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\begin{document}

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\section*{dualizing object in a closed category}

\begin{quote}%
This entry is about dualizing objects in closed (monoidal) categories in the sense of [[homological algebra]] and [[stable homotopy theory]] (e.g. [[dualizing modules]]). For a more general concept see at \emph{[[dualizing object]]}.


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

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

[[!include category theory - contents]]

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

[[!include duality - 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{in_closed_monoidal_categories}{In closed monoidal categories}\dotfill \pageref*{in_closed_monoidal_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{InACartesianClosedCategory}{In a cartesian closed category}\dotfill \pageref*{InACartesianClosedCategory} \linebreak
\noindent\hyperlink{in_the_category_of_spectra__anderson_duality}{In the category of spectra -- Anderson duality}\dotfill \pageref*{in_the_category_of_spectra__anderson_duality} \linebreak
\noindent\hyperlink{in_the_category_of_suplattices}{In the category of suplattices}\dotfill \pageref*{in_the_category_of_suplattices} \linebreak
\noindent\hyperlink{chu_spaces}{Chu spaces}\dotfill \pageref*{chu_spaces} \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 \emph{dualizing object} $D$ in a [[closed category]] $\mathcal{C}$ is an [[object]] such that the [[internal hom]] $[-,D] \colon \mathcal{C} \to \mathcal{C}^{op}$ into it serves as an [[involution|involutive]] [[duality]] operation on $\mathcal{C}$, or at least on a suitable [[full subcategory]] $\mathcal{C}_D \hookrightarrow \mathcal{C}$, i.e. it induces an [[equivalence of categories]] $[-,D] \colon \mathcal{C}_D \to \mathcal{C}_D^{op}$.

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

Let $\mathcal{C}$ be a [[closed category]], $D\in \mathcal{C}$ an object, and $\mathcal{C}_D \hookrightarrow \mathcal{C}$ a [[full subcategory]]. We say that $D$ is a \textbf{dualizing object on $\mathcal{C}_D$} if the induced functor

\begin{displaymath}
[-,D] : \mathcal{C}_D \to \mathcal{C}_D^{op}
\end{displaymath}
is an [[equivalence of categories]]. Note that this includes the assumption that $[-,D]$ maps $\mathcal{C}_D$ into itself, which is not automatic.

Note that we do not in general assume $D\in \mathcal{C}_D$. In [[stable homotopy theory]] the subcategory in question is typically that of [[homotopy types with finite homotopy groups]] (e.g. for [[Anderson duality]]). Specifically in [[homological algebra]] one speaks also of \emph{[[dualizing modules]]}. (See for instance (\hyperlink{HeardStojanoska14}{Heard-Stojanoska 14, def. 3.1} and [[Representability Theorems|Lurie, section 4.2]]).) Notably in a [[Grothendieck-Verdier context]] $(f^\ast \dashv f_\ast)$, $(f_! \dashv f^!)$ of [[six operations]] the functor $f^!$ typically preserves dualizing objects in this sense, which is a crucial ingredient of [[Verdier duality]].

If $\mathcal{C} = \mathcal{C}_D$, we say that $D$ is a \textbf{global dualizing object}. A biclosed monoidal category $\mathcal{C}$ with a global dualizing object is one definition of a [[star-autonomous category]].

\hypertarget{in_closed_monoidal_categories}{}\subsubsection*{{In closed monoidal categories}}\label{in_closed_monoidal_categories}

If $\mathcal{C}$ is a [[closed symmetric monoidal category]], then $[-,D]$ is adjoint to itself on the right, i.e. we have a natural isomorphism $\mathcal{C}(A,[B,D]) \cong \mathcal{C}(B,[A,D])$. Thus, if $[-,D]$ maps $\mathcal{C}_D$ into itself, then its restriction to $\mathcal{C}_D$ is an equivalence of categories if and only if the unit and counit of this adjunction are isomorphisms. But these unit and counit are both the ``double-dualization'' map

\begin{displaymath}
A \to [[A,D],D]
\end{displaymath}
(the [[adjunct]] of the [[evaluation map]] $[A,D] \otimes A \to D$, which in turn is the adjunct of the identity map $[A,D] \to [A,D]$) for $A\in \mathcal{C}_D$; so $D$ is a dualizing object on $\mathcal{C}_D$ if and only if $[-,D]$ preserves $\mathcal{C}_D$ and these maps are isomorphisms for all $A\in \mathcal{C}_D$.

If $\mathcal{C}$ is a non-symmetric [[biclosed monoidal category]], with two internal-homs $(A\otimes -) \dashv [A,-]$ and $(-\otimes A) \dashv \langle A,-\rangle$, then $[-,D]$ is adjoint on the right not to itself but to $\langle-,D\rangle$. A similar argument then shows that $D$ is dualizing if and only if $[-,D]$ and $\langle-,D\rangle$ both map $\mathcal{C}_D$ to itself, and both double-dualization maps

\begin{displaymath}
A\to \langle [A,D],D\rangle\qquad and \qquad A \to [\langle A,D\rangle,D]
\end{displaymath}
are isomorphisms.

