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\begin{document}

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\section*{Isbell duality}

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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - contents]]

[[higher geometry]] $\leftarrow$ \textbf{Isbell duality} $\rightarrow$ [[higher algebra]]


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{respect_for_limits}{Respect for limits}\dotfill \pageref*{respect_for_limits} \linebreak
\noindent\hyperlink{isbell_selfdual_objects}{Isbell self-dual objects}\dotfill \pageref*{isbell_selfdual_objects} \linebreak
\noindent\hyperlink{isbell_envelope}{Isbell envelope}\dotfill \pageref*{isbell_envelope} \linebreak
\noindent\hyperlink{examples_and_similar_dualities}{Examples and similar dualities}\dotfill \pageref*{examples_and_similar_dualities} \linebreak
\noindent\hyperlink{FunctionAlgebrasOnPresheaves}{Function $T$-Algebras on presheaves}\dotfill \pageref*{FunctionAlgebrasOnPresheaves} \linebreak
\noindent\hyperlink{FuncCompDerStacks}{Function $k$-algebras on derived $\infty$-stacks}\dotfill \pageref*{FuncCompDerStacks} \linebreak
\noindent\hyperlink{function_algebras_on_stacks}{Function $T$-algebras on $\infty$-stacks}\dotfill \pageref*{function_algebras_on_stacks} \linebreak
\noindent\hyperlink{function_2algebras_on_algebraic_stacks}{Function 2-algebras on algebraic stacks}\dotfill \pageref*{function_2algebras_on_algebraic_stacks} \linebreak
\noindent\hyperlink{GelfandDuality}{Gelfand duality}\dotfill \pageref*{GelfandDuality} \linebreak
\noindent\hyperlink{serreswan_theorem}{Serre-Swan theorem}\dotfill \pageref*{serreswan_theorem} \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}

A general abstract [[adjunction]]

\begin{displaymath}
(\mathcal{O} \dashv Spec)
  : 
  CoPresheaves
  \underoverset{Spec}{O}{\leftrightarrows}
  Presheaves
\end{displaymath}
relates (higher) [[presheaves]] with (higher) [[copresheaves]] on a given ([[higher category theory|higher]]) [[category]] $C$: this is called \textbf{Isbell conjugation} or \textbf{Isbell duality} (after [[John Isbell]]).

To the extent that this adjunction descends to presheaves that are ([[higher topos theory|higher]]) [[sheaves]] and copresheaves that are ([[(infinity,1)-algebraic theory|higher]]) [[algebra over an algebraic theory|algebras]] this duality relates [[higher geometry]] with [[higher algebra]].

Objects preserved by the [[monad]] of this adjunction are called \textbf{Isbell self-dual}.

Under the interpretation of [[presheaves]] as generalized [[spaces]] and [[copresheaves]] as generalized [[quantities]] modeled on $C$ (\hyperlink{Lawvere86}{Lawvere 86}, see at \emph{[[space and quantity]]}), Isbell duality is the archetype of the [[duality]] between [[geometry]] and [[algebra]] that permeates mathematics (such as [[Gelfand duality]] or the [[embedding of smooth manifolds into formal duals of R-algebras]]).

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

Let $\mathcal{V}$ be a good enriching category (a [[cosmos]], i.e. a [[complete category|complete]] and [[cocomplete category|cocomplete]] [[closed monoidal category|closed]] [[symmetric monoidal category]]).

Let $\mathcal{C}$ be a [[small category|small]] $\mathcal{V}$-[[enriched category]].

Write $[\mathcal{C}^{op}, \mathcal{V}]$ and $[\mathcal{C}, \mathcal{V}]$ for the [[enriched functor categories]].

\begin{prop}
\label{}\hypertarget{}{}
There is a $V$-[[adjunction]]

\begin{displaymath}
(\mathcal{O} \dashv Spec)
  \colon
  [C, \mathcal{V}]^{op}
  \underoverset{Spec}{O}{\leftrightarrows}
  [C^{op}, \mathcal{V}]
\end{displaymath}
where

\begin{displaymath}
\mathcal{O}(X) \colon c \mapsto [C^{op}, \mathcal{V}](X, \mathcal{V}(-,c))
  \,,
\end{displaymath}
and

\begin{displaymath}
Spec(A) \colon c \mapsto [C, \mathcal{V}]^{op}(\mathcal{V}(c,-),A)
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
This is also called [[Isbell duality]]. Objects which are preserved by $\mathcal{O} \circ Spec$ or $Spec \mathcal{O}$ are called \textbf{Isbell self-dual}.

