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At Castle House Dental Practice we have the following team members.
Your Dentists
Dr. Iain Morton BDS – RETIRED
Iain retired in May 2017 and we wish him a happy and active retirement. We are delighted that he had chosen us to continue his patient care. Iain Joined the practice in September 2016 having previously worked in general practice for a number of years at Southside Dental Practice. "raised on Bute" he qualified in 1979. Iain is interested in restoring teeth to their nature beauty using conservative tooth coloured materials. Iain is a keen sportsman in particular competitive sailing. GDC NUMBER : 53190
DR. Chris Dall Bds
Chris graduated from Bristol University in 2008 and spent several years working in general practice in Bristol and Manchester. Making his move to Scotland with his wife, son and two dogs in March 2019. Chris has a keen interest in the outdoors. GDC NUMBER : 154153
Dr. Rosemary Hunter BDS
Rosy graduated from Liverpool Dental School in 2009 and after spending time working in North Wales moved to Scotland. She worked in Elgin, during which time she successfully completed her membership exams for The Royal College of Surgeons of Edinburgh and gained valuable experience in treating nervous patients and especially young children under sedation. She joined our team in November 2018. She lives with her husband and two young children and in her spare time, Rosy loves to sew, craft, paint and sing. GDC 176567
Dr. Annelise McDougall BDS
Having grown up in Forres, Annelise graduated from Glasgow Dental School in 1997 and worked in general practice in Inverness for 3 years before joining a Dental Practice in Dingwall & Invergordon. She joined Castle House in May 2017. She has three young children, keeps bees and hens and enjoys running, cycling and a good book. GDC NUMBER: 73061
Dr. Margaret Behrendt BDS
Dr. Behrendt qualified in 1985 and has worked in a variety of General Dental Practice environments throughout the North of Scotland. She undertook a period of voluntary work through VSO working as a Dentist in the Maldives. She has been with Castle House Dental Practice since 1996. Margeret's interests include literature and gardening, and she enjoys golf and skiing.GDC NUMBER: 60007
Dr. Paul White BDS
Paul graduated from Glasgow in 1991 and spent his early practicing career in Stornoway. He joined the practice in 2001. He has an enthusiastic interest in Cosmetic and Aesthetic dentistry (especially tooth coloured fillings) and endodontics. A keen follower of Gaelic music, he often supports his very talented musical family across Scotland. GDC NUMBER: 66598
Dr. Derek Brown BDS
Dr. Brown graduated from Dundee in 1997 and entered Dental Practice in Invergordon and Dingwall. He joined the practice in 2006. He has also worked as a Clinical Assistant in Orthodontics and continues his interest in Orthodontics within the practice. Derek has a young family, but still finds time to be an enthusiastic mountain biker.GDC NUMBER: 73351
Dr. Kevin Leeming BDS MFGDP MJDF DPDS
Having qualified in 1989 Kevin worked in a dental Practice in North Wales before taking time out to travel. On returning to the UK he worked as an Oral Surgery SHO gaining valuable surgical experience before returning to General Dental Practice. He has been working as a dentist in the Inverness area since 1994 and also worked as a Clinical Assistant in the Orthodontic Department at Raigmore Hospital, Inverness. On the retirement of Dr. George Black and Dr. Hilary Black in 1996 he moved to Castle House Dental Practice as the Principal. He passed his MFGDP diploma in 2005, achieving his MJDF qualification in 2007 and his DPDS in 2008. As well as Implants and cosmetic dentistry Kevin offers facial aesthetic treatments. Other interests include motorbikes, hill walking and scuba diving. GDC NUMBER: 64742
Your Hygienists
Fiona McCallum (CEB cert dental Hygiene 1986)
Fiona joined the practice in August 2018. Since qualifying from Edinburgh Dental hospital in 1986 she has been working in a number of practices in Inverness where she has gained her knowledge and experience. Keen on the outdoors Fiona loves spending time with her family. GDC NUMBER: 3156
Hannah Irvine (Therapist – Bsc Oral Health Science 2012)
Hannah joined the practice in September 2016. She qualified in 2012 with a degree in Oral Health Science from Inverness and has been working in Inverness practices since then. This allows her to work as a Dental Hygienist/Therapist. Hannah enjoys walking and exploring the Scottish Countryside with her two dogs when not working. GDC NUMBER: 222960
Alison Reid (RDH Diploma Dental Hygiene)
Alison qualified as a Hygienist from Dundee Dental Hospital in 1999. She has worked in a number of practice environments both in Inverness and in London. When not in the surgery she enjoys reading, caring for her pets, and holidays. GDC NUMBER: 5615
www.gdc-uk.org


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\section{Introduction}

Coherent categories are categories with finite limits, pullback-stable image factorisations and pullback-stable unions. Their importance was established in \cite{makkai} (under the name "logical category") as the class of small coherent categories and coherent functors (the structure-preserving ones) can be identified with many-sorted coherent (also called positive) theories and interpretations, see Section 3 for an overview.

A central result in categorical logic, Makkai's conceptual completeness states that if a coherent functor $F:\mathcal{C}\to \mathcal{D}$ induces an equivalence $F^*:\mathbf{Coh}(\mathcal{D},\mathbf{Set})\to \mathbf{Coh}(\mathcal{C},\mathbf{Set})$ (i.e.~if it is a Morita-equivalence), and $\mathcal{C}$ is a pretopos (a coherent category with disjoint unions and quotients of equivalence relations) then $F$ is an equivalence.

Providing a Quillen model structure is a way of doing axiomatic homotopy theory in categories. It consists of three class of arrows on some bicomplete category: weak equivalences, fibrations and cofibrations, satisfying certain diagrammatic axioms. A number of classical results generalise to this setting, in particular Whitehead's theorem takes the form, that in a model category if $X$ and $Y$ are both bifibrant, a map $f:X\to Y$ is a weak equivalence iff it is a homotopy equivalence.

It is therefore natural to look for a model structure whose weak equivalences are the Morita-equivalences, the bifibrant objects are the pretoposes and between pretoposes two maps are homotopic iff they are naturally isomorphic, as in this case we would get conceptual completeness as an instance of Whitehead's theorem. However the 1-category of small coherent categories and coherent functors is neither complete nor cocomplete, but it is in the (2,1)-categorical sense (as it is proved in \cite{adjoint}). Hence we should look for a (2,1)-model structure with these properties (in the sense of \cite{infmodel}), whose existence will be proved in Section 4.

Section 2 gives the proof of the (2,1)-categorical small object argument which will be used in the construction. It follows the standard 1-categorical proof (given e.g.~in \cite{hovey}), with naturally modified arguments for the higher categorical setting. It is not necessary for the pure existence of a model structure with the desired properties: since the full subcategory of pretoposes is known to be reflective in the (2,1)-sense, the generalisation of the theory of reflective factorisation systems would also suffice. However our approach has the advantage that it gives an explicit description for the generating cofibrations. Finally we will prove the model structure to be right proper.

I am grateful for the fruitful conversations with John Bourke, Jiří Rosický and P\'al Zs\'amboki.

\section{Small object argument for (2,1)-categories}

From now on let $\mathbf{C}$ denote a strict (2,1)-category, that is, a strict 2-category whose 2-cells are invertible. First we recall the notion of a (co)limit for (2,1)-categories.

\begin{definition}
Given a small 2-diagram $d_\bullet :\mathcal{I}\to \mathbf{C}$ (i.e.~a strict 2-functor of strict (2,1)-categories), its \emph{2-limit} is an object $d$ with a pseudonatural transformation $\pi :\Delta (d)\Rightarrow d_{\bullet }$ such that $\pi _*:\mathbf{C}(a,d)\to Nat(\Delta (a),d_{\bullet })$ is an equivalence of categories (groupoids in this case). (The codomain is the category of pseudonatural transformations from the constant $a$-valued functor to $d_{\bullet }$ with the modifications as the morphisms.)
\end{definition}

\begin{remark}
In elementary terms the 2-limit can be described as a cone

$ $\\
\adjustbox{scale=0.8,center}{
\begin{tikzcd}
	&&& d \\
	\\
	\\
	\\
	{d_i} &&&&&& {d_k} \\
	\\
	&&&& {d_j}
	\arrow["{f}"', from=5-1, to=7-5]
	\arrow["{g}"', from=7-5, to=5-7]
	\arrow["{h}"', from=5-1, to=5-7]
	\arrow["{p_i}"', from=1-4, to=5-1]
	\arrow["{p_j}", from=1-4, to=7-5]
	\arrow["{p_k}", from=1-4, to=5-7]
	\arrow["{\eta _f}"{description}, shift left=5, shorten <=34pt, shorten >=34pt, Rightarrow, from=5-1, to=7-5]
	\arrow["{\eta _g}"{description}, shift left=5, shorten <=17pt, shorten >=17pt, Rightarrow, from=7-5, to=5-7]
	\arrow["{\eta _h}"{description}, shift left=5, shorten <=53pt, shorten >=53pt, Rightarrow, from=5-1, to=5-7]
\end{tikzcd}
}
such that for each 2-cell $g\circ f \Rightarrow h$ in the diagram the above tetrahedron commutes. Moreover it has the following universal property: given another such cone $(e,(q_i)_{i\in \mathcal{I}_0},(\nu _f)_{f\in \mathcal{I}_1})$ there is a map $r:e\to d$, unique up to unique natural isomorphism such that there are 2-isomorphisms $\alpha _i : p_i r \Rightarrow q_i$ for which the composite of the 2-cells

\[\begin{tikzcd}
	&& e \\
	\\
	&& d \\
	{d_i} &&&& {d_j}
	\arrow["r"{description}, from=1-3, to=3-3]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{p_i}"{description}, from=3-3, to=4-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{p_j}"{description}, from=3-3, to=4-5]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{q_i}"{description}, curve={height=6pt}, from=1-3, to=4-1]
	\arrow[""{name=3, anchor=center, inner sep=0}, "{q_j}"{description}, curve={height=-6pt}, from=1-3, to=4-5]
	\arrow["f"{description}, from=4-1, to=4-5]
	\arrow["{\eta _f}"{description}, shorten <=13pt, shorten >=13pt, Rightarrow, from=0, to=1]
	\arrow["{\alpha _i}"', shorten <=6pt, shorten >=6pt, Rightarrow, from=3-3, to=2]
	\arrow["{\alpha _j}", shorten <=6pt, shorten >=6pt, Rightarrow, from=3-3, to=3]
\end{tikzcd}\]
is $\nu _f : f q_i \Rightarrow q_j$ (for each arrow $f$ of the diagram).

\end{remark}

In this section we will generalise the small object argument for locally small 2-cocomp\-lete strict (2,1)-categories. The proof follows the one given in \cite{hovey} for the 1-categorical setting.

\begin{definition}
Let $\lambda $ be an ordinal seen as a (2,1)-category with trivial 2-cells. Given a 2-colimit preserving diagram $\lambda \to \mathbf{C}$ with 2-colimit $\mathcal{X}$
\[\begin{tikzcd}
	&& {\mathcal{X}} \\
	{\mathcal{X}_0} & {\mathcal{X}_1} & {\mathcal{X}_2} & \dots
	\arrow["{f_0}"', from=2-1, to=2-2]
	\arrow["{f_1}"', from=2-2, to=2-3]
	\arrow["{f_2}"', from=2-3, to=2-4]
	\arrow["f", from=2-1, to=1-3]
	\arrow[from=2-2, to=1-3]
	\arrow[from=2-3, to=1-3]
\end{tikzcd}\]
the coprojection map $f:\mathcal{X}_0\to \mathcal{X}$ is called the \emph{transfinite composition} of the $\lambda $-sequence $(f_i)_{i<\lambda }$.
\end{definition}

\begin{definition}
Let $I\subset Arr(\mathbf{C})$ be a set. \emph{$I$-$cell$} is the class of maps that can be written as the transfinite composition of 2-pushouts from $I$. \emph{$I$-$inj$} is the class whose members ($f$) have the following right lifting property: given a square
\[\begin{tikzcd}
	{\mathcal{C}} && {\mathcal{X}} \\
	\\
	{\mathcal{D}} && {\mathcal{Y}}
	\arrow["g"', from=1-1, to=3-1]
	\arrow["{h'}", from=3-1, to=3-3]
	\arrow["f", from=1-3, to=3-3]
	\arrow["h", from=1-1, to=1-3]
	\arrow["\eta"{description}, shorten <=11pt, shorten >=11pt, Rightarrow, from=3-1, to=1-3]
\end{tikzcd}\]
with $g\in I$, there is a lifting
\[\begin{tikzcd}
	{\mathcal{C}} && {\mathcal{X}} \\
	\\
	{\mathcal{D}} && {\mathcal{Y}}
	\arrow["g"', from=1-1, to=3-1]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{h'}", from=3-1, to=3-3]
	\arrow["f", from=1-3, to=3-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, "h", from=1-1, to=1-3]
	\arrow["k"{description}, from=3-1, to=1-3]
	\arrow["{\nu_1}"{description}, shorten <=13pt, shorten >=9pt, Rightarrow, from=3-1, to=1]
	\arrow["{\nu_2}"{description}, shorten <=13pt, shorten >=9pt, Rightarrow, from=0, to=1-3]
\end{tikzcd}\]
such that the pasting of $\nu _1$ and $\nu _2$ is $\eta $, i.e.~$f\nu _1 \circ \nu _2g =\eta$.

\emph{$I$-$proj$} is the class whose members have the left lifting property wrt.~$I$. As usual we set \emph{$I$-$cof$}=($I$-$inj$)-$proj$, and \emph{$I$-$fib$}=($I$-$proj$)-$inj$.
\end{definition}

\begin{proposition}
$I$-cell $\subseteq I$-$cof$.
\end{proposition}

\begin{proof}
Clearly $I\subseteq I$-$cof$, hence it suffices to prove that $I$-$cof$ is closed under pushouts and transfinite compositions. First we show that given
\[\begin{tikzcd}
	\bullet && \bullet && \bullet \\
	\\
	\bullet && \bullet && \bullet
	\arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=3-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, "g", from=1-1, to=1-3]
	\arrow["{f'}"', from=1-3, to=3-3]
	\arrow["{g'}", from=3-1, to=3-3]
	\arrow["h", from=1-3, to=1-5]
	\arrow[""{name=2, anchor=center, inner sep=0}, "m", from=1-5, to=3-5]
	\arrow[""{name=3, anchor=center, inner sep=0}, "k", from=3-3, to=3-5]
	\arrow["\alpha", shorten <=7pt, shorten >=7pt, Rightarrow, from=1, to=0]
	\arrow["\beta", shorten <=7pt, shorten >=7pt, Rightarrow, from=2, to=3]
\end{tikzcd}\]
with the left square being a 2-pushout, there exists a lifting in the right square.

