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\section*{Gabriel-Ulmer duality}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{gabrielulmer_duality}{}\section*{{Gabriel--Ulmer duality}}\label{gabrielulmer_duality}

\noindent\hyperlink{the_idea}{The idea}\dotfill \pageref*{the_idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{the_idea}{}\subsection*{{The idea}}\label{the_idea}

\textbf{Gabriel--Ulmer duality} says that there is an [[equivalence of categories|equivalence]] of [[2-categories]] (or in other words, a [[biequivalence]])

\begin{displaymath}
\begin{matrix}
Lex^{op} & \to &  LFP \\
  C      & \mapsto & Lex(C, Set) 
\end{matrix}
\end{displaymath}
where [[Lex]] is the 2-category of:

\begin{itemize}%
\item small [[finitely complete categories]],


\item [[finite limit]]$\:$ [[preserved limit|preserving]] [[functors]], and


\item [[natural transformations]],



\end{itemize}
and LFP is the 2-category of

\begin{itemize}%
\item [[locally finitely presentable categories]],


\item finitary [[right adjoint|right]] [[adjoint functors]] and


\item [[natural transformations]].



\end{itemize}
The idea is that an object $C \in Lex$ can be thought of as an [[essentially algebraic theory]], which has a category of [[model|models]] $Lex(C,Set)$. Gabriel--Ulmer duality says that this category of models is locally finitely presentable, all LFP categories arise in this way, and we can recover the theory $C$ from its category of models. There are similar dualities for other classes of theory such as [[regular theories]].

A version of Gabriel--Ulmer duality for [[enriched category theory]] was proved by [[Max Kelly]] (see \hyperlink{LackTendas}{LackTendas}). Let $\mathcal{V}$ be a symmetric monoidal closed complete and cocomplete category which is locally finitely presentable as a closed category. Then let $\mathcal{V}$-$Lex$ be the 2-category of finitely complete $\mathcal{V}$-categories ($\mathcal{V}$-categories with finite weighted limits), finite limit preserving $\mathcal{V}$-functors, and $\mathcal{V}$-natural transformations, and $\mathcal{V}$-$LFP$ the 2-category of locally finitely presentable $\mathcal{V}$-categories, right adjoint $\mathcal{V}$-functors that preserve filtered colimits, and $\mathcal{V}$-natural transformations. Then there is a biequivalence

\begin{displaymath}
\begin{matrix}
\mathcal{V}-Lex^{op} & \to &  \mathcal{V}-LFP \\
  C      & \mapsto & Lex(C, \mathcal{V}). 
\end{matrix}
\end{displaymath}
For instance, in the [[truth value]]-enriched case, the duality is between [[meet semilattices]] and [[algebraic lattices]].

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

The original source is:

\begin{itemize}%
\item [[Peter Gabriel]], [[Friedrich Ulmer]], \emph{Lokal Praesentierbare Kategorien}, Springer Lecture Notes in Mathematics \textbf{221}, Berlin, 1971.

\end{itemize}
A careful discussion and proof of the biequivalence is in

\begin{itemize}%
\item [[Jiri Adamek]], Hans-Eberhard Porst, \emph{Algebraic Theories of Quasivarieties} , J. Algebra \textbf{208} (1998) pp.379-398.

\end{itemize}
Some other general treatments of Gabriel-Ulmer duality (and generalizations to other [[doctrine|doctrines]]):

\begin{itemize}%
\item C. Centazzo, [[E. M. Vitale]], \emph{A duality relative to a limit doctrine}, Theory and Appl. of Categories \textbf{10}, No. 20, 2002, 486--497, \href{http://www.tac.mta.ca/tac/volumes/10/20/10-20.pdf}{pdf}


\item [[Stephen Lack]], [[John Power]], \emph{Gabriel--Ulmer duality and Lawvere Theories enriched over a general base}, \href{http://maths.mq.edu.au/~slack/papers/jfp.pdf}{pdf}


\item [[M. Makkai]], [[A. Pitts]], \emph{Some results on locally finitely presentable categories}, Trans. Amer. Math. Soc. \textbf{299} (1987), 473-496, \href{http://www.ams.org/mathscinet-getitem?mr=869216}{MR88a:03162}, \href{http://dx.doi.org/10.2307/2000508}{doi}, \href{http://www.ams.org/journals/tran/1987-299-02/S0002-9947-1987-0869216-2/S0002-9947-1987-0869216-2.pdf}{pdf}



\end{itemize}
For a 2-dimensional analogue see the slides from a 2010 talk by Makkai: \href{http://www.math.mcgill.ca/makkai/scans/2010Washington0001.pdf}{pdf}

A formal-categorical account using [[Yoneda structure|Yoneda structures]] can be found in

\begin{itemize}%
\item Ivan Di Liberti, [[Fosco Loregian]], \emph{Accessibility and Presentability in 2-Categories} , arXiv:1804.08710 (2018). (\href{https://arxiv.org/abs/1804.08710}{abstract})

\end{itemize}
For a discussion of Gabriel--Ulmer duality and related dualities in the context of [[enriched category theory]] see

\begin{itemize}%
\item [[Stephen Lack]], Giacomo Tendas, \emph{Enriched Regular Theories}, (\href{https://arxiv.org/abs/1907.02301}{arXiv:1907.02301})

\end{itemize}
This discusses (see Theorem 2.1) Kelly's original result for $V$-[[enriched categories]], where $V$ is a closed symmetric monoidal category whose underlying category $V_0$ is locally small, complete and cocomplete, in section 9 (cf. theorem 9.8) of

\begin{itemize}%
\item [[Max Kelly]], \emph{Structures defined by finite limits in the enriched context}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle cat\'e{}goriques, \textbf{23} no. 1 (1982), pp. 3-42, \href{http://www.ams.org/mathscinet-getitem?mr=648793}{MR648793},\href{http://www.numdam.org/item?id=CTGDC_1982__23_1_3_0}{numdam}

\end{itemize}
For a connection to [[Tannaka duality]] theory see

\begin{itemize}%
\item nCaf\'e{} discussion \href{http://golem.ph.utexas.edu/category/2011/07/doctrinal_and_tannakian_recons.html}{here}


\item [[Brian Day]], \emph{Enriched Tannaka duality}, JPAA \textbf{108} (1996) pp.17-22, \href{http://www.ams.org/mathscinet-getitem?mr=1382240}{MR97d:18008} 



\end{itemize}
For a discussion of an $\infty$-version of Gabriel-Ulmer duality between finitely complete and idempotent complete $(\infty, 1)$-categories and locally finitely presentable $(\infty, 1)$-categories see this \href{https://mathoverflow.net/q/293031/447}{MO discussion}.

[[!redirects Gabriel-Ulmer duality]] [[!redirects Gabriel–Ulmer duality]] [[!redirects Gabriel--Ulmer duality]]



\end{document}