\end{remark}
The proof is mostly a tautology after the notation is unwinded. The mechanism of the proof may still be of interest and be relevant for generalizations and for less tautological variations of the setup. We therefore spell out several proofs.

\begin{proof}
Use the [[end]]-expression for the [[hom-object]]s of the [[enriched functor categories]] to compute

\begin{displaymath}
\begin{aligned}
     [C,\mathcal{V}]^{op}(\mathcal{O}(X), A)
     & :=
     \int_{c \in C} \mathcal{V}(A(c), \mathcal{O}(X)(c))
     \\
     & := 
     \int_{c \in C} \mathcal{V}(A(c), [C^{op}, \mathcal{V}](X, \mathcal{V}(-,c)))
     \\
     & :=
     \int_{c \in C} \int_{d \in C} \mathcal{V}(A(c), \mathcal{V}(X(d), \mathcal{V}(d,c)))
     \\
     & \simeq 
     \int_{d \in C} \int_{c \in C} \mathcal{V}(X(d), \mathcal{V}(A(c), \mathcal{V}(d,c)))
    \\
     & =:
     \int_{d \in C}  \mathcal{V}(X(d), [C,\mathcal{V}]^{op}(\mathcal{V}(d,-),A))
    \\
     & =:
     \int_{d \in C}  \mathcal{V}(X(d), Spec(A)(d))
    \\
     & =:
     [C^{op}, \mathcal{V}](X, Spec(A))
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Here apart from writing out or hiding the ends, the only step that is not a definition is precisely the middle one, where we used that $\mathcal{V}$ is a [[symmetric monoidal category|symmetric]] [[closed monoidal category]].

\end{remark}
The following proof does not use ends and needs instead slightly more preparation, but has then the advantage that its structure goes through also in great generality in [[higher category theory]].

\begin{proof}
Notice that

\textbf{Lemma 1:} $Spec(\mathcal{V}(c,-)) \simeq \mathcal{V}(-,c)$

because we have a natural isomorphism

\begin{displaymath}
\begin{aligned}
    Spec(\mathcal{V}(c,-))(d) 
    & :=
    [C,\mathcal{V}](\mathcal{V}(c,-), \mathcal{V}(d,-))
    \\
    & \simeq
    \mathcal{V}(d,c)
  \end{aligned}
\end{displaymath}
by the [[Yoneda lemma]].

From this we get

\textbf{Lemma 2:} $[C^{op}, \mathcal{V}](Spec \mathcal{V}(c,-), Spec A) \simeq 
    [C,\mathcal{V}](A, \mathcal{V}(c,-))$

by the sequence of natural isomorphisms

\begin{displaymath}
\begin{aligned}
     [C^{op}, \mathcal{V}](Spec \mathcal{V}(c,-), Spec A)
     & \simeq
     [C^{op}, \mathcal{V}](\mathcal{V}(-,c), Spec A)
     \\
     & \simeq (Spec A)(c)
     \\
     & := [C, \mathcal{V}](A, \mathcal{V}(c,-))
  \end{aligned}
  \,,
\end{displaymath}
where the first is Lemma 1 and the second the [[Yoneda lemma]].