Using that $f$ has the left lifting property wrt.~$m$ we get
\[\begin{tikzcd}
	\bullet && \bullet && \bullet \\
	\\
	\bullet && \bullet && \bullet
	\arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=3-1]
	\arrow["g", from=1-1, to=1-3]
	\arrow["{g'}", from=3-1, to=3-3]
	\arrow["h", from=1-3, to=1-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, "m", from=1-5, to=3-5]
	\arrow["k", from=3-3, to=3-5]
	\arrow["{l_1}", from=3-1, to=1-5]
	\arrow["\delta"', shorten <=11pt, shorten >=11pt, Rightarrow, from=1, to=3-3]
	\arrow["{\gamma }", shorten <=11pt, shorten >=11pt, Rightarrow, from=1-3, to=0]
\end{tikzcd}\]
such that the pasting of $\alpha $ and $\beta $ is the same as the pasting of $\gamma $ and $\delta $ (i.e.~$k\alpha \circ \beta g =\delta f \circ m\gamma$).

Using the universal property of the 2-pushout we get
\[\begin{tikzcd}
	\bullet && \bullet \\
	\\
	\bullet && \bullet \\
	&&& \bullet
	\arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=3-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, "g", from=1-1, to=1-3]
	\arrow["{f'}", from=1-3, to=3-3]
	\arrow["{g'}"', from=3-1, to=3-3]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{l_1}"', curve={height=12pt}, from=3-1, to=4-4]
	\arrow[""{name=3, anchor=center, inner sep=0}, "h", curve={height=-12pt}, from=1-3, to=4-4]
	\arrow["{l_2}"{description}, dashed, from=3-3, to=4-4]
	\arrow["\alpha", shorten <=7pt, shorten >=7pt, Rightarrow, from=1, to=0]
	\arrow["\nu", shorten >=4pt, Rightarrow, from=3-3, to=2]
	\arrow["\eta", shorten <=4pt, Rightarrow, from=3, to=3-3]
\end{tikzcd}\]
such that the pasting of $\alpha $, $\nu $ and $\eta $ is $\gamma $.

We should prove that $l_2$ is a splitting of $\beta $. It is enough to see that in

\[\begin{tikzcd}
	\bullet && \bullet &&& \bullet && \bullet \\
	&&&&&&&&& \bullet \\
	\bullet && \bullet &&& \bullet && \bullet \\
	&&& \bullet &&&& \bullet && \bullet \\
	&&&& \bullet
	\arrow[""{name=0, anchor=center, inner sep=0}, "f"', from=1-1, to=3-1]
	\arrow[""{name=1, anchor=center, inner sep=0}, "g", from=1-1, to=1-3]
	\arrow["{f'}", from=1-3, to=3-3]
	\arrow["{g'}"', from=3-1, to=3-3]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{l_1}"', curve={height=12pt}, from=3-1, to=4-4]
	\arrow[""{name=3, anchor=center, inner sep=0}, "h", curve={height=-12pt}, from=1-3, to=4-4]
	\arrow["{l_2}"{description}, dashed, from=3-3, to=4-4]
	\arrow["m", from=4-4, to=5-5]
	\arrow["{g'}", from=3-6, to=3-8]
	\arrow["{f'}", from=1-8, to=3-8]
	\arrow[""{name=4, anchor=center, inner sep=0}, "f"', from=1-6, to=3-6]
	\arrow[""{name=5, anchor=center, inner sep=0}, "g", from=1-6, to=1-8]
	\arrow["k", from=3-8, to=4-10]
	\arrow["h", from=1-8, to=2-10]
	\arrow["m", from=2-10, to=4-10]
	\arrow["{l_1}"', from=3-6, to=4-8]
	\arrow["m"', from=4-8, to=4-10]
	\arrow["\beta", shorten <=10pt, shorten >=10pt, Rightarrow, from=2-10, to=3-8]
	\arrow["\delta", Rightarrow, from=3-8, to=4-8]
	\arrow["\alpha", shorten <=7pt, shorten >=7pt, Rightarrow, from=1, to=0]
	\arrow["\nu", shorten >=4pt, Rightarrow, from=3-3, to=2]
	\arrow["\eta", shorten <=4pt, Rightarrow, from=3, to=3-3]
	\arrow["\alpha", shorten <=7pt, shorten >=7pt, Rightarrow, from=5, to=4]
\end{tikzcd}\]
the 2-cells filling the boundaries are identical as in this case both $ml_2$ and $k$ are suitable splittings, hence there is a unique natural isomorphism $\varepsilon:ml_2 \Rightarrow k$ for which $\varepsilon f' \circ m\eta =\beta $ and $m\nu \circ \varepsilon ^{-1}g' =\delta$ (and the first equality means that $l_2$ is a splitting). The equality of the 2-cells follows from the identities observed above.

Now assume that each $f_i$ ($i<\lambda $) has left lifting property wrt.~$m$ (and that $f_i$-s form a (co)continuous sequence). We have to prove that its transfinite composition $f$ has the same lifting property. We have a lift in
\[\begin{tikzcd}
	\bullet &&& \bullet \\
	\\
	\bullet && \bullet & \bullet
	\arrow["{f_0}"', from=1-1, to=3-1]
	\arrow["f"{pos=0.3}, from=1-1, to=3-3]
	\arrow[from=3-1, to=3-3]
	\arrow["h", from=1-1, to=1-4]
	\arrow["k", from=3-3, to=3-4]
	\arrow["m", from=1-4, to=3-4]
	\arrow["{h_1}"{description, pos=0.7}, dashed, from=3-1, to=1-4]
\end{tikzcd}\]
and this way one defines $h_i$ for each successor ordinal $i<\lambda $. At limit steps $h_i$ is induced by the universal property of the 2-colimit of the sequence. Compatibility of the 2-cells is checked as before.
\end{proof}

It is worth to spell out explicitly:

\begin{proposition}
Left lifting properties are preserved by 2-pushouts and transfinite compositions. Dually, right lifting properties are preserved by 2-pullbacks and transfinite cocompositions (2-limit of the reversed sequence). In particular (taking $\lambda =2$) $I$-$inj$ and $I$-$proj$ are subcategories.

\label{lifting}
\end{proposition}

\begin{proposition}
$I$-cell is closed under transfinite composition.
\end{proposition}

\begin{proof}
We need to prove that "the transfinite composition of transfinite compositions is a transfinite composition", i.e.~that if we have a sequential (2-)diagram then its colimit can be computed as the colimit of any cofinal subsequence. This is Proposition 4.1.1.8. in \cite{lurietopos}.
\end{proof}

\begin{proposition}
The homotopy pushout of a coproduct of maps from $I$ is in $I$-cell.
\end{proposition}

\begin{proof}
Let $g_j$ $(j\in J)$ be a family of arrows from $I$. Their coproduct is the induced map:
\[\begin{tikzcd}
	{\mathcal{C}_j} && {\cup _{j\in J} \mathcal{C}_j} \\
	\\
	{\mathcal{D}_j} && {\cup _{j\in J}\mathcal{D}_j}
	\arrow["{g_j}", from=1-1, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow["{\cup g_j}", dashed, from=1-3, to=3-3]
	\arrow["{\chi _j}"{description}, shorten <=11pt, shorten >=11pt, Rightarrow, from=1-3, to=3-1]
\end{tikzcd}\]
Now take the 2-pushout:
\[\begin{tikzcd}
	{\cup _j\mathcal{C}_j} && {\mathcal{X}} \\
	\\
	{\cup _j\mathcal{D}_j} && {\mathcal{Y}}
	\arrow["{\cup g_j}"', from=1-1, to=3-1]
	\arrow["{h_1}", from=3-1, to=3-3]
	\arrow["f", from=1-3, to=3-3]
	\arrow["{h_0}", from=1-1, to=1-3]
	\arrow["\eta"{description}, shorten <=11pt, shorten >=11pt, Rightarrow, from=1-3, to=3-1]
\end{tikzcd}\]
We will proceed by transfinite recursion and take: $X_0=X$, $\rho _0=f$ and $i_{0,0}=1_X$. In the successor step we form the 2-pushout of $g_j:\mathcal{C}_j\to \mathcal{D}_j$ and $\mathcal{C}_j\to \cup \mathcal{C}_j \xrightarrow{h_0} \mathcal{X} \xrightarrow{i_{0,j}}\mathcal{X}_j$ to get $X_{j+1}$ and induce $\rho _{j+1}$ by the universal property of the square. Hence we get a commutative cube (where the faces are filled with the obvious 2-cells):
\[\begin{tikzcd}
	{\mathcal{C}_j} &&&& {\mathcal{X}_j} \\
	& {\cup \mathcal{C}_j} & {\mathcal{X}} & {\mathcal{X}_j} \\
	\\
	& {\cup \mathcal{D}_j} && {\mathcal{Y}} \\
	{\mathcal{D}_j} &&&& {\mathcal{X}_{j+1}}
	\arrow["{h_0}", from=2-2, to=2-3]
	\arrow["{i_{0,j}}", from=2-3, to=2-4]
	\arrow["{\rho _j}"{description}, from=2-4, to=4-4]
	\arrow["{\cup g_j}"{description}, from=2-2, to=4-2]
	\arrow["{h_1}"{description}, from=4-2, to=4-4]
	\arrow[from=1-1, to=2-2]
	\arrow[Rightarrow, no head, from=1-5, to=2-4]
	\arrow[from=5-1, to=4-2]
	\arrow[dashed, from=5-5, to=4-4]
	\arrow[from=1-5, to=5-5]
	\arrow[from=1-1, to=1-5]
	\arrow["{g_j}"{description}, from=1-1, to=5-1]
	\arrow[from=5-1, to=5-5]
	\arrow["f"{description}, from=2-3, to=4-4]
\end{tikzcd}\]
When $j$ is a limit ordinal $\mathcal{X}_j$ is given by the transfinite composition
\[\begin{tikzcd}
	&&& {\mathcal{Y}} \\
	&&& {\mathcal{X}_j} \\
	{\mathcal{X}_0} && {\mathcal{X}_1} && \dots
	\arrow["{i_{0,1}}"', from=3-1, to=3-3]
	\arrow["{i_{1,2}}"', from=3-3, to=3-5]
	\arrow["{i_{0,j}}"{description}, from=3-1, to=2-4]
	\arrow["{i_{1,j}}"'{pos=0.7}, from=3-3, to=2-4]
	\arrow[dashed, from=2-4, to=1-4]
	\arrow["{\rho_1}"{description}, curve={height=-6pt}, from=3-3, to=1-4]
	\arrow["{\rho _0}"{description}, curve={height=-6pt}, from=3-1, to=1-4]
\end{tikzcd}\]
(the 3-cells are filled). We claim that with $\lambda =|J|$ the map $\mathcal{X}\to \mathcal{X}_{\lambda}$ is also a homotopy pushout for $\cup g_j$ along $h_0$. To see this we should find some 2-cells for

\[\begin{tikzcd}
	{\cup \mathcal{C}_j} &&& {\mathcal{X}} \\
	\\
	{\cup \mathcal{D}_j} &&& {\mathcal{X}_\lambda} \\
	&&&& {\mathcal{Y}}
	\arrow["{h_0}"{description}, from=1-1, to=1-4]
	\arrow["{i_{0,\lambda}}"{description}, from=1-4, to=3-4]
	\arrow["{\cup g_j}"{description}, from=1-1, to=3-1]
	\arrow["{\small{\cup \{\mathcal{D}_j\to \mathcal{X}_{j+1}\to \mathcal{X}_\lambda\}}}", from=3-1, to=3-4]
	\arrow["{\rho _{\lambda}}", from=3-4, to=4-5]
	\arrow["f"{description}, curve={height=-12pt}, from=1-4, to=4-5]
	\arrow["{h_1}"{description}, curve={height=12pt}, from=3-1, to=4-5]
\end{tikzcd}\]
whose composite is $\eta$. They can be found on the surface of the commutative 3-simplicial set

\[\begin{tikzcd}
	{\mathcal{C}_j} &&&& {\mathcal{X}_j} \\
	& {\cup \mathcal{C}_j} & {\mathcal{X}} & {\mathcal{X}_j} \\
	\\
	& {\cup \mathcal{D}_j} && {\mathcal{Y}} \\
	{\mathcal{D}_j} &&&& {\mathcal{X}_{j+1}} \\
	& {\cup\mathcal{D}_j} && {\mathcal{X}_{\lambda}}
	\arrow["{h_0}", from=2-2, to=2-3]
	\arrow["{i_{0,j}}", from=2-3, to=2-4]
	\arrow["{\rho _j}"{description}, from=2-4, to=4-4]
	\arrow["{\cup g_j}"{description}, from=2-2, to=4-2]
	\arrow["{h_1}"{description}, from=4-2, to=4-4]
	\arrow[from=1-1, to=2-2]
	\arrow[Rightarrow, no head, from=1-5, to=2-4]
	\arrow[from=5-1, to=4-2]
	\arrow[dashed, from=5-5, to=4-4]
	\arrow[from=1-5, to=5-5]
	\arrow[from=1-1, to=1-5]
	\arrow["{g_j}"{description}, from=1-1, to=5-1]
	\arrow[from=5-1, to=5-5]
	\arrow["f"{description}, from=2-3, to=4-4]
	\arrow[Rightarrow, no head, from=4-2, to=6-2]
	\arrow[from=5-1, to=6-2]
	\arrow["{\rho _\lambda}"{description, pos=0.3}, from=6-4, to=4-4]
	\arrow[from=5-5, to=6-4]
	\arrow[from=6-2, to=6-4]
\end{tikzcd}\]
\end{proof}

\begin{definition}
An object $\mathcal{X}$ of $\mathbf{C}$ is \emph{$\lambda $-small} wrt.~a subcategory $J$ if $\mathbf{C}(\mathcal{X},-)$ commutes with $\lambda $-filtered sequential 2-colimits formed in $J$. $\mathcal{X}$ is small if it is $\lambda $-small for some $\lambda $.
\end{definition}

\begin{theorem}[Small object argument]
Let $I\subset Arr(\mathbf{C})$ be a set, and assume that domains of $I$ are small relative to $I$-cell. Then for any map $f:\mathcal{X}\to \mathcal{Y}$ there are arrows $\mathcal{X}\xrightarrow{f'}\mathcal{Z}\xrightarrow{f''}\mathcal{Y}$ such that $f'\in I$-cell, $f''\in I$-$inj$ and $f'' \circ f'$ is isomorphic to $f$.
\label{smallob}
\end{theorem}

\begin{proof}
We proceed by transfinite recursion and take $\mathcal{Z}_0=\mathcal{X}$, $\rho _0 =f$ and $i_{0,0}=1_{\mathcal{X}}$.

