publishes as much as 13" x 19"; up to 1,830-sheet overall paper capacity; functions auto duplexing (print, copy, scan, fax) as well as auto media size discovery.
Rapid scan speed-- as much as 25 web pages per minute.
Open Platform ready-- smooth integration with web-based business applications such as PaperCut MF and also other workflows.
Duty cycle-- up to 75,000-page maximum regular monthly responsibility cycle.
Security features-- PIN number certification for work release; user control access; printer and network setups via Web Config with printer's IP address; SSL/TLS safety and security; IPsec.
Allowed for remote printer data collection-- compatible with MPS software program remedies from PrintFleet ®, ECi (FMAudit ®) and others.


text:
By WENDY WATTA
In a business community that has been said to be very close-knit, Nyambura has since come a long way from selling beaded jewellery in Westlands to running a popular store and being considered part of the community in Shela.
A Store That Sells From Trendy Textile to Fertility Dolls
Thursday, August 9, 2018 20:39
Nyambura Wahu's African Corner store in Lamu. PHOTO | COURTESY
Browsing through Nyambura Wahu's African Corner store in Lamu, shelves are filled with vibrant jewellery, cutlery, masks, bags, sandals, fabric and more, and there is good traffic from locals either stopping by to say a cheerful hello or tourists looking to bag some souvenirs.
"I ended up in Lamu in 2012. I was born and bred in Nairobi and decided to come look for better opportunities to sell my jewellery. A friend suggested I come check out Lamu and I was brave enough to save and come down by bus. I really wasn't looking to settle here at the time," she says.
"I started out in Lamu Old Town which wasn't very touristic at the time. Shella village had few shops offering what I had. I started with a small space, business picked up fast and I was able to expand in only six months," she explains.
Her store is so called because it stocks goods from all over the continent, particularly East and West Africa.
Luxury Scarves Inspired by Nature
Kiondo-Inspired Ecobags
Facials for the modern gentleman
How women want their men to dress
"I do all the jewellery, key rings and other products are all bought from other artists. The art is from Cameroonians and small soapstone carvings from Kisii. The most expensive would have to be the Bakuba cloth from Congo and the mud cloth from Ghana," she says.
In Mali for instance, Bogolanfini is a traditionally handmade cotton fabric dyed with fermented mud. The cloth is woven in narrow strips then sewn together by hand to make a larger fabric, a process that can take up to one month.
The cloth is treated in ink from leaves and branches of an indigenous tree then dried and painted with mud fermented over months. This is left to dry and cure, rinsed and repeated severally to deepen the colour, and each is often designed with symbols telling a story of the community.
"Our mud cloth is from Ghana and goes for Sh9,000 for two metres and Sh15,000 for 3.5 metres. The other item that costs Sh15,000 is a beaded lizard, also from Ghana,'' she says.
Other than fashion, African Corner also stocks soaps made from moringa, coconut and baobab. She also has Moringa energy bars which are nutritious and great for malnourished children.
"We also have other unique items like a fertility doll from Turkana- the community believed that if you couldn't bear children and you got one of these, it would happen for you. There's also a bigger fertility doll from Cameroon," she says, pointing to what looks like a voodoo doll.
"Our prices however vary, from really affordable earrings under Ksh 1,000 to black and white baskets from Rwanda and more."


The Agave toumeyana plant is very similar in appearance to Agave 'Victoria-Reginae' agave plant, except that the leaves are spread wider apart, and the white hairs and markings are more numerous. The Agave toumeyana plant is a very spellbinding plant and will catch the attention of all your friends and neighbors. The Agave toumeyana plant forms offsets freely and will form a rounded mound about 3 ft. in diameter.


View: Straw yellow with greenish reflections.
Smell: Intense fruity pineapple and apple with citrus notes.
Taste: Fresh with well-integrated acidity, persuasive mineral, persistent aftertaste with good olfactory correspondence.
Smell: Fruity of ripe red fruit like cherry and plum.
Taste: Round, pleasant, balanced with a warm and sweet aftertaste.
View: Intense, intense ruby red.
Smell: Fruity of ripe red fruit like morello cherry and plum.
Taste: Round, pleasant, balanced, tasty with a fruity aftertaste.
View: Bright intense ruby red.
View: Intense straw yellow with greenish reflections.
Smell: Delicate with floral scents and white pulp fruit.
Taste: fresh and sapid with a lingering aftertaste.
Smell: Fruity of ripe red fruit.
View: Pale straw yellow and a fine and persistent perlage that highlights at the time of mixing a very pleasant and evanescent foam.
Taste: The nose is distinguished by its strong hints of apple, followed by sharp memories of citrus and white floral notes.
Taste: Pleasant on the palate and fresh with hints of cut grass and artichoke, characterized by a bitter and spicy aftertaste.