Since (by what is sometimes called the [[co-Yoneda lemma]]) every object $X \in [C^{op}, \mathcal{V}]$ may be written as a [[colimit]]

\begin{displaymath}
X \simeq {\lim_\to}_i \mathcal{V}(-,c_i)
\end{displaymath}
over [[representable functor|representables]] $\mathcal{V}(-,c_i)$ we have

\begin{displaymath}
X \simeq {\lim_\to}_i Spec(\mathcal{V}(c_i,-))
  \,.
\end{displaymath}
In terms of the same diagram of representables it then follows that

\textbf{Lemma 3:}

\begin{displaymath}
\mathcal{O}(X) \simeq {\lim_{\leftarrow}}_i \mathcal{V}(c_i,-)
\end{displaymath}
because using the above colimit representation and the Yoneda lemma we have natural isomorphisms

\begin{displaymath}
\begin{aligned}
     \mathcal{O}(X)(d) 
     &=
     [C^{op}, \mathcal{V}](X, \mathcal{V}(-,c))
     \\
     & \simeq
     [C^{op}, \mathcal{V}]({\lim_\to}_i \mathcal{V}(-,c_i), \mathcal{V}(-,c))
     \\
     & \simeq
     {\lim_\leftarrow}_i [C^{op}, \mathcal{V}](\mathcal{V}(-,c_i), \mathcal{V}(-,c))
     \\
     & \simeq 
     {\lim_\leftarrow}_i \mathcal{V}(c_i,c)
  \end{aligned}
  \,.
\end{displaymath}
Using all this we can finally obtain the adjunction in question by the following sequence of natural isomorphisms

\begin{displaymath}
\begin{aligned}
     [C,\mathcal{V}]^{op}(\mathcal{O}(X), A)
     & 
     \simeq
     [C,\mathcal{V}](A, {\lim_\leftarrow}_i \mathcal{V}(c_i,-), )
     \\
     & \simeq 
     {\lim_{\leftarrow}}_i [C, \mathcal{V}](A, \mathcal{V}(c_i,-))
     \\
     & \simeq 
     {\lim_{\leftarrow}}_i [C^{op}, \mathcal{V}](Spec \mathcal{V}(c_i,-), Spec A)
     \\
     & \simeq
     [C^{op}, \mathcal{V}]({\lim_{\to}}_i  Spec \mathcal{V}(c_i,-), Spec A)
     \\
     & \simeq
     [C^{op}, \mathcal{V}](X, Spec A)     
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
The pattern of this proof has the advantage that it goes through in great generality also on [[higher category theory]] without reference to a higher notion of enriched category theory.

\begin{defn}
\label{}\hypertarget{}{}
An object $X$ or $A$ is \textbf{Isbell-self-dual} if

\begin{itemize}%
\item $A \stackrel{}{\to} \mathcal{O} Spec(A)$ is an [[isomorphism]] in $[C,\mathcal{V}]$;


\item $X \to Spec \mathcal{O} X$ is an [[isomorphism]] in $[C^{op}, \mathcal{V}]$, respectively.



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Under certain circumstances, Isbell duality can be extended to large $\mathcal{V}$-enriched categories $C$. For example, if $C$ has a small generating subcategory $S$ and a small cogenerating subcategory $T$, then for each $F: C^{op} \to \mathcal{V}$ and $G: C \to \mathcal{V}$, one may construct $\mathcal{O}(F)$ and $Spec(G)$ objectwise as appropriate subobjects in $\mathcal{V}$:

\begin{displaymath}
\mathcal{O}(F)(c) = [C^{op}, \mathcal{V}](F, C(-, c)) \hookrightarrow \int_{s: S} \mathcal{V}(F s, \hom(s, c))
\end{displaymath}
\begin{displaymath}
Spec(G)(c) = [C, \mathcal{V}](G, C(c, -)) \hookrightarrow \int_{t: T} \mathcal{V}(G t, \hom(c, t))
\end{displaymath}
\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{respect_for_limits}{}\subsubsection*{{Respect for limits}}\label{respect_for_limits}

Choose any [[class]] $L$ of [[limit]]s in $C$ and write $[C,\mathcal{V}]_\times \subset [C,\mathcal{V}]$ for the [[full subcategory]] consisting of those functors preserving these limits.