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

\hypertarget{InACartesianClosedCategory}{}\subsubsection*{{In a cartesian closed category}}\label{InACartesianClosedCategory}

A [[cartesian closed category]] that with a global dualizing object is necessarily just a [[preorder]]. This statement is often known as \emph{[[Joyal]]`s lemma}, recalled for instance in \hyperlink{Abramsky09}{Abramsky 09}. It can be slightly strengthened as follows:

\begin{prop}
\label{}\hypertarget{}{}
A cartesian closed category $C$ that is self-dual (carries an equivalence $N: C^{op} \to C$) is necessarily a preorder (whose posetal reflection is then a [[Heyting algebra]]). If the self-duality comes from a dualizing object, then the Heyting algebra is a Boolean algebra.

\end{prop}
\begin{proof}
For the first statement, let $1$ be terminal; then $N(1)$ is an initial object $0$, and similarly $N$ takes finite products to finite coproducts (coproducts are necessary for a Heyting algebra). So it remains to show $C$ is a preorder. Let $x, y$ be any two objects. The number of morphisms $x \to y$ is the number of morphisms $1 \to [x, y]$, which is the number of morphisms $N([x, y]) \to N(1) = 0$. But this is at most one since the initial object is strict (if there is any $z \to 0$, then $z$ is a retract of $0 \times z \cong 0$, hence $z \cong 0$; thus there is at most one morphism $z \to 0$).

The second statement is immediate: if $d$ is the dualizing object, then

\begin{displaymath}
d \cong [1, d] = N(1) \cong 0
\end{displaymath}
so that $N(x) = [x, 0]$ is the negation and the hypothesis becomes the condition that double negation on the Heyting algebra is the identity, i.e., the Heyting algebra is a Boolean algebra.

\end{proof}
\hypertarget{in_the_category_of_spectra__anderson_duality}{}\subsubsection*{{In the category of spectra -- Anderson duality}}\label{in_the_category_of_spectra__anderson_duality}

In the [[stable (infinity,1)-category of spectra]], the [[sphere spectrum]] (which induces [[Spanier-Whitehead duality]] on spectra which are [[dualizable objects]] with respect to the [[smash product of spectra]]) is not a dualizing object. However, the [[Anderson spectrum]] $I_{\mathbb{Z}}$ is a dualizing object on a suitable subcategory of finite spectra ([[Representability Theorems|Lurie, Example 4.3.9]]). The [[duality]] operation $[-,I_{\mathbb{Z}}]$ that it induces in [[Anderson duality]].

For instance, the Anderson dual of [[KU]] is (complex conjugation-equivariantly) the 4-fold [[suspension spectrum]] $\Sigma^4 KU$ (\hyperlink{HeardStojanoska14}{Heard-Stojanoska 14, theorem 8.2}); and [[tmf]]$[1/2]$ is Anderson dual to its 21-fold suspension (\hyperlink{Stojanoska12}{Stojanoska 12})

\hypertarget{in_the_category_of_suplattices}{}\subsubsection*{{In the category of suplattices}}\label{in_the_category_of_suplattices}

In the category [[Sup]] of [[suplattices]], the opposite $\Omega^{op}$ of the poset $\Omega$ of [[truth values]] is a global dualizing object (and hence $Sup$ is [[star-autonomous category|star-autonomous]]). The internal-hom $[A,B]$ is the suplattice of sup-preserving maps $A\to B$, hence $[A,\Omega^{op}]$ is the suplattice of contravariant maps $A \to \Omega^{op}$ that take suprema in $A$ to infima in $\Omega$. But by the [[adjoint functor theorem]] for posets, any such map is representable by an element of $A$; thus $[A,\Omega^{op}] \cong A^{op}$. The double-dualization is therefore isomorphic to the identity $A\cong (A^{op})^{op}$.

\hypertarget{chu_spaces}{}\subsubsection*{{Chu spaces}}\label{chu_spaces}

The [[Chu construction]] is a (co)universal way of ``making any object into a dualizing object''.

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

\begin{itemize}%
\item [[dual object in a closed category]]
\item [[star-autonomous category]]
\item [[Chu construction]]
\item [[ambimorphic object]]

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

General discussion is in

\begin{itemize}%
\item [[Mitya Boryachenko]], [[Vladimir Drinfeld]], \emph{A duality formalism in the spirit of Grothendieck and Verdier} (\href{http://arxiv.org/abs/1108.6020}{arXiv:1108.6020})


\item [[Jacob Lurie]], section 4.2 of \emph{[[Representability Theorems]]}



\end{itemize}
Reviews of the general concept and then discussion of [[Anderson duality]] is in

\begin{itemize}%
\item [[Vesna Stojanoska]], \emph{Duality for Topological Modular Forms}, Doc. Math. 17 (2012) 271-311 (\href{http://arxiv.org/abs/1105.3968}{arXiv:1105.3968})


\item [[Drew Heard]], [[Vesna Stojanoska]], \emph{K-theory, reality, and duality} (\href{http://arxiv.org/abs/1401.2581}{arXiv:1401.2581})



\end{itemize}
Discussion in the context of the [[linear logic]]/[[quantum logic]] of [[quantum physics]] is in

\begin{itemize}%
\item [[Samson Abramsky]], \emph{No-Cloning in categorical quantum mechanics}, (2008) in I. Mackie and S. Gay (eds), \emph{Semantic Techniques for Quantum Computation} , Cambridge University Press (\href{http://arxiv.org/abs/0910.2401}{arXiv:0910.2401})

\end{itemize}
[[!redirects dualizing objects in a closed category]] [[!redirects dualizing objects in closed categories]]



\end{document}