For a successor ordinal $j+1$ collect all squares

\[\begin{tikzcd}
	{\mathcal{A}_s} && {\mathcal{Z}_j} \\
	\\
	{\mathcal{B}_s} && {\mathcal{Y}}
	\arrow["{g_s}"{description}, from=1-1, to=3-1]
	\arrow["{h_s}"{description}, from=1-1, to=1-3]
	\arrow["{\rho _j}"{description}, from=1-3, to=3-3]
	\arrow["{k_s}"{description}, from=3-1, to=3-3]
	\arrow["{\eta _s}"{description}, shorten <=11pt, shorten >=11pt, Rightarrow, from=1-3, to=3-1]
\end{tikzcd}\]
with $g_s\in I$ to an $S$-indexed set, then form the 2-pushout of $\sqcup g_s$ and $\sqcup h_s$ and induce $\rho _{j+1}$:

\[\begin{tikzcd}
	{\sqcup \mathcal{A}_s} && {\mathcal{Z}_j} \\
	\\
	{\sqcup \mathcal{B}_s} && {\mathcal{Z}_{j+1}} \\
	&&& {\mathcal{Y}}
	\arrow["{\sqcup h_s}", from=1-1, to=1-3]
	\arrow["{\sqcup g_s}"', from=1-1, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow["{i_{j,j+1}}"', from=1-3, to=3-3]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\rho _j}", curve={height=-12pt}, from=1-3, to=4-4]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{\sqcup k_s}"', curve={height=12pt}, from=3-1, to=4-4]
	\arrow["{\rho _{j+1}}", dashed, from=3-3, to=4-4]
	\arrow[shorten <=17pt, shorten >=17pt, Rightarrow, from=1-3, to=3-1]
	\arrow[shorten <=4pt, shorten >=2pt, Rightarrow, from=0, to=3-3]
	\arrow[shorten <=2pt, shorten >=3pt, Rightarrow, from=3-3, to=1]
\end{tikzcd}\]
Note that the composition of the three 2-cells is the natural isomorphism induced by $\{ \eta _s : s\in S \}$. (*)

When $j$ is a limit ordinal we form the transfinite composition

\[\begin{tikzcd}
	&&&& {\mathcal{Y}} \\
	&&&& {\mathcal{Z}_j} \\
	{\mathcal{Z}_0} && {\mathcal{Z}_1} && \dots
	\arrow["{i_{0,1}}"{description}, from=3-1, to=3-3]
	\arrow["{i_{1,2}}"{description}, from=3-3, to=3-5]
	\arrow["{i_{0,j}}"{description, pos=0.4}, from=3-1, to=2-5]
	\arrow["{i_{1,j}}"{description, pos=0.4}, from=3-3, to=2-5]
	\arrow["{\rho _0}"{description}, curve={height=-12pt}, from=3-1, to=1-5]
	\arrow["{\rho_1}"{description}, curve={height=-6pt}, from=3-3, to=1-5]
	\arrow["{\rho _j}"', dashed, from=2-5, to=1-5]
\end{tikzcd}\]

Let $\lambda $ be a cardinal, such that domains of $I$ are $\lambda $-small. The composition $\mathcal{X} \xrightarrow{i_{0,\lambda}} \mathcal{Z}_{\lambda} \xrightarrow{\rho _{\lambda}} \mathcal{Y}$ is isomorphic to $f$ and $i_{0,\lambda} \in I$-cell by the previous propositions.

It remains to prove that $\rho _\lambda \in I$-$inj$. Take a square
\[\begin{tikzcd}
	{\mathcal{A}} && {\mathcal{Z}_\lambda} \\
	\\
	{\mathcal{B}} && {\mathcal{Y}}
	\arrow["h"{description}, from=1-1, to=1-3]
	\arrow["{\rho _\lambda}"{description}, from=1-3, to=3-3]
	\arrow["g"{description}, from=1-1, to=3-1]
	\arrow["k"{description}, from=3-1, to=3-3]
	\arrow["\eta"{description}, shorten <=11pt, shorten >=11pt, Rightarrow, from=1-3, to=3-1]
\end{tikzcd}\]
As $\mathcal{A}$ is $\lambda $-small, $h$ factors through some stage $\mathcal{Z}_j$ (up to isomorphism). This means that the back face of the left cube in

\[\begin{tikzcd}
	{\mathcal{A}} &&& {\mathcal{Z}_{j}} && {\mathcal{Z}_{\lambda}} \\
	& {\sqcup \mathcal{A}_s} & {\mathcal{Z}_{j}} \\
	& {\sqcup \mathcal{B}_s} & {\mathcal{Z}_{j+1}} \\
	{\mathcal{B}} &&& {\mathcal{Y}}
	\arrow["{h'}"{description}, from=1-1, to=1-4]
	\arrow["{\rho _j}"{description, pos=0.7}, from=1-4, to=4-4]
	\arrow["g"{description}, from=1-1, to=4-1]
	\arrow["k"{description}, from=4-1, to=4-4]
	\arrow["{\sqcup g_s}"', from=2-2, to=3-2]
	\arrow[from=3-2, to=3-3]
	\arrow[from=2-3, to=3-3]
	\arrow["{\sqcup h_s}", from=2-2, to=2-3]
	\arrow[from=1-1, to=2-2]
	\arrow[from=4-1, to=3-2]
	\arrow[from=3-3, to=4-4]
	\arrow[Rightarrow, no head, from=2-3, to=1-4]
	\arrow["{i_{j,\lambda }}"{description}, from=1-4, to=1-6]
	\arrow["{i_{j,\lambda }}"{description, pos=0.6}, from=2-3, to=1-6]
	\arrow["{i_{j+1,\lambda }}"{description}, from=3-3, to=1-6]
	\arrow["{\rho _{\lambda}}"{description, pos=0.7}, from=1-6, to=4-4]
	\arrow["{\rho _j}"{description}, color={rgb,255:red,110;green,110;blue,110}, curve={height=-6pt}, from=2-3, to=4-4]
	\arrow["{\sqcup k_s}"{description}, color={rgb,255:red,110;green,110;blue,110}, curve={height=6pt}, from=3-2, to=4-4]
	\arrow["h"{description}, curve={height=-12pt}, from=1-1, to=1-6]
\end{tikzcd}\]
was considered in the formation of $\mathcal{Z}_{j+1}$.
This face is just the gluing of
\[\begin{tikzcd}
	{\mathcal{A}} &&& {\mathcal{Z}_{j}} \\
	& {\mathcal{Z}_{\lambda}} \\
	{\mathcal{B}} &&& {\mathcal{Y}}
	\arrow["{h'}"{description}, from=1-1, to=1-4]
	\arrow["g"{description}, from=1-1, to=3-1]
	\arrow["k"{description}, from=3-1, to=3-4]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\rho _j}"{description}, from=1-4, to=3-4]
	\arrow[""{name=1, anchor=center, inner sep=0}, "h"{description}, from=1-1, to=2-2]
	\arrow["{i_{j,\lambda}}"{description}, from=1-4, to=2-2]
	\arrow["{\rho _\lambda}"{description}, from=2-2, to=3-4]
	\arrow[shorten <=4pt, shorten >=8pt, Rightarrow, from=2-2, to=3-1]
	\arrow[shorten <=17pt, shorten >=22pt, Rightarrow, from=0, to=2-2]
	\arrow[shorten <=36pt, shorten >=22pt, Rightarrow, from=1-4, to=1]
\end{tikzcd}\]

By (*) the left cube is a commutative (identical) 3-cell, and so is the cone over the $\mathcal{Z}_n$'s. Hence the lift $\mathcal{B}\to \sqcup \mathcal{B}_s \to \mathcal{Z}_{j+1} \to \mathcal{Z}_\lambda $ is a splitting of $\eta $.
\end{proof}

Finally we give the definition of a (2,1)-model structure. This is the special case of Definition 1.1 in \cite{infmodel} except that we require the existence of all limits and colimits not just the finite ones.

\begin{definition}
Let $\mathbf{C}$ be a 2-complete and 2-cocomplete (2,1)-category. Given three sub-2-categories $W$ (called weak equivalences), $Fib$ (called fibrations) and $Cof$ (called cofibrations) we say that $(\mathbf{C},W,Fib,Cof)$ is a \emph{model category} if the following axioms are satisfied:
\begin{itemize}
    \item Given $h\cong gf$ if two maps are weak equivalences then so is the third one.
    \item All three subcategories are closed under retracts, i.e.~given a diagram

\[\begin{tikzcd}
	&& \bullet \\
	\bullet && \bullet && \bullet \\
	\bullet &&&& \bullet
	\arrow[from=2-1, to=1-3]
	\arrow[from=1-3, to=2-5]
	\arrow[from=3-1, to=2-3]
	\arrow[from=2-3, to=3-5]
	\arrow["f", from=2-5, to=3-5]
	\arrow["g", from=1-3, to=2-3]
	\arrow["f"', from=2-1, to=3-1]
	\arrow["1"{description}, curve={height=12pt}, from=2-1, to=2-5]
	\arrow["1"{description}, curve={height=12pt}, from=3-1, to=3-5]
\end{tikzcd}\]
with strictly commuting front face, filled with the identity 3-cell, $g\in W/Fib/Cof$ implies $f\in W/Fib/Cof$.
\item Maps in $Fib $ have the right lifting property against $W\cap Cof$ and maps in $W\cap Fib$ have the right lifting property against $Cof$.
\item Every map $h$ can be written as $h\cong gf \cong g'f'$ with $f\in Cof $, $g\in W\cap Fib$ and $f'\in W\cap Cof$, $g'\in Fib$.
\end{itemize}
\end{definition}

\section{An overview of coherent categories}

\begin{definition}
A category $\mathcal{C}$ is \emph{coherent}, if it
\begin{itemize}
\item has finite limits,
\item has pullback-stable images, i.e.~every arrow can be factored as an effective epimorphism followed by a monomorphism, and effective epimorphisms are stable under pullbacks,
\item has pullback-stable finite unions.
\end{itemize}

A functor $F:\mathcal{C}\to \mathcal{D}$ is \emph{coherent} if it preserves finite limits, effective epimorphisms and finite unions.
\end{definition}

We will denote the (2,2)-category of small coherent categories, coherent functors and all natural transformations by $\mathbf{Coh}$ and the corresponding (2,1)-category whose 2-cells are the natural isomorphisms by $\mathbf{Coh}_{\sim }$. We will make use of the following results:

\begin{theorem}
The 1-category $\mathbf{Coh}_1$ is accessible.
\end{theorem}

\begin{proof}
By \cite{sketches} $\mathbf{Coh}_1$ is a finitary injectivity class in a presheaf category, hence it is accessible.
\end{proof}

It is well-known since \cite{flexible} that $\mathbf{Coh}_{\sim }$ is 2-complete. Using accessibility of $\mathbf{Coh}_1$, it follows by Section 9.3 of \cite{adjoint} that $\mathbf{Coh}_{ \sim}$ is 2-cocomplete as well, so giving:

\begin{theorem}
$\mathbf{Coh}_{\sim }$ is 2-complete and 2-cocomplete.
\end{theorem}

\begin{theorem}
Let $d_{\bullet }:\mathcal{\lambda }\to \mathbf{Coh}_{\sim }$ be a strict diagram (i.e.~all 2-cells are identical). Then its colimit $d$ in $\mathbf{Coh}_1$ is also the 2-colimit in $\mathbf{Coh}_{\sim }$.
\label{nicefiltered}
\end{theorem}

\begin{proof}
We must show that $\mathbf{Coh}_{\sim }(d,a)\to Strict(d_{\bullet },\Delta(a))\to Nat(d_{\bullet},\Delta(a))$ is an equivalence, or equivalently that the second map is an equivalence since the first one is an iso, or even just that the second map is essentially surjective since it is automatically fully faithful.  Here $Strict$ refers to strict cocones.  Consider a pseudo-natural transformation $d_{\bullet}\Rightarrow \Delta(a)$. Since filtered colimits are 2-colimits in $\mathbf{Cat}$ by Lemma 5.4.9 of \cite{accessible}, we can replace this by an isomorphic strict cocone in $\mathbf{Cat}$ --- but since being a coherent functor is isomorphism invariant, this means that the strict cocone belongs to $\mathbf{Coh}_{\sim }$ as well, proving the claim.
\end{proof}

This implies that in the inductive proof of Theorem \ref{smallob} the sequence $\mathcal{Z}_0 \xrightarrow{i_{0,1}} \dots $ can be chosen to be strict, so any $\lambda $ with $cf(\lambda )> sup \{\kappa _f :f\in I\}$ works, where $\kappa _f$ is the presentability rank of $dom(f)$. Hence we have:

\begin{theorem}
Let $I$ be a small set of coherent functors. Given a coherent functor $\mathcal{C}\xrightarrow{M} \mathcal{E}$ it is isomorphic to a composition $\mathcal{C}\xrightarrow{F}\mathcal{D}\xrightarrow{G}\mathcal{E}$ where $F\in I$-cell (moreover $F$ is the strict transfinite composition of 2-pushouts from $I$) and $G\in I$-$inj$. In particular $F\in I$-$cof$.
\label{smallforcoh}
\end{theorem}

Now we recall from \cite{makkai} the correspondence between coherent categories/coherent functors and coherent theories/interpretations.

\begin{definition}
Let $L $ be a (many-sorted) signature. An $L $-formula is called \emph{coherent} if it is built up from atomic formulas using finite $\wedge $ (including $\top$), finite $\vee$ (including $\bot $) and $\exists $. Their class is denoted by $L_{\omega \omega }^g$.

