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\begin{document}

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\section*{definable groupoid}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{internal_categories}{}\paragraph*{{Internal categories}}\label{internal_categories}

[[!include internal infinity-categories contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In a model $M$ of a first-order theory $T$ a definable set $G$ might have additional algebraic stucture (e.g. that of a [[group]], [[ring]], [[groupoid]], [[category]], etc.) also given by definable functions and predicates. Since this set with extra structure is given by some collection of formulas in the language of $T$, it is interpreted in every model of $T$, and is hence an invariant of the [[syntactic category]] (walking model) $\mathbf{Def}(T)$ of $T$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

To make this precise:

\begin{itemize}%
\item A definable group is just a [[group object]] in $\mathbf{Def}(T)$.
\item A definable groupoid is just a groupoid object, i.e. an [[internal groupoid]] in $\mathbf{Def}(T)$.
\item A definable ring is just a semigroup object in the category of abelian group objects of $\mathbf{Def}(T)$.
\item A definable category is just a category object, i.e. an [[internal category]] in $\mathbf{Def}(T)$.

\end{itemize}
Since groups, groupoids, rings, and categories can be given by algebraic theories, a definition (modulo having [[elimination of imaginaries|EI]]) of one of these in $T$ is just an interpretation of one of those theories in $T$ is just a logical functor from the walking models of one of these to $\mathbf{Def}(T)$.

\hypertarget{remarks}{}\subsubsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item Evaluating the unit of the [[Makkai duality]] adjunction at $T$ yields $\mathbf{Def}(T) \simeq \operatorname{Hom}_{\mathbf{Ult}}(\mathbf{Mod}(T), \mathbf{Set})$ the category of [[ultrafunctors]] from the category of models (logical functors $\mathbf{Def}(T) \to \mathbf{Set}$) to $\mathbf{Set}$ viewed as [[ultracategories]], so that one may identify a definable set (and definable sets with extra definable structure on them) $X \in \mathbf{Def}(T)$ with a [[functor of points]] $M \mapsto X(M)$ on the category of models.


\item The fact that the [[external axiom of choice]] (all epimorphisms split) holds in $\mathbf{Set}$ if and only if every [[fully faithful]] [[essentially surjective]] [[functor]] between [[small]] categories is a true [[equivalence of categories]] can be word-for-word internalized to $\mathbf{Def}(T)$. This means: if the theory has two constants, then $T$ has definable Skolem functions if and only if every fully faithful essentially surjective definable functor between definable categories admits a [[pseudoinverse]].


\item With a suitable amount of choice (definable Skolem functions, for example) Freyd's general adjoint functor theorem also carries over word-for-word to the setting of internal categories in $\mathbf{Def}(T)$.


\item In particular, there is much studied case of \textbf{definable groups}, cf. e.g. (\hyperlink{PeterzilPillay}{Peterzil-Pillay})



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{defn}
\label{}\hypertarget{}{}
There is a [[bijection|bijective]] correspondence between internal imaginary sorts of $T$ and definable concrete groupoids with a single isomorphism class, up to bi-interpretability over $T$ for the internal imaginary sorts and Hrushovski-equivalence for the definable concrete groupoids.

\end{defn}
This is (\hyperlink{Hrushovski}{Hrushovski 2006, Th.3.2}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[definability]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Y. Peterzil, A. Pillay, \emph{Generic sets in definably compact groups}, Fundamenta Mathematicae \textbf{193} (2007), pp. 153--170, \href{http://www.ams.org/mathscinet-getitem?mr=2282713}{MR2282713}, \href{http://dx.doi.org/10.4064/fm193-2-4}{doi}

\end{itemize}
\begin{itemize}%
\item Y. Peterzil, A. Pillay, S. Starchenko, \emph{Linear groups definable in o-minimal structures}, J. Algebra \textbf{247} (2002), no. 1, pp. 1--23, \href{http://www.ams.org/mathscinet-getitem?mr=1873380}{MR1873380}, \href{http://dx.doi.org/10.1006/jabr.2001.8861}{doi}


\item Alessandro Berarducci, \emph{Definable groups in o-minimal structures}, \href{http://www.dm.unipi.it/~dinasso/marian2004/berarducci.pdf}{pdf}; \emph{Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup}, J. Symbolic Logic \textbf{74}:3 (2009), 891-900, \href{http://www.ams.org/mathscinet-getitem?mr=2548466}{MR2548466}, \href{http://projecteuclid.org/euclid.jsl/1245158089}{euclid}, \href{http://dx.doi.org/10.2178/jsl/1245158089}{doi}, \emph{O-minimal spectra, infinitesimal subgroups and cohomology}, J. Symbolic Logic \textbf{72} (2007), no. 4, pp. 1177--1193, \href{http://www.ams.org/mathscinet-getitem?mr=2371198}{MR2371198}, \href{http://projecteuclid.org/euclid.jsl/1203350779}{euclid}, \href{http://dx.doi.org/10.2178/jsl/1203350779}{doi}


\item Margarita Otero, \emph{A survey on groups definable in o-minimal structures}, in: Model theory with applications to algebra and analysis. Vol. 2, 177--206, London Math. Soc. Lecture Note Ser. \textbf{350}, Cambridge Univ. Press 2008, \href{http://www.ams.org/mathscinet-getitem?mr=2436142}{MR2010b:03042}, \href{http://dx.doi.org/10.1017/CBO9780511735219.006}{doi}


\item [[Ehud Hrushovski]], \emph{Groupoids, imaginaries and internal covers} (2006), \href{http://arxiv.org/abs/math/0603413}{arxiv/math.LO/0603413}



\end{itemize}
\begin{itemize}%
\item [[Ehud Hrushovski]], \emph{On finite imaginaries}, \href{http://arxiv.org/abs/0902.0842}{arxiv/0902.0842}

\end{itemize}
[[!redirects definable groupoids]] [[!redirects definable category]] [[!redirects definable categories]] [[!redirects definable group]] [[!redirects definable groups]]



\end{document}