# Dualising Objects

When does a dualising object induce a dual adjunction?

**Definition:** Let $\mathcal A,\mathcal B$ be two categories with faithful functors to $Set$. $(\bar A,\bar B)$ is called a dualising object if both components have the same underlying set $|\bar A|=|\bar B|$.

**Remark:** In (Porst and Tholen), an ismorphism $|\bar A|\to|\bar B|$ is part of the structure.

**Example:** $2$ is a dualising object for many categories, for example, $Set$ and $Set$, or $Stone$ and $BoolAlg$, or, $Top$ and $Frame$, etc.

Under what conditions does the dualising object give rise to the following adjunction?

$$\mathcal B(-,\bar B)\dashv\mathcal A(-,\bar A):\mathcal A^{op}\to\mathcal B$$

The general category theoretic answer involves the notion of an [initial lift](https://ncatlab.org/nlab/show/final+lift):

**Definition:** If $U$ is a functor and $(f_i:X\to U C_i)_{i\in I}$ is a family of arrows (called a source), then $(g_i:A\to C_i)_{i\in I}$ is a ***lift*** of $(f_i)_{i\in I}$ if $Ug_i=f_i$. And $(g_i)_{i\in I}$ is an ***initial lift*** if it is initial in the category of lifts of $(f_i)_{i\in I}$.

**Remark:** If $U$ is faithful, as we assume in this section anyway, we can say that $A$ is a lift of the $f_i$ if there exists a source $g_i$ with $Ug_i=f_i$.

**Theorem:** We have
$$\mathcal B(-,\bar B)\dashv\mathcal A(-,\bar A):\mathcal A^{op}\to\mathcal B$$

if for all $A\in\mathcal A$, $B\in\mathcal B$ the sources given by evaluation $e_x(h)=h(x)$
$$(\,\mathcal A(A,\bar A)\stackrel{e_a}{\longrightarrow} |\bar B|\,)_{a\in|\bar A|}
\quad\quad\quad\quad
(\,\mathcal B(B,\bar B)\stackrel{e_a}{\longrightarrow} |\bar A|\,)_{b\in|\bar B|}$$

have an initial lift.

**Remark:** It will often happen that the functors $|-|$ preserve products. In this case the we can write the sources as maps. To make the notation more striking let us write $D$ for both of $\bar A$ and $\bar B$ and their underlying sets.
$$\mathcal A(A,D)\stackrel{}{\longrightarrow} D^{|A|}
\quad\quad\quad\quad
\mathcal B(B,D)\stackrel{}{\longrightarrow} D^{|B|}$$

- If $\mathcal A$ is a category of algebras then the induced subalgebra of $D^{|B|}$ will be the initial lift of $\mathcal B(B,D)$.
- If $\mathcal B$ is a category of topological spaces, then the induced subspace of the product space $D^{|A|}$ will be the initial lift of $\mathcal(A,D)$.

**Example:** Let $\mathcal A$ be the category of Boolean algebras and $\mathcal B$ be the category of Stone spaces and $\bar A=2$ the two-element Boolean algebra and $\bar B=2$ be the two-element Stone space. Now the adjunction follows from instantiating the remark above.

## In Order Enriched Category Theory

The theorem also works in the enriched setting.

For example, if the categories in question are order enriched and the functors $|-|$ are [P-faithful](https://hackmd.io/@alexhkurz/BywmMN9G8) and take values in posets, then the previous remark generalises to the situation where again the liftings are such that they give us isomoprhisms $\mathcal A(A,D)\stackrel{}{\longrightarrow} D^{|A|}$ and $\mathcal B(B,D)\stackrel{}{\longrightarrow} D^{|B|}$.

In particular, if $\mathcal B$ is a category ordered topological spaces with continuous order-preserving maps, we need to understand the "product space" $D^{|A|}$ in case $|A|$ is a poset.

By definition, the power $X^P$ for an ordered topological space $X\in OTop$ and a poset $P$ is given by the (necessarily unique up to iso) solution of the equation

$$OTop(Y,X^P)\cong Pos(P,OTop(Y,X))$$

**Prop:** Let $X$ be an ordered topological space. Let $P$ be a poset and $S$ its underlying set. Then $X^P$ is the set of monotone functions $P\to X$ topologised by the product topology on $X^S$.

*Proof:* From left to right, map $f\in OTop(Y,X^P)$ to $\lambda py.fyp$. We have to show that $\lambda y.fyp$ is continuous for all $p\in P$. Let $U$ be an open subset of $X$. We have to show that $\{y\in Y\mid fyp\in U\}$ is open in $Y$.
> The set of monotone functions $\{g:P\to X\mid g(p) \in U\}$ is an open subset of $X^P$ by definition of the product topology on $X^S$. The inverse image of $\{g:P\to X\mid g(p) \in U\}$ under $\lambda y. (\lambda p.fyp)$ is $\{y\in Y\mid fyp\in U\}$ which is open by assumption on $f$. QED.

From right to left, map $g\in Pos(P,OTop(Y,X))$ to $\lambda yp.gpy$. Let $U$ be an open subset of $X^P$. We have to show that $\{y\in Y\mid \lambda p.gpy\in U\}$ is an open subset of $Y$.
> Choose $y\in Y$ such that $\lambda p.gpy \in U$. There are finitely many $p_i$ and open subsets $V_i$ of $X$ such that $gp_iy \in V_i$. The $\{y\mid gp_iy \in V_i\}$ are open subset of $Y$ by assumption on $g$. Therefore $W=\bigcap_i\{y\mid gp_iy \in V_i\}$ is an open subset of $Y$ containing $y$. Moreover, by definition of the $V_i$, we have $W\subseteq \{y\in Y\mid \lambda p.gpy\in U\}$. QED.

Finally, the two functions "from left to right" and "from right to left" are order-isomorphism. QED.

**Remark:** Equivalently, $X^P$ can be seen as the subspace of $X^S$ consisting of all compatible tuples, where a tuple $(x_p)_{p\in P}$ is said to be compatible if $p\le p'$ implies $x_p\le x_{p'}$.

## References

The main reference is

- Porst, Tholen: Concrete Dualities (1991).

The definition of initial (or final) lift can be found in

- [nLab, final lift](https://ncatlab.org/nlab/show/final+lift).

Chapter VI.6 of

- Johnstone: Stone Spaces (1982).

is also an excellent source.

- Kurz, Velebil: <b><a href="papers/Kurz-Velebil-2013.pdf">Enriched logical connections</a></b>.

generalises Porst-Tholen to enriched categories.