A formula of the form $\forall x_1 \dots \forall x_n (\varphi \to \psi )$ with $\varphi ,\psi \in L_{\omega \omega }^g$ is called a \emph{coherent sequent} and it is written as $\varphi \Rightarrow \psi $. A \emph{coherent theory} is a set of coherent sequents.
\end{definition}

\begin{definition}
Given a signature $L$, an \emph{$L$-structure} in a category $\mathcal{C}$ with finite products, associates to each sort $X$ an object $M(X)$ of $\mathcal{C}$, to each relation symbol $R$ a subobject $M(R) \leq M(X_1)\times \dots \times M(X_n)$, and to a function symbol $f$ a morphism $M(f):M(X_1)\times \dots \times M(X_n)\to M(X)$.
\end{definition}

To define the models of a coherent theory internally to some coherent category $\mathcal{C}$ we need to interpret $L_{\omega \omega }^g$ formulas in an $L$-structure $M$ in $\mathcal{C}$:

\begin{definition}
Let $M$ be an $L$-structure in a coherent category $\mathcal{C}$. The \emph{interpretation} of a coherent formula is given by the following steps:
\begin{itemize}
\item If $\vec{x}=(x_1,\dots x_n)$ is a finite sequence of free variables, $x_i$ is of sort $X_i$, then $M(\vec{x})=M(X_1)\times \dots \times M(X_n)$.
\item If $t$ is a term (of sort $Y$) whose free variables are from $\vec{x}$, then $M_{\vec{x}}(t)$ will be an arrow $M(\vec{x})\to M(Y)$ in the following way:
\begin{itemize}
\item If $t=x_i$, then $M_{\vec{x}}(t)$ is the projection map $M(\vec{x})\to M(X_i)$.
\item If $t=f(t_1,\dots t_n)$, then $M_{\vec{x}}(t)$ is the composite $M(\vec{x})\xrightarrow {\langle M_{\vec{x}}(t_1),\dots \rangle} \prod M(Y_i) \xrightarrow{M(f)} M(Y)$
\end{itemize}
When $\mathcal{C}=\mathbf{Set}$, these are the functions which to a possible evaluation of $\vec{x}$ assign the induced value of $t$.
\item If $\varphi $ is a formula, whose free variables are among $\vec{x}=(x_1,\dots x_n)$, then its interpretation in the context $\vec{x}$ is a subobject $M_{\vec{x}}(\varphi )\leq M(\vec{x})$. It can be readily checked that in the case of $\mathbf{Set}$-models this gives precisely the set of evaluations of $\vec{x}$ which make $\varphi $ valid in $M$.

\begin{tabularx}{\textwidth}{XX} \begin{tikzpicture}
\node (1) at (0,0) {$M_{\vec{x}}(t_1\approx t_2) $};
\node[anchor=base,right=10mm of 1] (2) {$M(\vec{x})$};
\node[anchor=base,right=10mm of 2] (3) {$M(Y)$};
\draw[right hook ->] (1)--(2) node [midway,above] {\small{$e$}};
\draw[postaction={transform canvas={yshift=-1mm},draw}] [->] (2) -- (3) node [midway,above] {\small{$M_{\vec{x}}(t_1)$}} node [midway,below] {\small{$M_{\vec{x}}(t_2)$}};
\end{tikzpicture} &
is an equalizer.
\end{tabularx}

\begin{tabularx}{\textwidth}{XX}
\begin{tikzpicture}
\node (3) at (0,0) {$M_{\vec{x}}(R(t_1,\dots t_n))$};
\node[anchor=base,right=10mm of 3] (2a) {$M(\vec{x})$};
\node[anchor=base,below=10mm of 3] (2b) {$M(R)$};
\node[anchor=base,below=10mm of 2a] (1) {$\prod _{i=1}^n M(Y_i)$};
\draw[right hook->] (3)--(2a);
\draw[->] (3)--(2b);
\draw[->] (2a)--(1) node [midway,right] {$\langle M_{\vec{x}}(t_1),\dots \rangle$};
\draw[right hook->] (2b)--(1) node [midway,below] {$M(i)$};
\end{tikzpicture}
& \centering is a pullback.
\end{tabularx}
\begin{tabularx}{\textwidth}{X}
 $M_{\vec{x}}(\bigwedge \Theta)=\bigwedge\{M_{\vec{x}}(\theta):\theta\in \Theta\}$ \\
 \end{tabularx}
 \begin{tabularx}{\textwidth}{X}
 $M_{\vec{x}}(\bigvee \Theta)=\bigvee\{M_{\vec{x}}(\theta):\theta\in \Theta\}$ \\
 \end{tabularx}
 \begin{tabularx}{\textwidth}{X}
\makecell{ $M_{\vec{x}}(\exists y \varphi)$ (where $y$ is not in $\vec{x}$) is the eff.~epi-mono factorisation:
\\
\begin{tikzpicture}
\node (3) at (0,0) {$M_{\vec{x},y}(\varphi)$};
\node[anchor=base,right=10mm of 3] (2a) {$M(\vec{x},y)$};
\node[anchor=base,right=10mm of 2a] (2b) {$M(\vec{x})$};
\node[anchor=base,below=10mm of 2a] (1) {$M_{\vec{x}}(\exists y \varphi)$};
\draw[right hook->] (3)--(2a);
\draw[->] (2a)--(2b) node [midway,above] {$\pi _{\vec{x}}$};
\draw[->>] (3)--(1);
\draw[right hook->] (1)--(2b);
\end{tikzpicture}
}
\end{tabularx}

\end{itemize}
\label{intpret}
\end{definition}

\begin{definition}
The sequent $\varphi \Rightarrow \psi$ is \emph{valid} in the structure $M$ (in symbols: $M\models \varphi \Rightarrow \psi $), iff $ M_{\vec{x}}(\varphi ) \leq M_{\vec{x}}(\psi ) $ (where $\vec{x}$ is the collection of all free variables in $\varphi \Rightarrow \psi $).

$M$ is a \emph{model} of the theory $T$ if all the sequents from $T$ are valid in $M$.

A \emph{homomorphism} $\alpha :M\to M'$ of $T$-models consists of an arrow $\alpha _X: M(X)\to M'(X)$ for each sort $X$, for which the square

\[\begin{tikzcd}
	{M(X_1)\times \dots \times M(X_n)} && {M(Y)} \\
	\\
	{M'(X_1)\times \dots \times M'(X_n)} && {M'(Y)}
	\arrow["{M(f)}", from=1-1, to=1-3]
	\arrow["{\alpha _{X_1} \times \dots \times \alpha _{X_n}}"', from=1-1, to=3-1]
	\arrow["{\alpha _Y}", from=1-3, to=3-3]
	\arrow["{M'(f)}", from=3-1, to=3-3]
\end{tikzcd}\]
commutes and the dashed arrow in

\[\begin{tikzcd}
	{M(R)} && {M(X_1)\times \dots \times M(X_n)} \\
	\\
	{M'(R)} && {M'(X_1)\times \dots \times M'(X_n)}
	\arrow[hook, from=1-1, to=1-3]
	\arrow["{\alpha _{X_1} \times \dots \times \alpha _{X_n}}", from=1-3, to=3-3]
	\arrow[hook, from=3-1, to=3-3]
	\arrow[dashed, from=1-1, to=3-1]
\end{tikzcd}\]
exists.

The category of $T$-models and homomorphisms in a category $\mathcal{C}$ is denoted by $T$-$mod(\mathcal{C})$.
\label{modelincat}
\end{definition}

At first we will replace categories with theories:

\begin{definition}
The \emph{canonical language} of the category $\mathcal{C}$ has the signature $L=L_{\mathcal{C}}$ which contains a sort $\bar{A}$ for every object $A$ of $\mathcal{C}$, and a function symbol $\bar{f}:\bar{A}\to \bar{B}$ for every such arrow $f$ of $\mathcal{C}$. Then $\mathcal{C}$ is naturally an $L$-structure by the identical interpretation of $L$ (i.e.~sending $\bar{A}$ to $A$ and $\bar{f}$ to $f$). More generally; each functor $F:\mathcal{C}\to \mathcal{D}$ creates an $L$-structure in $\mathcal{D}$.
\end{definition}

The following theorem (2.4.5. in \cite{makkai}) says, that from inside, $\mathcal{C}$ looks similar to $\mathbf{Set}$.

\begin{theorem}
Assume, that $\mathcal{C}$ has finite limits. Then the following diagrams in $\mathcal{C}$ have the stated properties, iff the sequents on their right side (have interpretation and) are valid (in $\mathcal{C}$, as a structure over its canonical language, with the identical interpretation for the signature).

\begin{tabularx}{\textwidth}{cXX}
1. & $A\xrightarrow{f} A$ is the identity on $A$ & $\Rightarrow f(a)\approx a$ \\
\hline
2. &
\begin{tikzpicture}
\node (3) at (0,0) {$A$};
\node[anchor=base,right=5mm of 3] (2a) {$C$};
\node[anchor=base,below=5mm of 3] (2b) {$B$};
\draw[->] (3)--(2a) node [midway,above] {$h$};
\draw[->] (3)--(2b) node [midway,left] {$f$};
\draw[->] (2b)--(2a) node [midway,below] {$g$};
\end{tikzpicture} is commutative & $\Rightarrow gf(a)\approx h(a)$\\
\hline
3. & $A\xrightarrow{f} B$ is mono & $f(a)\approx f(a') \Rightarrow a\approx a'$ \\
\hline
4. & $A\xrightarrow{f} B$ is surjective & $\Rightarrow \exists a:f(a)\approx b$ \\
\hline
5. & $A$ is the terminal object & $\Rightarrow a\approx a'$ \\ & & $\Rightarrow \exists a: a\approx a$ \\
\hline
6. & $A$ is the initial object & $a\approx a \Rightarrow $ \\
\hline
7. & $A\xleftarrow{f} C\xrightarrow{g} B$ is a product diagram & \small{$f(c)\approx f(c') \wedge g(c)\approx g(c') \Rightarrow c\approx c'$} \\ & & $\Rightarrow \exists c (f(c)\approx a \wedge g(c)\approx b) $ \\
\hline
8. & \begin{tikzpicture}
\node (1) at (0,0) {$E$};
\node[anchor=base,right=5mm of 1] (2) {$A$};
\node[anchor=base,right=5mm of 2] (3) {$B$};
\draw[right hook->] (1)--(2) node [midway,above] {$\epsilon$};
\draw[postaction={transform canvas={yshift=-1mm},draw}] [->] (2) -- (3) node [midway,above] {$f$} node [midway,below] {$g$};
\end{tikzpicture} is an equalizer & $f(a)\approx g(a) \Leftrightarrow \exists e: \epsilon (e)\approx a$ \\
\hline
9. & \makecell[l]{$B\xhookrightarrow{g} X\xhookleftarrow{f_i} A_i$ ($i\in I$). \\ $B$ is the sup of $A_i$-s} & \small{$\bigvee _{i\in I} \exists a_i: f_i(a_i)\approx x \Leftrightarrow \exists b: g(b)\approx x$}\\
\hline
10. & \makecell[l]{$B\xhookrightarrow{g} X\xhookleftarrow{f_i} A_i$ ($i\in I$). \\ $B$ is the inf of $A_i$-s} & \small{$\bigwedge _{i\in I} \exists a_i: f_i(a_i)\approx x \Leftrightarrow \exists b: g(b)\approx x$} \\
\end{tabularx}
\label{diagram}
\end{theorem}

\begin{definition}
Let $\mathcal{C}$ be a coherent category. Its (coherent) \emph{internal theory} $T_{\mathcal{C}}$ (or $Th(\mathcal{C})$) over the signature $L_{\mathcal{C}}$ consists of those sequents which refer to identities, commutative triangles, finite limits, surjective arrows and finite unions (as it is described above).
\end{definition}

\begin{theorem}
The categories $T_{\mathcal{C}}$-$mod(\mathcal{E})$ and $\mathbf{Coh}(\mathcal{C},\mathcal{E})$ are isomorphic (and the isomorphism is given by $M_0^*$ where $M_0:L_{\mathcal{C}}\to \mathcal{C}$ is the identical interpretation).
\end{theorem}

Now we will replace theories with categories. The idea behind the syntactic category is that a model can be seen as a collection of definable sets (evaluations of formulas) together with definable functions (whose graphs are definable). The syntactic category is the natural parametrisation of this category, i.e.~the one whose functorial images are precisely these collections.

The notion of derivability ($\vdash $) refers to a deduction system which is sound wrt.~every coherent category and which is complete wrt.~$\mathbf{Set}$-models, see \cite{makkai} for the details.

\begin{definition}
Let $T$ be a coherent theory. Its \emph{syntactic category} $\mathcal{C}_T$ is defined as follows:
\begin{itemize}
\item The objects are equivalence classes of coherent formulas (in context) over the given signature, where $\varphi (\vec{x})\sim \psi (\vec{y})$, iff $\psi (\vec{y})=\varphi (\vec{y}/ \vec{x})$. Note that $[\varphi (\vec{x})]$ and $[\varphi (\vec{x},\vec{y})]$ (with $y$ being an extra variable not present in $\varphi$) corresponds to different objects. This technicality is not essential, as $[\varphi (\vec{x},y)]$ turns out to be isomorphic with $[\varphi (\vec{x}) \wedge y\approx y]$, hence if we require all variables in the context $\vec{x}$ to appear freely in $\varphi $ we get an equivalent category.
\item An arrow $[\varphi (\vec{x})]\xrightarrow{[\theta (\vec{x},\vec{y})]} [\psi (\vec{y})]$ is an equivalence class of formulas, having the following properties:
\begin{itemize}
\item $\vec{x}$ and $\vec{y}$ are disjoint (this can always be assumed, as we can find such representatives of the objects),
\item $T\vdash \theta(\vec{x},\vec{y}) \Rightarrow \varphi (\vec{x}) \wedge \psi (\vec{y})$,
\item $T\vdash \varphi (\vec{x}) \Rightarrow \exists \vec{y} \theta(\vec{x},\vec{y})$,
\item $T\vdash \theta(\vec{x},\vec{y}) \wedge \theta(\vec{x},\vec{y'}) \Rightarrow \vec{y}=\vec{y'}$.
\end{itemize}
$\theta(\vec{x},\vec{y}) \sim \theta'(\vec{x'},\vec{y'})$, iff $T\vdash \theta(\vec{x},\vec{y}) \Leftrightarrow \theta'(\vec{x},\vec{y})$.
\end{itemize}

The identity arrow $1_{[\varphi (\vec{x}) ]}$ is given as $[\varphi (\vec{x})]\xrightarrow{[\varphi (\vec{x})\wedge \vec{x}\approx \vec{x'}]} [\varphi (\vec{x'})]$. The composition of $[\varphi (\vec{x})]\xrightarrow{[\theta (\vec{x},\vec{y})]} [\psi (\vec{y})] \xrightarrow{[\mu (\vec{y},\vec{z})]} [\chi (\vec{z})]$ is represented by $\exists \vec{y}(\theta (\vec{x},\vec{y}) \wedge \mu (\vec{y},\vec{z}))$.
\end{definition}

\begin{remark}
The required properties for $\theta $ are often referred as being "T-provably functional". This is because these are exactly the conditions which can guarantee, that the interpretation of $\theta $ in a model $M$ is not merely a subobject of $M(\varphi )\times M(\psi )\leq M(\vec{x})\times M(\vec{y})$, but the graph of an arrow from $M(\varphi )$ to $M(\psi )$.
\end{remark}

\begin{theorem}
Given a coherent theory $T$ over a signature $L$, its syntactic category $\mathcal{C}_T$ is a well-defined coherent category. The interpretation $M_0:L\to \mathcal{C}_T$ which maps a sort $X$ to the object $[x\approx x]$ (where $x$ is a variable of sort $X$), an arrow $f:X_1\times \dots \times X_n\to Y$ to $[x_1\approx x_1 \wedge \dots \wedge x_n\approx x_n]\xrightarrow{[f(x_1,\dots x_n)\approx y]}[y\approx y]$ and a relation symbol $R\subseteq X_1\times \dots \times X_n$ to the subobject $[R(x_1,\dots x_n)]\xhookrightarrow{[R(x_1,\dots x_n) \wedge x_1\approx x_1' \wedge \dots x_n\approx x_n']}[x_1'\approx x_1' \wedge \dots x_n'\approx x_n']$ is a model of $T$ with the property that $T\vdash \varphi \Rightarrow \psi $ iff $M_0\models \varphi \Rightarrow \psi$.