\begin{prop}
\label{}\hypertarget{}{}
The $(\mathcal{O} \dashv Spec)$-adjunction does descend to this inclusion, in that we have an adjunction

\begin{displaymath}
(\mathcal{O} \dashv Spec)
  :
  [C, \mathcal{V}]_{\times}^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  [C^{op}, \mathcal{V}]
\end{displaymath}
\end{prop}
\begin{proof}
Because the [[hom-functor]]s preserves all [[limit]]s:

\begin{displaymath}
\begin{aligned}
    \mathcal{O}(X)({\lim_{\leftarrow}}_j c_j)
    & := [C^{op}, \mathcal{V}](X,\mathcal{V}(-,{\lim_{\leftarrow}}_j c_j))
    \\
    & \simeq [C^{op}, \mathcal{V}](X,{\lim_{\leftarrow}}_j  \mathcal{V}(-,c_j))
    \\
    & \simeq {\lim_{\leftarrow}}_j  [C^{op}, \mathcal{V}](X,\mathcal{V}(-,c_j))
    \\
    & =: {\lim_{\leftarrow}}_j \mathcal{O}(X)(c_j)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{isbell_selfdual_objects}{}\subsubsection*{{Isbell self-dual objects}}\label{isbell_selfdual_objects}

\begin{prop}
\label{}\hypertarget{}{}
All [[representable functor|representables]] are Isbell self-dual.

\end{prop}
\begin{proof}
By \hyperlink{ProofB}{Proof B , lemma 1} we have a [[natural isomorphism]]s in $c \in C$

\begin{displaymath}
Spec(\mathcal{V}(c,-)) \simeq \mathcal{V}(-,c)
  \,.
\end{displaymath}
Therefore we have also the natural isomorphism

\begin{displaymath}
\begin{aligned}
     \mathcal{O} Spec \mathcal{V}(c,-)(d)
     & \simeq
     \mathcal{O} \mathcal{V}(-,c) (d)
     \\
     & :=
      [C^{op}, \mathcal{V}](\mathcal{V}(-,c), \mathcal{V}(-,d))
     \\
     & \simeq
     \mathcal{V}(c,d)
  \end{aligned}
  \,,
\end{displaymath}
where the second step is the [[Yoneda lemma]]. Similarly the other way round.

\end{proof}
\hypertarget{isbell_envelope}{}\subsubsection*{{Isbell envelope}}\label{isbell_envelope}

See [[Isbell envelope]].

\hypertarget{examples_and_similar_dualities}{}\subsection*{{Examples and similar dualities}}\label{examples_and_similar_dualities}

Isbell duality is a template for many other [[space]]/[[algebra]]-[[dualities]] in [[mathematics]].

\hypertarget{FunctionAlgebrasOnPresheaves}{}\subsubsection*{{Function $T$-Algebras on presheaves}}\label{FunctionAlgebrasOnPresheaves}

Let $\mathcal{V}$ be any [[cartesian closed category]].

Let $C := T$ be the [[syntactic category]] of a $\mathcal{V}$-enriched [[Lawvere theory]], that is a $\mathcal{V}$-category with finite [[product]]s such that all objects are generated under products from a single object $1$.

Then write $T Alg := [C,\mathcal{V}]_\times$ for category of product-preserving functors: the category of $T$-algebras. This comes with the canonical forgetful functor

\begin{displaymath}
U_T : T Alg \to \mathcal{V} :   A \mapsto A(1)
\end{displaymath}
Write

\begin{displaymath}
F_T : T^{op} \hookrightarrow T Alg
\end{displaymath}
for the [[Yoneda embedding]].

\begin{defn}
\label{}\hypertarget{}{}
Call

\begin{displaymath}
\mathbb{A}_T := Spec(F_T(1))  \in [C^{op}, \mathcal{V}]
\end{displaymath}
the \textbf{$T$-line object}.

\end{defn}
\begin{lemma}
\label{}\hypertarget{}{}
For all $X \in [C^{op}, \mathcal{V}]$ we have

\begin{displaymath}
\mathcal{O}(X) \simeq
  [C^{op}, \mathcal{V}](X, Spec(F_T(-)))
  \,.
\end{displaymath}
In particular

\begin{displaymath}
U_T(\mathcal{O}(X)) \simeq [C^{op}, \mathcal{V}](X,\mathbb{A}_T)
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
We have isomorphisms natural in $k \in T$

\begin{displaymath}
\begin{aligned}
  [C^{op}, \mathcal{V}](X, Spec(F_T(k)))
  & \simeq
  T Alg(F_T(k), \mathcal{O}(X))
  \\
  & \simeq
  \mathcal{O}(X)(k)
\end{aligned}
\end{displaymath}
by the above adjunction and then by the [[Yoneda lemma]].

