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An object that is both a monoid object and a comonoid in a braided monoidal category, with $\Delta ,\epsilon$ preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

The categorical dual of a monoid object: an object with coassociative comultiplication and a counit, with the ambient constraints included.

For a commutative monoid object $(M,\ast ,e)$ in preframes, the indicated subset is a frame. Its bottom is $e$, its binary join is $\ast$, and its directed joins and finite meets are inherited. Every homomorphism from a frame regarded as a join monoid factors uniquely through this subset. This gives a coreflective subcategory.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

A preframe has finite meets and joins of nonempty directed sets, with finite meets distributing over these joins. A preframe homomorphism preserves both operations. Its preframe tensor product $P\#Q$ has generators $p\#q$ and relations making $(p,q)\mapsto p\#q$ a homomorphism separately in each variable. Equivalently,In particular $1\#q=p\#1=1$. The tensor unit is the free preframe on one generator: classically it is $2$, and constructively it is the frame $\Omega$ of truth values, with generator $0$.

Regard a frame $L$ as a commutative monoid object in preframes, with multiplication $x\ast y=x\vee y$ and unit $0$. Binary join preserves finite meets and directed joins separately. Conversely take a commutative monoid object $(M,\ast ,e)$ in preframes and putThis set contains $e$ and $1$, since preservation of the empty meet gives $1\ast x=1$. If $x,y\in R(M)$, then $x\ast y\ge x,y$, andThus finite meets are inherited. If $D\subseteq R(M)$ is nonempty and directed, separate preservation of directed joins givesbecause a common upper bound $t\in D$ gives $d\ast d^{\prime }\le t\ast t=t$. Directed joins are inherited too. Associativity and commutativity show that $x\ast y$ is idempotent; it is the least idempotent upper bound of $x,y$. Therefore $e$ is the bottom and $\ast$ is binary join on $R(M)$. Arbitrary joins are directed joins of finite products, including the empty product $e$. Finallyso binary distributivity, together with directed distributivity, proves that $R(M)$ is a frame.

A homomorphism from a frame join monoid to $M$ sends every element to an idempotent above $e$ and therefore factors uniquely through $R(M)$. The factor preserves finite joins and directed joins, hence arbitrary joins. Conversely the inclusion $R(M)\to M$ preserves the preframe and monoid operations. Also a preframe monoid homomorphism between frames is automatically a frame homomorphism. This proves the full embedding and its right adjoint: $\mathbf{Frm}$ is a coreflective subcategory of $\mathbf{CMon}(\mathbf{PFrm})$, with coreflector $R$.

The coproduct of commutative monoid objects in a symmetric monoidal category is their tensor: the maps are $x\mapsto x\#e_N$, $y\mapsto e_M\#y$, and a pair $f,g$ induces $x\#y\mapsto f(x)\ast g(y)$. For two frames $L,K$, the induced multiplication on pure generators isEvery pure generator is idempotent and above $e$. Those properties are preserved by finite meets and directed joins, as just proved, so hold throughout the generated preframe. It is therefore already a frame, rather than requiring a further coreflection. ConsequentlyRectangles in the localic product are instead $i_L(a)\wedge i_K(b)=(a\#0)\wedge (0\#b)$.

For a filtered colimit in a category of frames, form the colimit $P$ in preframes, with stage maps $i_j$. It can be presented by stage elements, all their finite-meet and directed-join relations, and the transition relations. Since tensoring is a left adjoint, it preserves colimits. Filteredness makes the diagonal diagram cofinal in the product index diagram, so the stage multiplications induce $P\#P\to P$. The units agree. Each stage element is idempotent above the common unit; generation again implies $P=R(P)$. The resulting frame is the filtered colimit in a category in $\mathbf{Frm}$; its binary join is induced by computing two representatives at a common later stage. Thus binary coproducts and filtered colimits of frames are computed in preframes, with the indicated induced join operation.

We now prove compactness. For a compact locale, its frame $L$ has a top test $q_L:L\to \Omega$, $q_L(a)=[a=1]$, preserving finite meets and directed joins. For compact $L,K$, tensor the tests and use the unit isomorphism $\Omega \#\Omega \cong \Omega$ to obtain a preframe map $q:L\#K\to \Omega$ withIt reflects top on every pure generator. The property $q(x)=1⟹x=1$ is closed under finite meets and directed joins: for a directed join, preservation of the join says that its truth value is the existential disjunction of the individual truth values, so some term has value $1$. Since pure generators generate the preframe, $q$ reflects top everywhere. Applying $q$ to a directed cover of $1$ therefore shows that a term of that cover is $1$. This proves compactness of the binary coproduct frame, hence of the binary product locale.

For a filtered diagram of compact frames, define on stage $j$These truth-valued maps preserve finite meets: witnesses can be taken to a common later stage, using filteredness. They preserve directed joins: compactness at the witnessing later stage selects a member of the directed family. They are compatible with transitions, again by filteredness, so induce a preframe map $q:P\to \Omega$. On stage generators, $q(i_j(a))=1$ implies $i_j(a)=1$. The same generation argument proves that $q$ reflects top throughout $P$. Thus the filtered colimit frame is compact.

Finally the arbitrary coproduct of the frames $\mathcal{O}(X_i)$ is the filtered colimit of their finite coproducts, indexed by finite subsets of the index set; the empty finite coproduct is $\Omega$, itself compact. Finite coproducts are compact by the binary argument, and their filtered colimit is compact by the preceding paragraph. Opposing the categories givesThis proves the localic Tychonoff theorem through preframes, without choosing points or invoking the axiom of choice.

Use the specified self-dual-pairing meaning of “coseparable”; categorically this is the Frobenius monoid structure relevant here. The induced monoid object on $FA$ has mapsIts associativity follows from associativity of $m$ and the lax monoidal functor axiom for $\phi$: both iterated products are $F$ of the corresponding triple product, preceded by the same coherent iterated tensor comparison. The unit laws follow from those of $j,m$ and the two monoidal unit axioms. Thus $(FA,J,M)$ is a monoid object.

Its pairing is exactly the evaluation transported in the preceding part:If $n:I\to A\otimes A$ is the original coevaluation morphism, the corresponding new one is ${\psi }_{A,A}F(n){\phi }_0$. The preceding Frobenius monoidal functor calculation proves both snake identities. Hence the pairing of the induced monoid remains self-dual, giving the claimed canonical structure on $FA$. The terminology in the paper imposes this duality condition; it does not add a separability or semisimplicity condition.