The categories $T$-$mod(\mathcal{E})$ and $\mathbf{Coh}(\mathcal{C}_T,\mathcal{E})$ are equivalent. The equivalence is given by $M_0^*$ which takes a coherent functor $F:\mathcal{C}_T\to \mathcal{E}$ to the $L$-structure $F\circ M_0$ which is a model of $T$. The restriction of a natural transformation $\alpha :F\Rightarrow G$ to the image of $M_0$ yields a homomorphism of $L$-structures. Moreover $M_0^*$ is surjective on objects (not just essentially surjective).

If $\mathcal{C}$ is coherent then the interpretation of its canonical language $\mathcal{C}=L_{\mathcal{C}}\to \mathcal{C}_{Th(\mathcal{C})}$ is an equivalence.
\label{bigthm}
\end{theorem}

As a consequence we can prove that the forgetful 2-functor $\mathbf{U}:\mathbf{Coh}_{\sim }\to \mathbf{Cat}_{\sim }$ has a (2-categorical) left adjoint.

\begin{definition}
Let $\mathcal{A}$ be an ordinary category. We can see it as a signature whose sorts are the objects of $\mathcal{A}$ and whose unary function symbols are the morphisms. Then form the theory $Th(\mathcal{A})$ which consists of the sequents corresponding to the commutative triangles and identities in $\mathcal{A}$ (see Theorem \ref{diagram}). Let $\mathbf{F}(\mathcal{A})$ be its syntactic category and $\eta _{\mathcal{A}}:\mathcal{A}\to \mathbf{F}(\mathcal{A})$ be the interpretation of the signature (which is a functor when $\mathcal{A}$ is seen as a category).
\end{definition}

\begin{proposition}
$\mathbf{F}$ extends to a 2-functor $\mathbf{Cat}_{\sim}\to \mathbf{Coh}_{\sim }$ which is left adjoint to $\mathbf{U}$.
\end{proposition}

\begin{proof}
By the previous theorem $\eta _{\mathcal{A}}^*:\mathbf{Coh}(\mathbf{F}(\mathcal{A}), \mathcal{C})\to \mathbf{Cat}(\mathcal{A},\mathbf{U}(\mathcal{C}))\cong Th(\mathcal{A})$-$mod(\mathcal{C})$ is an equivalence of categories. As it reflects isomorphisms we can see it as an equivalence $\mathbf{Coh}_{\sim }(\mathbf{F}(\mathcal{A}), \mathcal{C})\to \mathbf{Cat}_{\sim}(\mathcal{A},\mathbf{U}(\mathcal{C}))$.
\end{proof}

We briefly list the constructions for finite limits, unions and image-facto\-ri\-sations in the syntactic category:

The terminal object is $[\top ]$. Binary products are given as
\[\begin{tikzcd}
	&& {[\chi (\vec{z})]} \\
	\\
	&& {[\varphi (\vec{x}) \wedge \psi (\vec{y})]} \\
	\\
	{[\varphi (\vec{x'})]} &&&& {[\psi (\vec{y'})]}
	\arrow["{[\varphi (\vec{x}) \wedge \psi (\vec{y}) \wedge \vec{x}\approx \vec{x'}]}"{description}, from=3-3, to=5-1]
	\arrow["{[\varphi (\vec{x}) \wedge \psi (\vec{y}) \wedge \vec{y}\approx \vec{y'}]}"{description}, from=3-3, to=5-5]
	\arrow["{[\mu (\vec{z},\vec{x'})]}"', curve={height=18pt}, from=1-3, to=5-1]
	\arrow["{[\nu (\vec{z},\vec{y'})]}", curve={height=-18pt}, from=1-3, to=5-5]
	\arrow[dashed, from=1-3, to=3-3]
\end{tikzcd}\]
where the dashed arrow is represented by $\mu (\vec{z},\vec{x}) \wedge \nu (\vec{z},\vec{y})$.

Equalizers are given as

\[\begin{tikzcd}
	{[\exists y (\mu (x,y)\wedge \nu (x,y))]} &&&& {[\varphi (x')]} && {[\psi (y)]} \\
	\\
	& {[\chi (z)]}
	\arrow["{[\exists y (\mu (x,y)\wedge \nu (x,y)) \wedge x\approx x']}", from=1-1, to=1-5]
	\arrow["{[\mu (x',y)]}", shift left=2, from=1-5, to=1-7]
	\arrow["{[\nu (x',y)]}"', shift right=2, from=1-5, to=1-7]
	\arrow["{[\tau (z,x')]}"', from=3-2, to=1-5]
	\arrow["{[\tau (z,x)]}", dashed, from=3-2, to=1-1]
\end{tikzcd}\]

It can be proved that every subobject of $[\psi (\vec{x})]$ can be represented by a monomorphism of the form $[\varphi  (\vec{x'})]\xrightarrow{[\varphi (\vec{x'}) \wedge \vec{x'}\approx \vec{x}]} [\psi (\vec{x})]$. Then unions are given as  $[\bigvee _i\varphi _i (\vec{x'})]\xrightarrow{[\bigvee _i\varphi _i(\vec{x'}) \wedge \vec{x'}\approx \vec{x}]} [\psi (\vec{x})]$.

Finally image-factorisations can be constructed as
\[\begin{tikzcd}
	{[\varphi (\vec{x})]} &&&& {[\psi (\vec{y})]} \\
	&& {[\exists \vec{x} \mu (\vec{x},\vec{y'})]}
	\arrow["{[\mu (\vec{x},\vec{y})]}", from=1-1, to=1-5]
	\arrow["{[\mu (\vec{x},\vec{y'})]}"'{pos=0.4}, from=1-1, to=2-3]
	\arrow["{[\exists \vec{x} \mu (\vec{x},\vec{y'}) \wedge \vec{y'}\approx \vec{y}]}"'{pos=0.6}, from=2-3, to=1-5]
\end{tikzcd}\]

We recall the idea of pretopos completion from \cite{makkai}.

\begin{definition}
A coherent functor $F:\mathcal{C}\to \mathcal{D}$ is said to be a \emph{weak equivalence} if $F^*:\mathbf{Coh}(\mathcal{D},\mathbf{Set})\to \mathbf{Coh}(\mathcal{C},\mathbf{Set})$ is an equivalence (i.e.~when $F$ is a Morita-equivalence). The class of weak equivalences is denoted by $W$.
\end{definition}

\begin{definition}
A coherent category $\mathcal{C}$ is a pretopos if it has
\begin{itemize}
    \item (finite) disjoint coproducts,
    \item quotients by equivalence relations, i.e.~given a subobject $j:R\hookrightarrow A\times A$ such that the axioms of reflectivity, symmetry and transitivity (see Definition \ref{jdef}) are valid in $\mathcal{C}$ (under the identical interpretation), the coequalizer of $\pi _1 j$ and $\pi _2j $ exists.
\end{itemize}
\end{definition}

\begin{theorem}
Given a coherent category $\mathcal{C}$ there exists a pretopos $\mathbf{R}\mathcal{C}$ and a coherent functor $\rho _{\mathcal{C}}:\mathcal{C}\to \mathbf{R}\mathcal{C}$ such that for any pretopos $\mathcal{S}$ the dashed arrow in
\[\begin{tikzcd}
	{\mathcal{C}} && {\mathbf{R}\mathcal{C}} \\
	\\
	&& {\mathcal{S}}
	\arrow["{\rho _{\mathcal{C}}}", from=1-1, to=1-3]
	\arrow[""{name=0, anchor=center, inner sep=0}, "F"', from=1-1, to=3-3]
	\arrow["\mathbf{R}F", dashed, from=1-3, to=3-3]
	\arrow["\alpha", shorten <=6pt, shorten >=6pt, Rightarrow, from=1-3, to=0]
\end{tikzcd}\]
exists and it is essentially unique: given $F'$ and $\beta :F'\rho _{\mathcal{C}}\Rightarrow F$ there is a unique natural isomorphism $\eta :\mathbf{R}F\Rightarrow F'$ such that the pasting of $\beta $ and $\eta $ is $\alpha $. Moreover $\rho _{\mathcal{C}}$ is a weak equivalence.
\end{theorem}

(This appears as Theorem 8.4.1.~of \cite{makkai}. The fact that $\rho _{\mathcal{C}}$ is a weak equivalence can be found as Proposition 9., Lecture 13.~in \cite{lurie}.)

\begin{corollary}
The full (2,1)-subcategory of pretoposes is reflective in the (2,1)-categorical sense.
\end{corollary}

We recall Makkai's conceptual completeness theorem:

\begin{theorem}
Let $\mathcal{C}$ be a pretopos. A coherent functor $F:\mathcal{C}\to \mathcal{D}$ is an equivalence iff it is a weak equivalence.
\label{conceptual}
\end{theorem}

This can be reformulated as:

\begin{theorem}
A coherent functor $F:\mathcal{C}\to \mathcal{D}$ is a weak equivalence iff $\mathbf{R}F$ is an equivalence.
\end{theorem}

\begin{proof}
Since $W$ clearly has the 2-for-3 property $F$ is a weak equivalence iff $\mathbf{R}F$ is a weak equivalence. By the conceptual completeness theorem the latter is equivalent to $\mathbf{R}F$ being an equivalence.
\end{proof}

As $\mathbf{R}$ is a left adjoint it preserves 2-colimits hence we get:

\begin{corollary}
$W$ is closed under transfinite composition, pushouts and retracts.
\label{wclosed}
\end{corollary}

\section{The model structure}

We will apply the previous results to provide a (2,1)-model structure for $\mathbf{Coh}_{\sim }$, using the (2,1)-categorical small object argument. However, there is another possibility. Theorem 3.3.~of \cite{reflective} proves in the 1-categorical context that given a finitely well-complete category $\mathbf{C}$ (i.e.~finitely complete with all intersections) and a reflective subcategory $\mathbf{A}$, then taking $W$ to be the class of maps inverted by the reflector, the pair $(W,W$-$inj$ $)$ is a factorisation system and any such (so-called reflective) factorisation system results a model structure by taking $W$ to be the class of weak equivalences, $W$-$inj$ to be the class of fibrations and all maps to be cofibrations. Following Remark 3.8.~and Example 2.4.~of \cite{cellular} $W$ is proved to be cofibrantly generated in this case.
The advantage of our approach is that it gives an explicit description of the generating trivial cofibrations and the (2,1)-categorical small object argument should be of independent interest. Using the results of \cite{makkai} we will also prove the resulting model structure to be right proper.

\begin{definition}
Take the signature $L_R $ with one sort $A$ and a binary relation symbol $R\subseteq A\times A$. Let $\mathcal{C}_R$ be the syntactic category of the theory

\begin{tabularx}{\textwidth}{XX}
$\top \Rightarrow R(a,a)$ & ($R$ is reflective) \\
$R(a,a')\Rightarrow R(a',a)$ & ($R$ is symmetric) \\
$R(a,a')\wedge R(a',a'') \Rightarrow R(a,a'')$ & ($R$ is transitive) \\
\end{tabularx}
We set $M_0:L_R \to \mathcal{C}_R$ to denote the canonical interpretation of the signature.

Then add a sort $B$ and a function symbol $p:A\to B$ to the signature (to form $L_{A/R}$) and add the axioms

\begin{tabularx}{\textwidth}{XX}
$\top \Rightarrow \exists a (b\approx p(a))$ & ($p$ is surjective) \\
$R(a,a')\Leftrightarrow p(a)\approx p(a')$ & ($p$ identifies precisely the $R$-equivalent elements) \\
\end{tabularx}
to the theory. Now form its syntactic category $\mathcal{C}_{A/R}$ with the canonical interpretation $M_0':L_{A/R}\to \mathcal{C}_{A/R}$ of the extended signature. Let $I:L_R\to L_{A/R}$ be the inclusion of the signatures. As $M_0'\circ I$ is a model of the defining theory of $\mathcal{C}_R$ by the surjectivity of $M_0^*$ on objects (Theorem \ref{bigthm}) we have a commutative square:
\[\begin{tikzcd}
	{L_R} && {L_{A/R}} \\
	& = \\
	{\mathcal{C}_R} && {\mathcal{C}_{A/R}}
	\arrow["I", from=1-1, to=1-3]
	\arrow["{M_0}"', from=1-1, to=3-1]
	\arrow["{M_0'}", from=1-3, to=3-3]
	\arrow["{M_{A/R}}"', dashed, from=3-1, to=3-3]
\end{tikzcd}\]
(and $M_{A/R}$ is unique up to isomorphism with the property that the above square commutes up to isomorphism, as $M_0^*$ is fully faithful).

Finally take the signature with sorts $A,B,S$ and unary function symbols $i_1:S\to A$ and $i_2:S \to B$. Let $\mathcal{C}_{A\cap B}$ be the syntactic category of the theory
\begin{tabularx}{\textwidth}{XX}
$i_{1,2}(s)\approx i_{1,2}(s') \Rightarrow s\approx s'$ & ($i_1$ and $i_2$ are monic)
\end{tabularx}
Now extend the signature with a new sort $X$ and unary function symbols $j_1:A\to X$, $j_2:B\to X$. Let $\mathcal{C}_{cov}$ be the syntactic category of the theory extending the previous sequents with
\begin{tabularx}{\textwidth}{XX}
$j_{1}(a)\approx j_{1}(a')\Rightarrow a\approx a'$ & ($j_1$ is monic) \\
$j_{2}(b)\approx j_{2}(b')\Rightarrow b\approx b'$ & ($j_2$ is monic) \\
 $ \top \Rightarrow \exists a (x\approx j_1(a)) \vee \exists b (x\approx j_2(b))$  & ($j_1$ and $j_2$ jointly cover $X$) \\
$j_1(a)\approx j_2(b) \Leftrightarrow \exists s: a\approx i_1(s) \wedge b\approx i_2(s)$ & \makecell{($j_1$ and $j_2$ identify precisely \\ the elements of $S$)} \\
\end{tabularx}
Let $M_{cov}:\mathcal{C}_{A\cap B}\to \mathcal{C}_{cov}$ be the evident interpretation (induced as before).