\end{proof}
All this generalizes to the following case:

instead of setting $C := T$ let more generally

\begin{displaymath}
T \subset C \subset T Alg^{op}
\end{displaymath}
be a [[small category|small]] [[full subcategory]] of $T$-algebras, containing all the free $T$-algebras.

Then the original construction of $\mathcal{O} \dashv Spec$ no longer makes sense, but that in terms of the line object still does

\begin{prop}
\label{}\hypertarget{}{}
Set

\begin{displaymath}
Spec A : B \mapsto T Alg(A,B)
\end{displaymath}
and

\begin{displaymath}
\mathcal{O}(X) : k \mapsto 
  [C^{op}, \mathcal{V}](X, Spec(F_T(k)))
  \,.
\end{displaymath}
Then we still have an adjunction

\begin{displaymath}
(\mathcal{O} \dashv Spec)
  :
  T Alg^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  [C^{op}, \mathcal{V}]
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
\begin{displaymath}
\begin{aligned}
    T Alg^{op}(\mathcal{O}(X), A)
    & :=
    \int_{k \in T} \mathcal{V}( A(k), \mathcal{O}(X)(k) )
    \\
    & := 
    \int_{k \in T} \mathcal{V}( A(k), [C^{op}, \mathcal{V}](X, Spec(F_T(k))) )
    \\
    & := \int_{k \in T} \int_{B \in C}
    \mathcal{V}(A(k), \mathcal{V}(X(B), T Alg(F_T(k), B) ))
    \\
    & \simeq \int_{k \in T} \int_{B \in C}
    \mathcal{V}(A(k), \mathcal{V}(X(B), B(k) ))
    \\
    & \simeq \int_{k \in T} \int_{B \in C}
    \mathcal{V}(X(B), \mathcal{V}(A(k), B(k) ))
    \\
    & =: \int_{B \in C} \mathcal{V}(X(B), T Alg(A,B))
    \\
    & =:
    \int_{B \in C} \mathcal{V}(X(B), Spec(A)(B))
    \\
    & =:
    [C^{op}, Set](X,Spec(A))
  \end{aligned}
  \,.
\end{displaymath}
The first step that is not a definition is the [[Yoneda lemma]]. The step after that is the symmetric-closed-monoidal structure of $\mathcal{V}$.

\end{proof}
\hypertarget{FuncCompDerStacks}{}\subsubsection*{{Function $k$-algebras on derived $\infty$-stacks}}\label{FuncCompDerStacks}

The structure of our \hyperlink{ProofB}{Proof B} above goes through in higher category theory.

Formulated in terms of [[derived stack]]s over the [[(∞,1)-category]] of [[dg-algebra]]s, this is essentially the argument appearing on \href{http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.3636v1.pdf#page=23}{page 23} of (\hyperlink{Ben-ZviNadler}{Ben-ZviNadler}).

\hypertarget{function_algebras_on_stacks}{}\subsubsection*{{Function $T$-algebras on $\infty$-stacks}}\label{function_algebras_on_stacks}

for the moment see at \emph{[[function algebras on ∞-stacks]]}.

\hypertarget{function_2algebras_on_algebraic_stacks}{}\subsubsection*{{Function 2-algebras on algebraic stacks}}\label{function_2algebras_on_algebraic_stacks}

see [[Tannaka duality for geometric stacks]]

\hypertarget{GelfandDuality}{}\subsubsection*{{Gelfand duality}}\label{GelfandDuality}

[[Gelfand duality]] is the [[equivalence of categories]] between (nonunital) commutative [[C-star algebra|C\emph{-algebras]] and ([[locally compact space|locally]]) [[compact topological spaces]]. See there for more details.}

\hypertarget{serreswan_theorem}{}\subsubsection*{{Serre-Swan theorem}}\label{serreswan_theorem}

