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\begin{document}

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\section*{horizontal categorification}

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

\hypertarget{categorification}{}\paragraph*{{Categorification}}\label{categorification}

[[!include categorification - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{further_discussion}{Further discussion}\dotfill \pageref*{further_discussion} \linebreak


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

\emph{Horizontal categorification} or \emph{Oidification} describes the process by which

\begin{enumerate}%
\item a concept is realized to be equivalent to a certain type of [[category]] with a \emph{single [[object]]};


\item and then this concept is generalized -- or oidified -- by passing to instances of such types of categories with more than one object.



\end{enumerate}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item This is to be contrasted with [[vertical categorification]].


\item It can be argued that the term `categorification' should be reserved for [[vertical categorification]], since we can use `oidification' for the horizontal concept.


\item It has rightly been remarked that [[groupoid]]s are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The horizontal categorification of [[group]]s are [[groupoid]]s: [[category|categories]] in which every morphism is invertible.


\item A horizontal categorification of [[algebra]]s are \emph{algebroids}: [[enriched category|enriched categories]] in the category of vector spaces.



\end{itemize}
[[David Roberts]]: How do Lie algebroids fit into this framework?

[[Urs Schreiber]]: at a rough level it is clear that the base space of a Lie algebroid has to be regardedas the ``space of objects''. Certainly a Lie algebroid over a point is precisely a Lie algebra.

But for a more precise statement one needs a more conceptual way to think of Lie algebroids. I am claiming at [[schreiber:∞-Lie algebroid]] that there is a way to regard Lie algebroids precisely as certain kinds of synthetically smooth groupoids, namely those all whose morphisms have ``infinitesimal extension'' in some sense. In such a picture Lie algebroids are on the same footing as Lie groupoids and are precisely the many-object version of Lie algebras = infinitesimal Lie groups.

\begin{itemize}%
\item A horizontal categorification of rings are [[ringoid]]s: [[enriched category|enriched categories]] over the category of abelian groups. (\href{http://golem.ph.utexas.edu/category/2006/09/ringoids.html}{blog})


\item A horizontal categorification of $C^*$-algebras hence ought to be known as \emph{$C^*$--algebroids} but is usually known as [[C-star-category|C\emph{-categories]].}


\item Since, by the [[Gelfand-Naimark theorem]], [[C-star algebra]]s are dual to [[topological space]]s, Paolo Bertozzini et. al proposed to define [[spaceoid]]s to be entities dual to $C^*$-categories (\href{http://golem.ph.utexas.edu/category/2008/01/spaceoids.html}{blog}).


\item And finally the \textbf{exception to the rule}: a many-object [[monoid]] is \emph{not} called a \emph{monoidoid} -- but is called a [[category]]! :-)



\end{itemize}
[[Mike Shulman|Mike]]: Is there (and do we want there to be) a general rule about whether an X-oid means an [[internal category]] whose one-object version is an X, or an [[enriched category]] whose one-object version is an X? The examples above seem to be taking the ``internal'' side, but the Cafe discussion on ``ringoids'' was about the ``enriched'' version; the two are very different! And ``dg-algebroid'' was suggested for [[dg-category]], which is an enriched oidification of a [[dg-algebra]], but a [[Hopf algebroid]] is an internal oidification of a [[Hopf algebra]].

[[Urs Schreiber|Urs]]: good point. We should say this at the beginning and split the list of examples in two sorts

\hypertarget{further_discussion}{}\subsection*{{Further discussion}}\label{further_discussion}

Related $n$-Caf\'e{}-discussion is in

\begin{itemize}%
\item [[John Baez]], \href{http://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html}{\emph{What is categorification?}}


\item [[John Baez]], \href{http://golem.ph.utexas.edu/category/2006/09/ringoids.html}{\emph{Ringoids}}



\end{itemize}
[[!redirects Oidification]] [[!redirects oidification]] [[!redirects oidifications]] [[!redirects horizontal categorifications]]



\end{document}