We set $J=\{M_{A/R}, M_{cov} \}$.
\label{jdef}
\end{definition}

\begin{definition}
The elements of $J$-$cof$ are called \emph{trivial cofibrations}, and the elements of $J$-$inj$ are called \emph{fibrations}. A coherent category $\mathcal{C}$ is called \emph{fibrant} if $\mathcal{C}\to *$ is a fibration.
\end{definition}

\begin{theorem}
Any coherent functor $M:\mathcal{C}\to \mathcal{D}$ factors as $M\cong GF$ where $F$ is a trivial cofibration (moreover $F\in J$-cell) and $G$ is a fibration.
\end{theorem}

\begin{proposition}
Trivial cofibrations are closed under 2-pushouts. Fibrations are closed under 2-pullbacks.
\end{proposition}

\begin{proof}
Follows from Proposition \ref{lifting}.
\end{proof}

\begin{proposition}
Let $F:\mathcal{C}\to \mathcal{D}$ be a trivial cofibration. Then there is a map $G:\mathcal{C}\to \mathcal{D}'$ in $J$-cell such that $F$ is the retract of $G$.
\label{retract}
\end{proposition}

\begin{proof}
We can write $F\cong HG$ where $G$ is the strict transfinite composition of pushouts from $J$ and $H$ is a fibration. Then there is a lifting in
\[\begin{tikzcd}
	{\mathcal{C}} && {\mathcal{D}'} \\
	\\
	{\mathcal{D}} && {\mathcal{D}}
	\arrow["F"', from=1-1, to=3-1]
	\arrow[""{name=0, anchor=center, inner sep=0}, "G", from=1-1, to=1-3]
	\arrow["H", from=1-3, to=3-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, Rightarrow, no head, from=3-1, to=3-3]
	\arrow[dashed, from=3-1, to=1-3]
	\arrow[shorten <=18pt, shorten >=18pt, Rightarrow, from=0, to=3-1]
	\arrow[shorten <=22pt, shorten >=13pt, Rightarrow, from=1-3, to=1]
\end{tikzcd}\]
which exhibits $F$ as the retract of $G$.
\end{proof}

\begin{theorem}
$J$-$cof$$=W$
\end{theorem}

\begin{proof}
$\subseteq $ By Corollary \ref{wclosed} it is enough to prove $J\subseteq W$. Let $N:\mathcal{C}_{R}\to \mathbf{Set}$ be a coherent functor. Then $NM_0$ gives a set $N(A)$ with an equivalence relation $N(R)$ on it. If we want to extend this to an $L_{A/R}$-structure which is a model of the additional axioms we are forced to interpret $p$ as the coequalizer of $\restr{\pi _1}{N(R)}$ and $\restr{\pi _2}{N(R)}$. A homomorphism of $L_R$-structures (which are models of the axioms on $R$) is a commutative diagram
\[\begin{tikzcd}
	{N(R)} && {N(A)\times N(A)} \\
	& {=} \\
	{N'(R)} && {N'(A)\times N'(A)}
	\arrow[hook, from=1-1, to=1-3]
	\arrow[hook, from=3-1, to=3-3]
	\arrow["{h\times h}"{description}, from=1-3, to=3-3]
	\arrow[dashed, from=1-1, to=3-1]
\end{tikzcd}\]
which induces a unique map between the coequalizers. The case of $M_{cov}$ is analogous.

$\supseteq $ Let $H:\mathcal{C}\to \mathcal{D}$ be a weak equivalence. We know from \cite{makkai} that $H$ is fully faithful, conservative, full wrt.~subobjects and each object $x\in \mathcal{D}$ is finitely covered by $\mathcal{C}$ via $H$, i.e.~there are objects $y_1,\dots y_n$ in $\mathcal{C}$, subobjects $b_k\hookrightarrow H(y_k)$ and maps $p_k:b_k\to x$ such that $x=\bigcup \exists _{p_k}b_k$. Since $H$ is full wrt.~subobjects we can take $b_k=H(y_k)$.

Let $(d_i)_{i<\lambda }$ be a well-ordering of objects of $\mathcal{D}$. By transfinite recursion we will give factorisations $H\cong \mathcal{C}\xrightarrow{F_i}\mathcal{D}_i\xrightarrow{G_i}\mathcal{D}$ such that $F_i\in J$-cell and for each $j<i$ $d_j$ is contained in the essential image of $G_i$. This is sufficient as $J-cell \subseteq W $ so by the 2-out-3 property $G_{\lambda }$ is an essentially surjective weak equivalence, i.e.~an equivalence.

We take $\mathcal{D}_0=\mathcal{C}, F_0=1_{\mathcal{C}}, G_0=h$. When $i$ is a limit ordinal let $F_i$ be the transfinite composition of $(F_j)_{j<i}$ and $G_i$ be the induced map. Now assume that $\mathcal{C}\xrightarrow{F_i}\mathcal{D}_i\xrightarrow{G_i}\mathcal{D}$ is given. We can find $y_1,\dots y_n$ in $\mathcal{D}_i$ such that in
\[\begin{tikzcd}
	{G_i(y_1)} \\
	\dots && {d_i} \\
	{G_i(y_n)}
	\arrow["{p_1}"{description}, from=1-1, to=2-3]
	\arrow["{p_n}"{description}, from=3-1, to=2-3]
\end{tikzcd}\]
$d_i=\bigcup _k \exists _{p_k}G_i(y_k)$. First we glue the quotient maps $q_i:G_i(y_k)\twoheadrightarrow \exists _{p_k}G_i(y_k)$ to $\mathcal{D}_i$, i.e.~take the 2-pushout and the induced map in
\[\begin{tikzcd}
	{\mathcal{C}_R} & {\mathcal{D}_i} \\
	{\mathcal{C}_{A/R}} & {\mathcal{D}_i^1} \\
	&&& {\mathcal{D}}
	\arrow["{M_{A/R}}"', from=1-1, to=2-1]
	\arrow[from=1-1, to=1-2]
	\arrow[from=1-2, to=2-2]
	\arrow[from=2-1, to=2-2]
	\arrow[curve={height=6pt}, from=2-1, to=3-4]
	\arrow["{G_i}", curve={height=-6pt}, from=1-2, to=3-4]
	\arrow["{G_i^1}"{description}, dashed, from=2-2, to=3-4]
\end{tikzcd}\]
where $R\hookrightarrow A$ is mapped to the subobject $R'\hookrightarrow y_1$ whose $G_i$-image is the kernel pair of $q_1$ (and which is an equivalence relation as $G_i$ is bijective on the subobject lattices) and $A\xrightarrow{p}B$ is mapped to $q_1$. The dashed arrow is a weak equivalence and $q_1$ lies in its essential image. Iterating it $n$ times yields a factorisation $H\cong \mathcal{C}\xrightarrow{F_i^n}\mathcal{D}_i^n\xrightarrow{G_i^n} \mathcal{D}$ with $F_i^n\in J$-cell and with $(d_j)_{j<i}$ and $\exists _{p_1}G_i(y_1),\dots \exists _{p_n}G_i(y_n)$  all lying in the essential image of $G_i^n $.

Using that $G_i^n$ is a weak equivalence and hence it is full wrt.~subobjects and it is fully faithful we have a covering of $d_i$
\[\begin{tikzcd}
	&& {d_i} \\
	{G_i^n(z_1)} && {G_i^n(z_2)} && \dots \\
	{G_i^n(z_{12})} & {G_i^n(z_1)\cap G_i^n(z_2)} & {G_i^n(z_{12}')}
	\arrow["{G_i^n(j_{12})}", hook, from=3-1, to=2-1]
	\arrow[hook, from=2-1, to=1-3]
	\arrow[hook, from=2-3, to=1-3]
	\arrow[hook', from=3-2, to=2-1]
	\arrow[hook, from=3-2, to=2-3]
	\arrow["\cong", from=3-1, to=3-2]
	\arrow["\cong", from=3-2, to=3-3]
	\arrow["{G_i^n(j_{12}')}"', hook, from=3-3, to=2-3]
	\arrow["{G_i^n(\varphi)}"{description}, curve={height=12pt}, from=3-1, to=3-3]
\end{tikzcd}\]
Now we just glue the unions of the elements of the cover to $\mathcal{D}_i^n$. Take the pushout and the induced map in
\[\begin{tikzcd}
	{\mathcal{C}_{A\cap B}} & {\mathcal{D}_i^n} \\
	{\mathcal{C}_{cov}} & {\mathcal{D}_i^{n+1}} \\
	&&& {\mathcal{D}}
	\arrow["{M_{cov}}"', from=1-1, to=2-1]
	\arrow[from=1-1, to=1-2]
	\arrow[from=1-2, to=2-2]
	\arrow[from=2-1, to=2-2]
	\arrow[curve={height=6pt}, from=2-1, to=3-4]
	\arrow["{G_i^n}", curve={height=-6pt}, from=1-2, to=3-4]
	\arrow["{G_i^{n+1}}"{description}, dashed, from=2-2, to=3-4]
\end{tikzcd}\]
where $A\leftarrow S\to B$ is sent to $z_1\xleftarrow{j_{12}}z_{12}\xrightarrow{j'_{12}\varphi }z_2$ and $A\to X\leftarrow B$ is mapped to $G_i^n(z_1)\to d_i \leftarrow G_i^n(z_2)$. It follows that $d_i$ is covered by $n-1$ elements from the essential image of $G_i^{n+1}$. We can take $\mathcal{D}_{i+1}=\mathcal{D}_i^{2n-1}$, $F_{i+1}=F_i^{2n-1}$ and $G_{i+1}=G_i^{2n-1}$.
\end{proof}

\begin{corollary}
Every map $F\in W\cap J$-$inj$ is an equivalence.
\end{corollary}

\begin{proof}
As $W=J$-$cof$ we have a lift in
\[\begin{tikzcd}
	\bullet & \bullet \\
	\bullet & \bullet
	\arrow["F"', from=1-1, to=2-1]
	\arrow[Rightarrow, no head, from=1-1, to=1-2]
	\arrow["F", from=1-2, to=2-2]
	\arrow[Rightarrow, no head, from=2-1, to=2-2]
	\arrow[dashed, from=2-1, to=1-2]
\end{tikzcd}\]
\end{proof}

Following the proof of Proposition 2.3.~in \cite{torsion} we get:

\begin{proposition}
The lift in the square
\[\begin{tikzcd}
	{\mathcal{A}} && {\mathcal{C}} \\
	\\
	{\mathcal{B}} && {\mathcal{D}}
	\arrow["F"', from=1-1, to=3-1]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{H_1}", from=1-1, to=1-3]
	\arrow["G", from=1-3, to=3-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{H_2}"', from=3-1, to=3-3]
	\arrow["K"{description}, from=3-1, to=1-3]
	\arrow["\alpha", shorten <=18pt, shorten >=13pt, Rightarrow, from=3-1, to=0]
	\arrow["\beta"', shorten <=13pt, shorten >=18pt, Rightarrow, from=1, to=1-3]
\end{tikzcd}\]
with $F\in W$, $G\in J$-$inj$ is essentially unique: given $K':\mathcal{B}\to \mathcal{C}$ and $\alpha '$, $\beta '$ with $\beta F \circ G\alpha =\beta 'F \circ G \alpha '$ there is a unique natural isomorphism $\gamma :K\Rightarrow K'$ for which $\alpha =\gamma F \circ \alpha '$ and $\beta '=G\gamma \circ \beta $.
\end{proposition}

\begin{proof}
The 2-cells
\[\begin{tikzcd}
	{\mathcal{A}} & {\mathcal{B}} &&&& {\mathcal{A}} & {\mathcal{B}} \\
	{\mathcal{B}} & {\mathcal{B}'} && {\mathcal{B}} & {\mathcal{D}} & {\mathcal{B}} & {\mathcal{B}'} && {\mathcal{C}} & {\mathcal{D}}
	\arrow["F"', from=1-1, to=2-1]
	\arrow[""{name=0, anchor=center, inner sep=0}, "F", from=1-1, to=1-2]
	\arrow["{I_1}", from=1-2, to=2-2]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{I_2}", from=2-1, to=2-2]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{1_{\mathcal{B}}}"'{pos=0.8}, curve={height=24pt}, from=2-1, to=2-4]
	\arrow[""{name=3, anchor=center, inner sep=0}, "{1_{\mathcal{B}}}", curve={height=-6pt}, from=1-2, to=2-4]
	\arrow[""{name=4, anchor=center, inner sep=0}, "{F'}"{description}, dashed, from=2-2, to=2-4]
	\arrow["{H_2}", from=2-4, to=2-5]
	\arrow[""{name=5, anchor=center, inner sep=0}, "F", from=1-6, to=1-7]
	\arrow["F"', from=1-6, to=2-6]
	\arrow["{I_1}", from=1-7, to=2-7]
	\arrow[""{name=6, anchor=center, inner sep=0}, "{I_2}", from=2-6, to=2-7]
	\arrow[""{name=7, anchor=center, inner sep=0}, "{K'}"{description}, curve={height=18pt}, from=2-6, to=2-9]
	\arrow[""{name=8, anchor=center, inner sep=0}, "K"{description}, curve={height=-6pt}, from=1-7, to=2-9]
	\arrow[""{name=9, anchor=center, inner sep=0}, "R"{description}, dashed, from=2-7, to=2-9]
	\arrow["G", from=2-9, to=2-10]
	\arrow[""{name=10, anchor=center, inner sep=0}, "{H_2}", curve={height=-12pt}, from=1-7, to=2-10]
	\arrow[""{name=11, anchor=center, inner sep=0}, "{H_2}"', curve={height=30pt}, from=2-6, to=2-10]
	\arrow["push"{description}, Rightarrow, draw=none, from=0, to=1]
	\arrow["push"{description}, Rightarrow, draw=none, from=5, to=6]
	\arrow["{\varepsilon_1}"{description}, Rightarrow, draw=none, from=3, to=2-2]
	\arrow["{\mu_1}"{description}, Rightarrow, draw=none, from=8, to=2-7]
	\arrow["{\mu_2}"{description, pos=0.4}, Rightarrow, draw=none, from=9, to=7]
	\arrow["{\varepsilon _2}"{description}, Rightarrow, draw=none, from=4, to=2]
	\arrow["\beta"{description}, Rightarrow, draw=none, from=10, to=2-9]
	\arrow["{\beta '}"{description}, Rightarrow, draw=none, from=2-9, to=11]
\end{tikzcd}\]
are identical, i.e.~we have $H_2F\xRightarrow{\beta 'F} GK'F \xRightarrow{G \alpha ^{-1} \circ G \alpha '}GKF \xRightarrow{\beta ^{-1}F} H_2F$ equals $1_{H_2F}$. By the universal property of the pushout there is a unique 2-cell
\[\begin{tikzcd}
	{\mathcal{B}'} && {\mathcal{C}} \\
	\\
	{\mathcal{B}} && {\mathcal{D}}
	\arrow["{F'}"', from=1-1, to=3-1]
	\arrow["R", from=1-1, to=1-3]
	\arrow["G", from=1-3, to=3-3]
	\arrow["{H_2}"', from=3-1, to=3-3]
	\arrow["\eta"{description}, shorten <=11pt, shorten >=11pt, Rightarrow, from=3-1, to=1-3]
\end{tikzcd}\]
with $H_2\xRightarrow{\beta } GK \xRightarrow{ G\mu _1}GRI_1 \xRightarrow{\eta ^{-1}I_1}H_2F'I_1$ equal to $H_2\varepsilon _1$ and with $H_2\xRightarrow{\beta '}GK \xRightarrow{G\mu _2} GRI_2\xRightarrow{ \eta ^{-1}I_2}H_2F'I_2 $ equal to $H_2\varepsilon _2$.