The [[Serre-Swan theorem]] says that suitable [[modules]] over an commutative [[C-star algebra|C\emph{-algebra]] are equivalently modules of [[sections]] of [[vector bundles]] over the \hyperlink{GelfandDuality}{Gelfand-dual} topological space.}

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

\begin{itemize}%
\item [[function algebra]]


\item [[function algebras on ∞-stacks]]


\item [[nucleus of a profunctor]]



\end{itemize}
[[!include Isbell duality - table]]

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

The original articles on Isbell duality and the [[Isbell envelope]] are

\begin{itemize}%
\item [[John Isbell]], \emph{Structure of categories}, Bulletin of the American Mathematical Society 72 (1966), 619-- 655. (\href{http://projecteuclid.org/euclid.bams/1183528163}{project euclid})


\item [[John Isbell]], \emph{Normal completions of categories}, Reports of the Midwest Category Seminar, vol. 47, Springer, 1967, 110--155.



\end{itemize}
More recent discussion is in

\begin{itemize}%
\item [[William Lawvere]], p. 17 of \emph{Taking categories seriously}, Revista Colombiana de Matematicas, XX (1986) 147-178, reprinted as: Reprints in Theory and Applications of Categories, No. 8 (2005) pp. 1-24 (\href{http://tac.mta.ca/tac/reprints/articles/8/tr8abs.html}{web})


\item [[Michael Barr]], John Kennison, R. Raphael, Isbell Duality, Theory and Applications of Categories, Vol. 20, 2008, No. 15, pp 504-542. (\href{http://www.tac.mta.ca/tac/volumes/20/15/20-15abs.html}{web})


\item [[Richard Garner]], \emph{The Isbell monad}, Advances in Mathematics \textbf{274} (2015) pp.516-537. (\href{http://comp.mq.edu.au/~rgarner/Papers/Isbell.pdf}{draft})


\item [[Vaughan Pratt]], \emph{Communes via Yoneda, from an elementary perspective}, Fundamenta Informaticae 103 (2010), 203--218.


\item Ivan Di Liberti, [[Fosco Loregian]], \emph{On the Unicity of Formal Category Theories}, arXiv:1901.01594 (2019). (\href{https://arxiv.org/abs/1901.01594}{abstract})



\end{itemize}
Isbell conjugacy for [[(∞,1)-presheaves]] over the [[(∞,1)-category]] of duals of [[dg-algebra]]s is discussed around page 32 of

\begin{itemize}%
\item [[David Ben-Zvi]], [[David Nadler]], \emph{Loop spaces and connections} (\href{http://arxiv.org/abs/1002.3636}{arXiv:1002.3636})

\end{itemize}
in

\begin{itemize}%
\item [[nLab:Bertrand Toën]], \emph{Champs affines} (\href{http://arxiv.org/abs/math/0012219}{arXiv:math/0012219})

\end{itemize}
Isbell self-dual [[∞-stack]]s over duals of commutative [[associative algebra]]s are called \emph{affine stacks}. They are characterized as those objects that are \emph{small} in a sense and local with respect to the [[cohomology]] with coefficients in the canonical [[line object]].

A generalization of this latter to $\infty$-stacks over duals of [[algebra over a Lawvere theory|algebras over arbitrary abelian Lawvere theories]] is the content of

\begin{itemize}%
\item [[Herman Stel]], \emph{$\infty$-Stacks and their function algebras -- with applications to $\infty$-Lie theory}, master thesis (2010) ([[schreiber:master thesis Stel|web]])

\end{itemize}
See also

\begin{itemize}%
\item MathOverflow: \href{http://mathoverflow.net/questions/84641/theme-of-isbell-duality}{theme-of-isbell-duality}


\item R.J. Wood, \emph{Some remarks on total categories}, J. Algebra \textbf{75\_:2, 1982, 538--545 }



\end{itemize}
[[!redirects Isbell dualities]]

[[!redirects Isbell conjugation]] [[!redirects Isbell conjugations]]

[[!redirects Isbell adjunction]] [[!redirects Isbell adjunctions]]



\end{document}