As weak equivalences are closed under 2-pushouts and by the 2-for-3 property the map $F'$ is a weak equivalence, hence we have a lift
\[\begin{tikzcd}
	{\mathcal{B}'} && {\mathcal{C}} \\
	\\
	{\mathcal{B}} && {\mathcal{D}}
	\arrow[""{name=0, anchor=center, inner sep=0}, "R", from=1-1, to=1-3]
	\arrow["G", from=1-3, to=3-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{H_2}"', from=3-1, to=3-3]
	\arrow["L"{description}, from=3-1, to=1-3]
	\arrow["{F'}"', from=1-1, to=3-1]
	\arrow["{\nu _1}", shorten <=18pt, shorten >=13pt, Rightarrow, from=3-1, to=0]
	\arrow["{\nu _2}"', shorten <=18pt, shorten >=13pt, Rightarrow, from=1, to=1-3]
\end{tikzcd}\]
with $G\nu _1 \circ \nu _2 F'= \eta $.

Then we have isomorphisms:
\[
\gamma : K\xRightarrow{\mu _1} RI_1 \xRightarrow{\nu _1^{-1}I_1} LF'I_1 \xRightarrow{L\varepsilon _1^{-1}} L \xRightarrow{L\varepsilon _2} LF'I_2 \xRightarrow{\nu _1 I_2} RI_2 \xRightarrow{\mu _2^{-1}} K'
\]
for which $\alpha =\gamma F \circ \alpha '$ and $\beta '=G\gamma \circ \beta $ is easily checked.

Given $\gamma '$ with these properties we get $\delta \neq \delta ':K\Rightarrow L$ then $\eta \neq G\eta _1 \circ G \delta ^{-1}\delta ' F' \circ \nu _2 F'$ are both compatible 2-cells which contradicts the universal property of the 2-pushout.
\end{proof}

\begin{proposition}
Assume $GF\cong H$. Then if $G,H$ are fibrations, the map $F$ is also a fibration.
\label{cancellation}
\end{proposition}

\begin{proof}
We have a lift $L$ in
\[\begin{tikzcd}
	&& \bullet \\
	\bullet \\
	&& \bullet \\
	\bullet & {=} \\
	&& \bullet
	\arrow["{M \in J}"', from=2-1, to=4-1]
	\arrow["F", from=1-3, to=3-3]
	\arrow["G", from=3-3, to=5-3]
	\arrow["K"{description}, from=2-1, to=1-3]
	\arrow["H"{description}, from=4-1, to=3-3]
	\arrow["GH"{description}, from=4-1, to=5-3]
	\arrow["\alpha", shorten <=28pt, shorten >=28pt, Rightarrow, from=4-1, to=1-3]
	\arrow["L"{description}, curve={height=12pt}, dashed, from=4-1, to=1-3]
\end{tikzcd}\]
with isomorphisms $\nu _1:LM\Rightarrow K$, $\nu _2:GH\Rightarrow GFL$ such that $\nu _2 M \circ GF\nu _1 =G\alpha $. When seeing it as a square whose right edge is $G$ we have two liftings $FL$ with 2-cells $\nu _1$, $\nu _2$ and $H$ with 2-cells $\alpha $, $1_{GH}$. By the previous proposition we get a unique isomorphism $\gamma :H\Rightarrow FL$ with $\gamma M \circ F\nu _1=\alpha $ and $G\gamma =\nu _2$ so $(L,\nu _1, \gamma )$ is a splitting of the 2-cell $\alpha $.
\end{proof}

\begin{definition}
A \emph{path object} for a coherent category $\mathcal{C}$ is the factorisation of $\Delta :\mathcal{C}\to \mathcal{C}\times \mathcal{C}$ as $\mathcal{C}\xrightarrow{I_0} \mathcal{C}' \xrightarrow{\langle P_0,P_1 \rangle } \mathcal{C}\times \mathcal{C}$ where $I_0$ is a weak equivalence and $\langle P_0,P_1 \rangle $ is a fibration.

Let $\mathcal{C},\mathcal{D}$ be fibrant. A \emph{homotopy} between two coherent functors $F,G:\mathcal{C}\to \mathcal{D}$ is a coherent functor $K:\mathcal{C}\to \mathcal{D}'$ with $P_0K\cong F$ and $P_1K\cong G$. We say that $F$ and $G$ are homotopic (and write $F\simeq G$) if there is a homotopy between them with some path object.
\end{definition}

\begin{theorem}
There is a (2,1)-model structure on $\mathbf{Coh}_{\sim }$ with $W$ being the class of weak equivalences, $Fib=J$-$inj$ and with all maps being cofibrations. A coherent category is fibrant iff it is a pretopos. Two maps $F,G:\mathcal{C}\to \mathcal{D}$ between pretoposes are homotopic iff they are naturally isomorphic.
\label{mainthm}
\end{theorem}

\begin{proof}
$W$ satisfies the 2-out-3 and the retract axioms, the elements of $J$-$inj$ have the right lifting property wrt.~$W\cap Cof =W=J$-$cof$, equivalences (i.e.~elements of $W\cap Fib$) have the right lifting property wrt.~anything and the existence of the nontrivial factorisation system has already been noticed and it follows from the (2,1)-categorical small object argument.

$\mathcal{C}\to *$ is a fibration iff it reflects quotients by equivalence relations and pushouts of monomorphisms along monomorphisms which are also pullbacks, equivalently if these constructions exist in $\mathcal{C}$. Taking monomorphisms with initial domain this implies the existence of disjoint coproducts (which are pullback-stable by the stability of unions). To see the converse we will show that any map out of a pretopos is a fibration. I.e.~let
\[\begin{tikzcd}
	{\mathcal{C}} && {\mathcal{P}} \\
	& {\mathcal{Q}} \\
	{\mathcal{D}} && {\mathcal{E}}
	\arrow["F", from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow["{F'}"', from=1-3, to=2-2]
	\arrow[from=3-1, to=2-2]
	\arrow["G", from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
	\arrow[dashed, from=2-2, to=3-3]
\end{tikzcd}\]
be a (homotopy) commutative square where $\mathcal{P}$ is a pretopos and $F$ is a weak equivalence, with $\mathcal{Q}$ being the 2-pushout. Then $F'$ is a weak equivalence, hence it is an equivalence as $\mathcal{P}$ is a pretopos (using Makkai's conceptual completeness: Theorem 7.1.8.~in \cite{makkai}). Its quasi-inverse gives a splitting of the original 2-cell.

It follows that when $\mathcal{D}$ is a pretopos the diagonal $\Delta :\mathcal{D}\to \mathcal{D}\times \mathcal{D}$ is a fibration, hence $(1_{\mathcal{D}},\Delta )$ is a path object for $\mathcal{D}$.

\end{proof}

\begin{remark}
The (2,1)-categorical analogue of Whitehead's theorem says that a map between objects that are both fibrant and cofibrant, is a weak equivalence iff it is a homotopy equivalence. Therefore the given model structure exhibits conceptual completeness (Theorem \ref{conceptual}, for coherent functors where the codomain is also a pretopos) as an instance of Whitehead's theorem.

Fibrant replacement gives pretopos completion.
\end{remark}

\begin{remark}
It is not surprising that there is a set $J$ of generating trivial cofibrations for which pretoposes are precisely the fibrant objects. Indeed, the full subcategory of pretoposes is an accessibly embedded accessible subcategory of $\mathbf{Coh}_{\sim }$ which is closed under finite products, and $\mathbf{Coh}_{\sim }$ is locally presentable in the 2-dimensional sense. Then by a 2-dimensional version of Theorem 4.8.~of \cite{rosicky} pretoposes form a small-injectivity class.
\end{remark}

\begin{proposition}
Fully faithful functors are closed under 2-pullbacks in $\mathbf{Coh}_{\sim }$.
\label{fullf}
\end{proposition}

\begin{proof}
By \cite{codescent} bijective-on-objects and fully faithful functors form a 2-categorical orthogonal factorisation system on $\mathbf{Cat}_{\sim }$, hence the right class is stable under 2-pullbacks. By \cite{flexible} the forgetful functor $\mathbf{U}:\mathbf{Coh}_{\sim }\to \mathbf{Cat}_{\sim }$ preserves and reflects 2-limits.
\end{proof}

\begin{proposition}
$F:\mathcal{C}\to \mathcal{D}$ is a fibration iff
\[\begin{tikzcd}
	{\mathcal{C}} && {\mathbf{R}\mathcal{C}} \\
	\\
	{\mathcal{D}} && {\mathbf{R}\mathcal{D}}
	\arrow["F"', from=1-1, to=3-1]
	\arrow["{\rho _{\mathcal{C}}}", from=1-1, to=1-3]
	\arrow["\mathbf{R}F", from=1-3, to=3-3]
	\arrow["{\rho _{\mathcal{D}}}"', from=3-1, to=3-3]
\end{tikzcd}\]
is a 2-pullback.
\end{proposition}

\begin{proof}
Since any map out of a pretopos is a fibration the "if" direction is immediate. For the converse assume that $F$ is a fibration and take the pullback
\[\begin{tikzcd}
	{\mathcal{C}} \\
	& S && {\mathbf{R}\mathcal{C}} \\
	\\
	& {\mathcal{D}} && {\mathbf{R}\mathcal{D}}
	\arrow["F"', curve={height=12pt}, from=1-1, to=4-2]
	\arrow["{\rho _{\mathcal{C}}}", curve={height=-12pt}, from=1-1, to=2-4]
	\arrow["\mathbf{R}F", from=2-4, to=4-4]
	\arrow["{\rho _{\mathcal{D}}}"', from=4-2, to=4-4]
	\arrow["{F'}"', from=2-2, to=4-2]
	\arrow["U"', from=2-2, to=2-4]
	\arrow["V"', dashed, from=1-1, to=2-2]
\end{tikzcd}\]

We should prove that $V$ is an equivalence. As both $F$ and $F'$ are fibrations $V$ is a fibration by Proposition \ref{cancellation}. Hence it suffices to prove that $V$ is a weak equivalence.

By \cite{makkai} any weak equivalence is fully faithful. By Proposition \ref{fullf} $U$ is fully faithful and since fully faithful functors form the right class of a factorisation system on $\mathbf{Cat}_1$ we get that so is $V$. Then it suffices to prove essential surjectivity.

Take $x\in S$. Referring to \cite{makkai} again $Ux$ is finitely covered by $\mathcal{C}$ via $\rho _{\mathcal{C}}$, i.e.~there are objects $c_1,\dots c_n\in \mathcal{C}$ and maps $f_i:\rho _{\mathcal{C}}(c_i)\to Ux$ such that $\bigvee _i \exists _{f_i}\rho _{\mathcal{C}}c_i =Ux$. Using that $U$ is fully faithful we get maps $g_i:Vc_i\to x$ with the same property.

As $\rho _{\mathcal{C}}$ is full wrt.~subobjects the composite $Sub(c)\hookrightarrow Sub(Vc)\hookrightarrow Sub(UVc)$ is surjective, therefore both maps are bijections, in particular $V$ is full wrt.~subobjects. Taking $k_i\hookrightarrow Vc_i\times Vc_i $ to be the kernel pair of $g_i$ we have that it is coming from an equivalence relation on $c_i\times c_i$. As $V$ has the right lifting property against $M_{A/R}$ we can choose $g_i$'s to be monomorphisms. Then orthogonality against $M_{cov}$ completes the proof.

\end{proof}

\begin{corollary}
The model structure of Theorem \ref{mainthm} is right proper.
\end{corollary}

\begin{proof}
By the pasting law of pullbacks and by the previous proposition $\rho _{\mathcal{B}} U'$ is a weak equivalence.

\[\begin{tikzcd}
	{\mathcal{S}} && {\mathcal{B}} && {\mathbf{R}\mathcal{B}} & {\mathbf{R}\mathcal{S}} \\
	& pb && pb \\
	{\mathcal{A}} && {\mathcal{C}} && {\mathbf{R}\mathcal{C}} & {\mathbf{R}\mathcal{A}}
	\arrow["U"', from=3-1, to=3-3]
	\arrow["{F'}"', from=1-1, to=3-1]
	\arrow["{U'}", from=1-1, to=1-3]
	\arrow["F"', from=1-3, to=3-3]
	\arrow["{\rho _{\mathcal{B}}}", from=1-3, to=1-5]
	\arrow["{\rho _{\mathcal{C}}}"', from=3-3, to=3-5]
	\arrow["{\mathbf{R}F}", from=1-5, to=3-5]
	\arrow["{\mathbf{R}F'}", from=1-6, to=3-6]
	\arrow["\simeq", from=1-5, to=1-6]
	\arrow["\simeq"', from=3-5, to=3-6]
\end{tikzcd}\]
\end{proof}

\begin{corollary}
Any square
\[\begin{tikzcd}
	{\mathcal{S}} && {\mathcal{B}} \\
	& \cong \\
	{\mathcal{A}} && {\mathcal{C}}
	\arrow["U"', from=3-1, to=3-3]
	\arrow["{U'}", from=1-1, to=1-3]
	\arrow["F", from=1-3, to=3-3]
	\arrow["{F'}"', from=1-1, to=3-1]
\end{tikzcd}\]
with $U,U'$ being weak equivalences and $F,F'$ being fibrations is a 2-pullback.
\end{corollary}

\begin{proof}
Let $V:\mathcal{S}\to \mathcal{S}'$ be the connecting map to the 2-pullback of $F$ and $U$. Then by the cancellation property of fibrations and by the 2-for-3 property of weak equivalences we get $V\in W\cap J$-$inj$ hence it is an equivalence.
\end{proof}

\printbibliography
\end{document}

redpajama_set_name: RedPajamaArXiv


text:
£21.7m supported around 5,000 jobs each month in winter
£21.7m was paid out to businesses between September to December last year as part of the Government's job retention scheme - with hospitality taking the biggest slice of support.
The latest batch of stats from the Government give an insight into how local business fared during the winter months, with December topping the numbers with over a third of the payments.
The Co-Funded Payroll scheme was initially set up in April 2020, to help support businesses impacted by pandemic restrictions by offering to give employers a subsidy to help pay wages, if they could prove "material detriment."
The Government later released a breakdown of the payments, showing that between March and August 2020, £81m was paid out to local businesses, supporting 67,137 jobs.
Pictured: Hospitality made up the most payments for this quarter, with £9.5m being handed out.
With the scheme then extended into the winter months of 2020, the Government have now released a report giving an updated list of payments for September to December 2020, which saw around 5,000 jobs supported each month.
December - which saw all licensed premises close in a sudden 'circuit breaker' response to rising cases - makes up for a substantial amount of this quarter's payments, totalling at £7.6m for over 1,500 businesses.
Across the four months, hotels, restaurants and bars were once more the main claimants, being given £9,512,967.38 in payments.
However, unlike March to August, when construction was ranked the second most supported sector, things have now changed, with Education, Health and Other Services now in that spot with £2,718,577.46.
Of the claimants, the highest was the Royal Yacht, which claimed £600,387.60 to support up to 170 jobs over 4 months.
The Government also said that they have recovered £3,321,666.45 in repayments from 331 companies.
However, they have not confirmed whether this is part of the initial £5.3m they requested back from the first round of payments, or whether this is a separate amount from the winter payments.
The Co-funded Payroll Scheme is currently due to expire at the end of August. However, with high levels of covid in the community now having an impact, the recent push back of the final covid relaxations, businesses have been questioning whether it will be extended further.
Economic Development Minister Senator Lyndon Farnham told Express yesterday that he was discussing the possibility with his officers.
Sector breakdown for all businesses receiving payroll support between September and December...
Hotels, restaurants and bars - £9,512,967.38
Education, Health and Other Services - £2,718,577.46
Miscellaneous Business Activities - £2,285,390.15
Construction - £2,129,949.67
Wholesale and Retail - £2,058,081.34
Transport and Storage - £1,834,493.56
Manufacturing - £351,922.67
Information and Communication - £273,150.75
Agriculture and Fisheries - £266,781.95
Finance and Legal Activities - £146,977.21
Utilities & Waste - £80,183.79
The top 50 claimants...
Royal Yacht Hotel - £600,387.60 supporting up to 170 jobs over 4 months.
Hotel De France Limited - £383,185.26 supporting up to 89 jobs over 4 months.
The Grand Holdings Ltd - £326,349.07 supporting up to 81 jobs over 4 months.
Hotel L'Horizon Limited - Trading As L'Horizon Beach Hotel & Spa - £306,431.83 supporting up to 73 jobs over 4 months.
Randalls Limited - £283,689.85 supporting up to 238 jobs in December.
Swissport Jersey Ltd - £272,670.17 supporting up to 99 jobs over 4 months.
Modern Hotels (Holdings) Ltd - £272,066.40 supporting up to 69 jobs over 4 months.
Seymours (Services) Limited - £271,484.77 supporting up to 82 jobs over 4 months.
Pomme D'Or Hotel (1932) Limited - £262,629.26 supporting up to 63 jobs over 4 months.
St. Brelade's Bay Hotel Properties Limited - £248,416.50 supporting up to 78 jobs over 3 months.
Blue Islands Ltd - £240,191.46 supporting up to 53 jobs over 4 months.
Jersey Rugby Football Club (2016) Trading - £229,886.98 supporting up to 50 jobs over 4 months.
Radisson Blu Waterfront Hotel - £228,539.52 supporting up to 65 jobs over 4 months.
Liberation Group - £217,055.81 supporting up to 144 jobs over December.
Angorabesco Ltd - £212,497.69 supporting up to 56 jobs over 4 months.
Longueville Manor Hotel Ltd - £211,970.01 supporting up to 60 jobs over 4 months.
Merton Hotel Limited - £198,518.26 supporting up to 53 jobs over 4 months.
The Club Hotel & Spa - £163,301.86 supporting up to 39 jobs over 4 months.
A De Gruchy & Co Ltd - £161,603.18 supporting up to 109 jobs over 2 months.
Cimandis Limited - £139,177.67 supporting up to 30 jobs over 4 months.
J & C Dry Lining Contractors Ltd - £132,390.32 supporting up to 28 jobs over 4 months.
JP Restaurants - £128,528.86 supporting up to 62 jobs over 2 months.
Airline Services (CI) Limited - £125,013.82 supporting up to 52 jobs over 4 months.
Hotel Savoy - £123,855.77 supporting up to 31 jobs over 4 months.
J P Mauger Ltd - £119,115.71 supporting up to 62 jobs over 2 months.
Somerville Hotel - £112,745.65 supporting up to 33 jobs over 3 months.
Hotel La Place - £107,775.14 supporting up to 31 jobs over 3 months.
Pizza Express (Jersey) Limited - £106,794.06 supporting up to 38 jobs over 4 months.
AC Mauger & Son Limited - £103,433.58 supporting up to 26 jobs over 4 months.
Apollo Hotel - £101,615.23 supporting up to 30 jobs over 4 months.
Gordon Electrical Services Limited - £98,937.41 supporting up to 27 jobs over 4 months.
Mange Tout - £98,849.93 supporting up 27 jobs over 4 months.
Mailmate Business Forms (C.I.) Limited - £96,644.99 supporting up to 31 jobs over 3 months.
Les Mielles Golf and Country Club - £95,699.22 supporting up to 31 jobs over 4 months.
C I Travel Group Limited - £92,680.99 supporting up to 37 jobs over 4 months.
Ommaroo Hotel Limited - £90,758.08 supporting up to 24 jobs over 4 months.
Hotel Revere - £90,703.96 supporting up to 29 jobs over 4 months.
Les Ormes Golf & Leisure Village - £89,694.49 supporting up to 45 jobs over 2 months.
Jersey Society For The Prevention Of Cruelty To Animals - £88,474.44 supporting up to 31 jobs over 3 months.
Jersey Oyster Company Limited - £87,155.65 supporting up to 27 jobs over 3 months.
Royal Hotel Limited - £83,928.46 supporting up to 22 jobs over 4 months.
The Atlantic Hotel - £83,231.01 supporting up to 36 jobs over 2 months.
WDFG Jersey Lted - £82,118.36 supporting up to 20 jobs over 4 months.
Capital Properties Ltd T/A Wildfire Restaurant & Bars - £81,904.26 supporting up to 27 jobs over 4 months.
Hotel De Normandie - £76,082.45 supporting up to 34 jobs over 3 months.
Hotel Ambassadeur (1984) Ltd - £75,648.19 supporting up to 19 jobs over 4 months.
Chateau La Chaire & The Rozel Pub & Dining - £74,726.06 supporting up to 19 jobs over 4 months.
Premier Inn - £72,928.52 supporting up to 24 jobs over 4 months.
Serco (Jersey) Ltd - £71,405.75 supporting up to 42 jobs over 3 months.
The Mansell Collection - £69,913.52 supporting up to 17 jobs over 4 months.
Posted by	Keith Marsh	on	Jul 16th, 2021
This is fine, providing the government checks the accounts of all those who received tax payer funding to ensure that a true need was in fact there.
Politician pushes to reverse hospital visitor ban
Guernsey Chief Minister blasts Private Eye 'Covid Hypocrite' award
France lifts travel ban for Channel Islands travellers

redpajama_set_name: RedPajamaCommonCrawl


How the economy should be dealt with.
Do we choose economics or ethics?
Women should be paid equally as men.
My concern with the countries economic problem.
Poverty is a major issue in today's society, but we are combating it in the wrong ways.
The gender wage gap and the rich poor gap are making our economy unequal and the minimum wage should be raised.
Our Economy is damaged, do you have the power and courage to change it for the better, or are you going to just let it slip?
I am here to talk to you (the future president) about the us economy.


Herewith a Pak 128 savegame. It's abt 1,2 Mb, now running with P508 and one of the latest nightlies ( R2344).
There are 9 cities with a total of 450.000 inhabitants.
Every city has at least 1 supermarket, 1 cardealer and 1 fuelstation and for every kind of product there is only one factory. See screenshot.
Everything connected by abt. 650 vehicles in 140 lines.
Hope you enjoy it, all comments are welcome!
Thanks I will try, perhaps not before monday.
It should be fun to see all the vehicules running.
I don't understand why do you build this kind of station access ... (see left part of the screenshot).
Thank you for downloading and observing my savegame!
For multiplatform stations I mostly use a choosesignal at the entrance. So trains can arrive and leave from any platform. I hate trains making zigzag movements, now they depart in a straight line.
Were you mistaken or ... ?
I have tried your save game and it's really impressive.
Not to mention that you'll need at least last release of 128.
Thanks for your info! I am currently using Pak 128 v1.4.4 on Simutrans v101. Is that sufficient?
By "these fields slowly grow", do you mean the production capacity will also increase, alongside with increasing number of surrounding grain fields? Besides, is it necessary that transportation links be established for the whole industry chain before the grain field can grow?
Yes, iirc 1.4.4 has at least 40 fields per farm (the number rose from 10 to 20, then 40 and finally 80?).
I think they appear regardless of anything. The only factor is - they must be on the same height as farm building.
Thanks for your feekback, VS. Have tried myself and it's really nice to see a vast expanse of golden grain fields in Autumn, so awesomely beautiful! Many thanks to those who painted such nice fields and to those who made fields possible to grow.
BTW, I have tried leaving the grain field alone without supplying transport connections, and it will not grow (have watched for >10 Simutrans years already). And it's good to see that the production units per month also increase with increased number of fields!
I agree that it's not the fastest but I used this train because it can transport passengers and mail.
You're right, there are three bridges and one of them was not changed after upgrading the line.


text:
Susan Howe Reading
Posted in Announcements, Beinecke Collections, Poetry at Yale, Readings at Beinecke, Readings at Yale by beineckepoetry on March 22, 2012
Susan Howe, Poetry Reading &
Performance with Musician David Grubbs
Thursday, April 5th, 4:00pm
Poet Susan Howe, winner of the 2011 Bollingen Prize for American Poetry, and musician David Grubbs will perform a collaborative piece based on Howe's award-winning volume That This. This is the third collaborative work Howe and Grubbs have created together; they performed their second collaboration, "Souls of the Labadie Tract," at Beinecke Library in 2009 (see a description of the event here: http://beineckepoetry.library.yale.edu/2009/01/21/performance-souls-of-the-labadie-tract/).
Poet Susan Howe is the author of numerous books of poems including: That This (winner of the 2011 Bollingen Prize for American Poetry awarded by the Yale University Library) , Souls of the Labadie Tract, The Midnight, Pierce-Arrow, and Singularities.
Musician David Grubbs has made many solo records, played in a number of groups (Squirrel Bait, Bastro, Gastr del Sol, Red Krayola, Wingdale Community Singers), and frequently collaborates with writers and artists. He is an Associate Professor in the Conservatory of Music at Brooklyn College, CUNY. He teaches in Brooklyn College's MFA program in Performance and Interactive Media Arts (PIMA) and Brooklyn College's MFA program in Creative Writing, and is a member of the faculty of the Brooklyn College Center for Computer Music (BC-CCM).
Women Make Modern
Posted in Announcements, Beinecke Collections by beineckepoetry on March 22, 2012
Ruth Stephan Papers / Tiger's Eye Records
Posted in Announcements, Beinecke Collections, Poetry at Yale by beineckepoetry on March 19, 2012
Generalists, rejoice! Students and researchers whose interests scatter all over history and to every corner of the map will now have greater access to the papers of a literary woman with similarly diverse tastes. Twentieth-century American writer, editor, and translator Ruth Stephan's son, John J. Stephan, has made a generous donation to Beinecke that will highlight his mother's archive, the Ruth Stephan Papers and The Tiger's Eye Records, by providing broad funding for new acquisitions, exhibitions, and student projects, as well as fellowships for graduate students and international scholars.
The Ruth Stephan Papers (http://hdl.handle.net/10079/fa/beinecke.stephan) consist of 148 boxes of various papers, photographs, and audiovisual material. A visitor to Beinecke who pages 5 boxes at random from the collection could find correspondence with Thornton Wilder, annotated drafts of Stephan's poetry, research files on Queen Christina of Sweden, translations of Quechua songs, and materials from the filming of a documentary on Zen Buddhism. Stephan was born Charlotte Ruth Walgreen, and her papers also include a small amount of material documenting the history of the drugstore chain that her father started.
The Walgreen family had little use for contemporary art, and Stephan found her artistic wings clipped by their expectation that she should devote herself wholly to a role as mother and wife. She divorced her first husband Justin Dart in 1939 and asserted her independence through poetic output, publishing her first verses in venues such as Harper's, Poetry, and Forum. She would find out, upon her father's death, that she had been disinherited.
In the meantime, Stephan married the painter John Stephan, who encouraged her writing and with whom she founded the little magazine The Tiger's Eye in 1947. The Stephans founded the publication to promote challenging new art. "The selection of material will be based on these questions," they wrote in the first issue: "Is it alive? Is it valid as art? How brave is its originality? How does it enter the imagination?" In its two-year run, the groundbreaking magazine served as an important site for aesthetic discussion and featured the work of literary and artistic heavyweights like T.S. Eliot and Picasso, as well as of previously unknown creators. Records of production and distribution of The Tiger's Eye can be found in Stephan's papers and in the Tiger's Eye Records (http://hdl.handle.net/10079/fa/beinecke.tigerseye).
With the closing of The Tiger's Eye, Stephan traveled extensively in Europe, Japan, and southeast Asia. In addition to further volumes of poetry, her work includes two novels based on the life of Queen Christina of Sweden, The Flight (1956) and My Crown, My Love (1960); a volume of translated Quechua stories and songs, The Singing Mountaineers (1957); an audio compilation, The Spoken Anthology of American Literature (1963); and a documentary film, Zen in Ryoko-in (1971). Her philanthropic work included establishing a poetry center at the University of Arizona in 1960 (http://poetry.arizona.edu/). –Charlotte Parker, Y2013.
Images: John Stephan, Photograph of Ruth Stephan at her desk, (1960); The Tiger's Eye, number 1 (1947).